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From: <da...@us...> - 2008-04-30 05:09:54
|
Revision: 867
http://axiom.svn.sourceforge.net/axiom/?rev=867&view=rev
Author: daly
Date: 2008-04-29 22:09:55 -0700 (Tue, 29 Apr 2008)
Log Message:
-----------
20080428 tpd src/input/schaum19.input post-mortem fixes
20080428 tpd src/input/schaum23.input post-mortem fixes
20080428 tpd src/input/schaum24.input post-mortem fixes
20080428 tpd src/input/schaum27.input post-mortem fixes
20080428 tpd src/input/schaum28.input post-mortem fixes
20080428 tpd src/input/schaum29.input post-mortem fixes
20080428 tpd src/input/schaum33.input post-mortem fixes
20080428 tpd src/input/schaum6.input post-mortem fixes
20080428 tpd src/input/schaum7.input post-mortem fixes
20080428 tpd src/input/schaum9.input post-mortem fixes
Modified Paths:
--------------
trunk/axiom/changelog
trunk/axiom/src/input/schaum19.input.pamphlet
trunk/axiom/src/input/schaum23.input.pamphlet
trunk/axiom/src/input/schaum24.input.pamphlet
trunk/axiom/src/input/schaum27.input.pamphlet
trunk/axiom/src/input/schaum28.input.pamphlet
trunk/axiom/src/input/schaum29.input.pamphlet
trunk/axiom/src/input/schaum33.input.pamphlet
trunk/axiom/src/input/schaum6.input.pamphlet
trunk/axiom/src/input/schaum7.input.pamphlet
trunk/axiom/src/input/schaum9.input.pamphlet
Modified: trunk/axiom/changelog
===================================================================
--- trunk/axiom/changelog 2008-04-28 06:16:48 UTC (rev 866)
+++ trunk/axiom/changelog 2008-04-30 05:09:55 UTC (rev 867)
@@ -1,3 +1,20 @@
+20080428 tpd src/input/schaum19.input post-mortem fixes
+20080428 tpd src/input/schaum23.input post-mortem fixes
+20080428 tpd src/input/schaum24.input post-mortem fixes
+20080428 tpd src/input/schaum27.input post-mortem fixes
+20080428 tpd src/input/schaum28.input post-mortem fixes
+20080428 tpd src/input/schaum29.input post-mortem fixes
+20080428 tpd src/input/schaum33.input post-mortem fixes
+20080428 tpd src/input/schaum6.input post-mortem fixes
+20080428 tpd src/input/schaum7.input post-mortem fixes
+20080428 tpd src/input/schaum9.input post-mortem fixes
+20080427 tpd src/input/schaum34.input show Schaums-Axiom equivalence
+20080427 tpd src/input/schaum33.input show Schaums-Axiom equivalence
+20080427 tpd src/input/schaum32.input show Schaums-Axiom equivalence
+20080427 tpd src/input/schaum31.input show Schaums-Axiom equivalence
+20080427 tpd src/input/schaum30.input show Schaums-Axiom equivalence
+20080427 tpd src/input/schaum29.input show Schaums-Axiom equivalence
+20080427 tpd src/input/schaum28.input fix Schaums-Axiom equivalence
20080426 tpd src/input/schaum28.input show Schaums-Axiom equivalence
20080426 tpd src/input/schaum27.input show Schaums-Axiom equivalence
20080426 tpd src/input/schaum26.input show Schaums-Axiom equivalence
Modified: trunk/axiom/src/input/schaum19.input.pamphlet
===================================================================
--- trunk/axiom/src/input/schaum19.input.pamphlet 2008-04-28 06:16:48 UTC (rev 866)
+++ trunk/axiom/src/input/schaum19.input.pamphlet 2008-04-30 05:09:55 UTC (rev 867)
@@ -1569,7 +1569,7 @@
--R Type: Expression Integer
--E
---S 125 14:416 Schaums and Axiom differ by a constant
+--S 125 14:416 Schaums and Axiom agree
dd:=expandLog cc
--R
--R (4) 0
Modified: trunk/axiom/src/input/schaum23.input.pamphlet
===================================================================
--- trunk/axiom/src/input/schaum23.input.pamphlet 2008-04-28 06:16:48 UTC (rev 866)
+++ trunk/axiom/src/input/schaum23.input.pamphlet 2008-04-30 05:09:55 UTC (rev 867)
@@ -658,6 +658,34 @@
--S 47
t1:=integrate(1/(p+q*sin(a*x)),x)
+--R
+--R (2)
+--R [
+--R log
+--R +-------+
+--R 2 2 2 | 2 2
+--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p
+--R +
+--R 2 3 3 2 3 2
+--R (- p q + p )sin(a x) + (- q + p q)cos(a x) - q + p q
+--R /
+--R q sin(a x) + p
+--R /
+--R +-------+
+--R | 2 2
+--R a\|q - p
+--R ,
+--R +---------+
+--R | 2 2
+--R (p sin(a x) + q cos(a x) + q)\|- q + p
+--R 2atan(-----------------------------------------)
+--R 2 2 2 2
+--R (q - p )cos(a x) + q - p
+--R - ------------------------------------------------]
+--R +---------+
+--R | 2 2
+--R a\|- q + p
+--R Type: Union(List Expression Integer,...)
--E
--S 48
Modified: trunk/axiom/src/input/schaum24.input.pamphlet
===================================================================
--- trunk/axiom/src/input/schaum24.input.pamphlet 2008-04-28 06:16:48 UTC (rev 866)
+++ trunk/axiom/src/input/schaum24.input.pamphlet 2008-04-30 05:09:55 UTC (rev 867)
@@ -2318,10 +2318,10 @@
--R 2
--R Type: Expression Complex Integer
--E
+@
And we get the surprising result that they are not equal.
In fact, they differ by a complex value depending on x.
Likely there is a branch-cut issue lurking somewhere.
-@
\section{\cite{1}:14.506~~~~~$\displaystyle
\int{x^m\cot^{-1}\frac{x}{a}}~dx$}
Modified: trunk/axiom/src/input/schaum27.input.pamphlet
===================================================================
--- trunk/axiom/src/input/schaum27.input.pamphlet 2008-04-28 06:16:48 UTC (rev 866)
+++ trunk/axiom/src/input/schaum27.input.pamphlet 2008-04-30 05:09:55 UTC (rev 867)
@@ -355,7 +355,7 @@
--R Type: RewriteRule(Integer,Integer,Expression Integer)
--E
---S 27 14:548 Axiom cannot simplify this expression
+--S 27 14:548 Schaums and Axiom agree
ff:=sinhcoshrule ee
--R
--R (9) 0
Modified: trunk/axiom/src/input/schaum28.input.pamphlet
===================================================================
--- trunk/axiom/src/input/schaum28.input.pamphlet 2008-04-28 06:16:48 UTC (rev 866)
+++ trunk/axiom/src/input/schaum28.input.pamphlet 2008-04-30 05:09:55 UTC (rev 867)
@@ -410,7 +410,9 @@
--R 2 2
--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) + a
--R Type: Expression Integer
---S 32
+--E
+
+--S 32 14:571 Schaums and Axiom differ by a constant
dd:=complexNormalize cc
--R
--R 1
@@ -526,7 +528,7 @@
--R Type: RewriteRule(Integer,Integer,Expression Integer)
--E
---S 41 14:572 Schaums and Axiom agree
+--S 41 14:572 Axiom cannot simplify this expression
ff:=sinhcoshrule ee
--R
--R (9)
@@ -787,9 +789,9 @@
--S 61
sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y)))
--R
---I %B sinh(y + x) - %B sinh(y - x)
---I (6) %B cosh(y)sinh(x) == -------------------------------
---R 2
+--I %BC sinh(y + x) - %BC sinh(y - x)
+--I (6) %BC cosh(y)sinh(x) == -------------------------------
+--R 2
--R Type: RewriteRule(Integer,Integer,Expression Integer)
--E
@@ -809,9 +811,9 @@
--S 63
sinhsinhrule:=rule(sinh(x)*sinh(y)==1/2*(cosh(x+y)-cosh(x-y)))
--R
---I %B cosh(y + x) - %B cosh(y - x)
---I (8) %B sinh(x)sinh(y) == -------------------------------
---R 2
+--I %BD sinh(y + x) - %BD sinh(y - x)
+--I (8) %BD cosh(y)sinh(x) == -------------------------------
+--R 2
--R Type: RewriteRule(Integer,Integer,Expression Integer)
--E
@@ -1485,7 +1487,7 @@
--R Type: Expression Integer
--E
---S 107 14:580 Axiom cannot compute this integral
+--S 107 14:580 Axiom cannot simplify this expression
cc:=aa-bb
--R
--R (3)
@@ -2681,7 +2683,7 @@
--R Type: Expression Integer
--E
---S 134 14:584 Axiom cannot compute this integral
+--S 134 14:584 Axiom cannot simplify this expression
cc2:=aa-bb2
--R
--R (5)
Modified: trunk/axiom/src/input/schaum29.input.pamphlet
===================================================================
--- trunk/axiom/src/input/schaum29.input.pamphlet 2008-04-28 06:16:48 UTC (rev 866)
+++ trunk/axiom/src/input/schaum29.input.pamphlet 2008-04-30 05:09:55 UTC (rev 867)
@@ -77,7 +77,7 @@
--R Type: RewriteRule(Integer,Integer,Expression Integer)
--E
---S 7 14:590 Schaums and Axiom agree
+--S 7 14:590 Schaums and Axiom differ by a constant
ee:=coshsqrrule dd
--R
--R 1
@@ -209,6 +209,7 @@
--E
--S 16 14:593 Axiom cannot simplify this expression
+cc:=aa-bb
--R
--R (3)
--R - cosh(a x)sinh(n log(cosh(a x))) - cosh(a x)cosh(n log(cosh(a x)))
@@ -727,6 +728,7 @@
--E
--S 47 14:598 Axiom cannot simplify this expression
+cc:=aa-bb
--R
--R (3)
--R 4 3
Modified: trunk/axiom/src/input/schaum33.input.pamphlet
===================================================================
--- trunk/axiom/src/input/schaum33.input.pamphlet 2008-04-28 06:16:48 UTC (rev 866)
+++ trunk/axiom/src/input/schaum33.input.pamphlet 2008-04-30 05:09:55 UTC (rev 867)
@@ -666,7 +666,7 @@
--R Type: Expression Integer
--E
---S 38 14:642 Axiom cannot simplify this expression
+--S 38 14:642 Schaums and Axiom differ by a constant
hh:=complexNormalize gg
--R
--R - log(- 1) + log(- 2)
Modified: trunk/axiom/src/input/schaum6.input.pamphlet
===================================================================
--- trunk/axiom/src/input/schaum6.input.pamphlet 2008-04-28 06:16:48 UTC (rev 866)
+++ trunk/axiom/src/input/schaum6.input.pamphlet 2008-04-30 05:09:55 UTC (rev 867)
@@ -467,7 +467,7 @@
--R Type: Expression Integer
--E
---S 38 14:134 Schaums and Axiom differ by a constant
+--S 38 14:134 Schaums and Axiom agree
cc:=aa-bb
--R
--R (3) 0
Modified: trunk/axiom/src/input/schaum7.input.pamphlet
===================================================================
--- trunk/axiom/src/input/schaum7.input.pamphlet 2008-04-28 06:16:48 UTC (rev 866)
+++ trunk/axiom/src/input/schaum7.input.pamphlet 2008-04-30 05:09:55 UTC (rev 867)
@@ -410,9 +410,9 @@
--S 35 14:150 Schaums and Axiom agree
cc:=aa-t2
-
- (7) 0
- Type: Expression Integer
+--R
+--R (7) 0
+--R Type: Expression Integer
--E
@
Modified: trunk/axiom/src/input/schaum9.input.pamphlet
===================================================================
--- trunk/axiom/src/input/schaum9.input.pamphlet 2008-04-28 06:16:48 UTC (rev 866)
+++ trunk/axiom/src/input/schaum9.input.pamphlet 2008-04-30 05:09:55 UTC (rev 867)
@@ -392,12 +392,12 @@
--S 28 14:188 Schaums and Axiom differ by a constant
ee:=complexNormalize dd
-
- log(- 1)
- (5) --------
- 3
- 2a
- Type: Expression Integer
+--R
+--R log(- 1)
+--R (5) --------
+--R 3
+--R 2a
+--R Type: Expression Integer
--E
@
This was sent by the SourceForge.net collaborative development platform, the world's largest Open Source development site.
|
|
From: <gi...@ax...> - 2008-04-30 05:08:52
|
changelog | 17 +++++++++++++++++
src/input/schaum19.input.pamphlet | 2 +-
src/input/schaum23.input.pamphlet | 28 ++++++++++++++++++++++++++++
src/input/schaum24.input.pamphlet | 2 +-
src/input/schaum27.input.pamphlet | 2 +-
src/input/schaum28.input.pamphlet | 22 ++++++++++++----------
src/input/schaum29.input.pamphlet | 4 +++-
src/input/schaum33.input.pamphlet | 2 +-
src/input/schaum6.input.pamphlet | 2 +-
src/input/schaum7.input.pamphlet | 6 +++---
src/input/schaum9.input.pamphlet | 12 ++++++------
11 files changed, 74 insertions(+), 25 deletions(-)
New commits:
commit 3e015b1d862ffc56f5a846871ca5adeb6ed87188
Author: Tim Daly <root@localhost.localdomain>
Date: Sat Apr 26 20:45:52 2008 -0400
20080428 tpd src/input/schaum19.input post-mortem fixes
20080428 tpd src/input/schaum23.input post-mortem fixes
20080428 tpd src/input/schaum24.input post-mortem fixes
20080428 tpd src/input/schaum27.input post-mortem fixes
20080428 tpd src/input/schaum28.input post-mortem fixes
20080428 tpd src/input/schaum29.input post-mortem fixes
20080428 tpd src/input/schaum33.input post-mortem fixes
20080428 tpd src/input/schaum6.input post-mortem fixes
20080428 tpd src/input/schaum7.input post-mortem fixes
20080428 tpd src/input/schaum9.input post-mortem fixes
|
|
From: Dwayne T. <eo...@br...> - 2008-04-29 16:34:39
|
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|
From: <da...@us...> - 2008-04-28 06:16:56
|
Revision: 866
http://axiom.svn.sourceforge.net/axiom/?rev=866&view=rev
Author: daly
Date: 2008-04-27 23:16:48 -0700 (Sun, 27 Apr 2008)
Log Message:
-----------
20080426 tpd src/input/schaum28.input show Schaums-Axiom equivalence
20080426 tpd src/input/schaum27.input show Schaums-Axiom equivalence
20080426 tpd src/input/schaum26.input show Schaums-Axiom equivalence
20080425 tpd src/input/schaum25.input show Schaums-Axiom equivalence
20080425 tpd src/input/schaum24.input show Schaums-Axiom equivalence
20080424 tpd src/input/schaum23.input show Schaums-Axiom equivalence
20080423 tpd src/input/schaum22.input show Schaums-Axiom equivalence
Modified Paths:
--------------
trunk/axiom/src/input/schaum28.input.pamphlet
trunk/axiom/src/input/schaum29.input.pamphlet
trunk/axiom/src/input/schaum30.input.pamphlet
trunk/axiom/src/input/schaum31.input.pamphlet
trunk/axiom/src/input/schaum32.input.pamphlet
trunk/axiom/src/input/schaum33.input.pamphlet
trunk/axiom/src/input/schaum34.input.pamphlet
Modified: trunk/axiom/src/input/schaum28.input.pamphlet
===================================================================
--- trunk/axiom/src/input/schaum28.input.pamphlet 2008-04-27 06:01:45 UTC (rev 865)
+++ trunk/axiom/src/input/schaum28.input.pamphlet 2008-04-28 06:16:48 UTC (rev 866)
@@ -234,6 +234,10 @@
\section{\cite{1}:14.569~~~~~$\displaystyle
\int{\cosh^2{ax}}~dx$}
$$\int{\cosh^2{ax}}=
+\frac{x}{2}+\frac{\sinh{ax}\cosh{ax}}{2a}
+$$
+Note that the Schaums print edition (1968 printing 3) has a typo:
+$$\int{\cosh^2{ax}}=
\frac{x}{2}+\frac{\sinh{ax}\cosh{ax}}{2}
$$
<<*>>=
@@ -250,40 +254,21 @@
--E
--S 18
-bb:=x/2+(sinh(a*x)*cosh(a*x))/2
+bb:=x/2+(sinh(a*x)*cosh(a*x))/(2*a)
--R
---R cosh(a x)sinh(a x) + x
---R (2) ----------------------
---R 2
+--R cosh(a x)sinh(a x) + a x
+--R (2) ------------------------
+--R 2a
--R Type: Expression Integer
--E
---S 19
+--S 19 14:569 Schaums and Axiom agree
cc:=aa-bb
--R
---R (- a + 1)cosh(a x)sinh(a x)
---R (3) ---------------------------
---R 2a
+--R (3) 0
--R Type: Expression Integer
--E
---S 20
-sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y)))
---R
---I %B sinh(y + x) - %B sinh(y - x)
---I (4) %B cosh(y)sinh(x) == -------------------------------
---R 2
---R Type: RewriteRule(Integer,Integer,Expression Integer)
---E
-
---S 21 14:569 SCHAUMS AND AXIOM DISAGREE?
-dd:=sinhcoshrule cc
---R
---R (- a + 1)sinh(2a x)
---R (5) -------------------
---R 4a
---R Type: Expression Integer
---E
@
\section{\cite{1}:14.570~~~~~$\displaystyle
@@ -294,7 +279,7 @@
<<*>>=
)clear all
---S 22
+--S 20
aa:=integrate(x*cosh(a*x)^2,x)
--R
--R
@@ -306,7 +291,7 @@
--R Type: Union(Expression Integer,...)
--E
---S 23
+--S 21
bb:=x^2/4+(x*sinh(2*a*x))/(4*a)-cosh(2*a*x)/(8*a^2)
--R
--R 2 2
@@ -317,7 +302,7 @@
--R Type: Expression Integer
--E
---S 24
+--S 22
cc:=aa-bb
--R
--R (3)
@@ -332,7 +317,7 @@
--R Type: Expression Integer
--E
---S 25
+--S 23
sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2)
--R
--R 2 cosh(2x) - 1
@@ -341,7 +326,7 @@
--R Type: RewriteRule(Integer,Integer,Expression Integer)
--E
---S 26
+--S 24
dd:=sinhsqrrule cc
--R
--R (5)
@@ -353,7 +338,7 @@
--R Type: Expression Integer
--E
---S 27
+--S 25
coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2)
--R
--R 2 cosh(2x) + 1
@@ -362,7 +347,7 @@
--R Type: RewriteRule(Integer,Integer,Expression Integer)
--E
---S 28
+--S 26
ee:=coshsqrrule dd
--R
--R - x sinh(2a x) + 2x cosh(a x)sinh(a x)
@@ -371,7 +356,7 @@
--R Type: Expression Integer
--E
---S 29
+--S 27
sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y)))
--R
--I %S sinh(y + x) - %S sinh(y - x)
@@ -380,7 +365,7 @@
--R Type: RewriteRule(Integer,Integer,Expression Integer)
--E
---S 30 14:570 Schaums and Axiom agree
+--S 28 14:570 Schaums and Axiom agree
ff:=sinhcoshrule ee
--R
--R (9) 0
@@ -396,7 +381,7 @@
<<*>>=
)clear all
---S 31
+--S 29
aa:=integrate(1/cosh(a*x)^2,x)
--R
--R
@@ -407,7 +392,7 @@
--R Type: Union(Expression Integer,...)
--E
---S 32
+--S 30
bb:=tanh(a*x)/a
--R
--R tanh(a x)
@@ -416,7 +401,7 @@
--R Type: Expression Integer
--E
---S 33
+--S 31
cc:=aa-bb
--R
--R 2 2
@@ -425,157 +410,14 @@
--R 2 2
--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) + a
--R Type: Expression Integer
---E
-
---S 34
-tanhrule:=rule(tanh(x) == sinh(x)/cosh(x))
+--S 32
+dd:=complexNormalize cc
--R
---R sinh(x)
---R (4) tanh(x) == -------
---R cosh(x)
---R Type: RewriteRule(Integer,Integer,Expression Integer)
---E
-
---S 35
-dd:=tanhrule cc
---R
---R (5)
---R 3 2 2
---R - sinh(a x) - 2cosh(a x)sinh(a x) + (- cosh(a x) - 1)sinh(a x)
---R +
---R - 2cosh(a x)
---R /
---R 2 2 3
---R a cosh(a x)sinh(a x) + 2a cosh(a x) sinh(a x) + a cosh(a x) + a cosh(a x)
+--R 1
+--R (4) - -
+--R a
--R Type: Expression Integer
--E
-
---S 36
-sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2)
---R
---R 2 cosh(2x) - 1
---R (6) sinh(x) == ------------
---R 2
---R Type: RewriteRule(Integer,Integer,Expression Integer)
---E
-
---S 37
-ee:=sinhsqrrule dd
---R
---R (7)
---R 3 2
---R - 2sinh(a x) + (- 2cosh(a x) - 2)sinh(a x) - 2cosh(a x)cosh(2a x)
---R +
---R - 2cosh(a x)
---R /
---R 2 3
---R 4a cosh(a x) sinh(a x) + a cosh(a x)cosh(2a x) + 2a cosh(a x)
---R +
---R a cosh(a x)
---R Type: Expression Integer
---E
-
---S 38
-coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2)
---R
---R 2 cosh(2x) + 1
---R (8) cosh(x) == ------------
---R 2
---R Type: RewriteRule(Integer,Integer,Expression Integer)
---E
-
---S 39
-ff:=coshsqrrule ee
---R
---R (9)
---R 3
---R - 2sinh(a x) + (- cosh(2a x) - 3)sinh(a x) - 2cosh(a x)cosh(2a x)
---R +
---R - 2cosh(a x)
---R /
---R 3
---R (2a cosh(2a x) + 2a)sinh(a x) + a cosh(a x)cosh(2a x) + 2a cosh(a x)
---R +
---R a cosh(a x)
---R Type: Expression Integer
---E
-
---S 40
-sinhcuberule:=rule(sinh(x)^3 == 1/4*sinh(3*x)-3/4*sinh(x))
---R
---R 3 sinh(3x) - 3sinh(x)
---R (10) sinh(x) == -------------------
---R 4
---R Type: RewriteRule(Integer,Integer,Expression Integer)
---E
-
---S 41
-gg:=sinhcuberule ff
---R
---R (11)
---R - sinh(3a x) + (- 2cosh(2a x) - 3)sinh(a x) - 4cosh(a x)cosh(2a x)
---R +
---R - 4cosh(a x)
---R /
---R 3
---R (4a cosh(2a x) + 4a)sinh(a x) + 2a cosh(a x)cosh(2a x) + 4a cosh(a x)
---R +
---R 2a cosh(a x)
---R Type: Expression Integer
---E
-
---S 42
-coshcuberule:=rule(cosh(x)^3 == 1/4*cosh(3*x)-3/4*cosh(x))
---R
---R 3 cosh(3x) - 3cosh(x)
---R (12) cosh(x) == -------------------
---R 4
---R Type: RewriteRule(Integer,Integer,Expression Integer)
---E
-
---S 43
-hh:=coshcuberule gg
---R
---R (13)
---R - sinh(3a x) + (- 2cosh(2a x) - 3)sinh(a x) - 4cosh(a x)cosh(2a x)
---R +
---R - 4cosh(a x)
---R /
---R (4a cosh(2a x) + 4a)sinh(a x) + a cosh(3a x) + 2a cosh(a x)cosh(2a x)
---R +
---R - a cosh(a x)
---R Type: Expression Integer
---E
-
---S 44
-coshcoshrule:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y)))
---R
---I %U cosh(y + x) + %U cosh(y - x)
---I (14) %U cosh(x)cosh(y) == -------------------------------
---R 2
---R Type: RewriteRule(Integer,Integer,Expression Integer)
---E
-
---S 45
-ii:=coshcoshrule hh
---R
---R (15)
---R - sinh(3a x) + (- 2cosh(2a x) - 3)sinh(a x) - 2cosh(3a x) - 6cosh(a x)
---R ----------------------------------------------------------------------
---R (4a cosh(2a x) + 4a)sinh(a x) + 2a cosh(3a x)
---R Type: Expression Integer
---E
-
---S 46 14:571 SCHAUMS AND AXIOM DISAGREE?
-jj:=complexNormalize ii
---R
---R a x 2
---R - 2(%e ) - 2
---R (16) ---------------
---R a x 2
---R 2a (%e ) - a
---R Type: Expression Integer
---E
@
\section{\cite{1}:14.572~~~~~$\displaystyle
@@ -586,7 +428,7 @@
<<*>>=
)clear all
---S 47
+--S 33
aa:=integrate(cosh(a*x)*cosh(p*x),x)
--R
--R
@@ -597,7 +439,7 @@
--R Type: Union(Expression Integer,...)
--E
---S 48
+--S 34
bb:=(sinh(a-p)*x)/(2*(a-p))+(sinh(a+p)*x)/(2*(a+p))
--R
--R (p - a)x sinh(p + a) + (p + a)x sinh(p - a)
@@ -607,7 +449,7 @@
--R Type: Expression Integer
--E
---S 49
+--S 35
cc:=aa-bb
--R
--R (3)
@@ -627,7 +469,7 @@
--R Type: Expression Integer
--E
---S 50
+--S 36
sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2)
--R
--R 2 cosh(2x) - 1
@@ -636,7 +478,7 @@
--R Type: RewriteRule(Integer,Integer,Expression Integer)
--E
---S 51
+--S 37
dd:=sinhsqrrule cc
--R
--R (5)
@@ -653,7 +495,7 @@
--R Type: Expression Integer
--E
---S 52
+--S 38
coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2)
--R
--R 2 cosh(2x) + 1
@@ -662,7 +504,7 @@
--R Type: RewriteRule(Integer,Integer,Expression Integer)
--E
---S 53
+--S 39
ee:=coshsqrrule dd
--R
--R (7)
@@ -675,7 +517,7 @@
--R Type: Expression Integer
--E
---S 54
+--S 40
sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y)))
--R
--I %V sinh(y + x) - %V sinh(y - x)
@@ -684,7 +526,7 @@
--R Type: RewriteRule(Integer,Integer,Expression Integer)
--E
---S 55 14:572 Schaums and Axiom agree
+--S 41 14:572 Schaums and Axiom agree
ff:=sinhcoshrule ee
--R
--R (9)
@@ -706,7 +548,7 @@
<<*>>=
)clear all
---S 56
+--S 42
aa:=integrate(cosh(a*x)*sin(p*x),x)
--R
--R
@@ -724,7 +566,7 @@
--R Type: Union(Expression Integer,...)
--E
---S 57
+--S 43
bb:=(a*sinh(a*x)*sin(p*x)-p*cosh(a*x)*cos(p*x))/(a^2+p^2)
--R
--R a sin(p x)sinh(a x) - p cos(p x)cosh(a x)
@@ -734,7 +576,7 @@
--R Type: Expression Integer
--E
---S 58
+--S 44
cc:=aa-bb
--R
--R (3)
@@ -749,7 +591,7 @@
--R Type: Expression Integer
--E
---S 59
+--S 45
coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2)
--R
--R 2 cosh(2x) + 1
@@ -758,7 +600,7 @@
--R Type: RewriteRule(Integer,Integer,Expression Integer)
--E
---S 60
+--S 46
dd:=coshsqrrule cc
--R
--R (5)
@@ -772,7 +614,7 @@
--R Type: Expression Integer
--E
---S 61
+--S 47
sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2)
--R
--R 2 cosh(2x) - 1
@@ -781,7 +623,7 @@
--R Type: RewriteRule(Integer,Integer,Expression Integer)
--E
---S 62 14:573 Schaums and Axiom agree
+--S 48 14:573 Schaums and Axiom agree
ee:=sinhsqrrule dd
--R
--R (7) 0
@@ -797,7 +639,7 @@
<<*>>=
)clear all
---S 63
+--S 49
aa:=integrate(cosh(a*x)*cos(p*x),x)
--R
--R
@@ -815,7 +657,7 @@
--R Type: Union(Expression Integer,...)
--E
---S 64
+--S 50
bb:=(a*sinh(a*x)*cos(p*x)+p*cosh(a*x)*sin(p*x))/(a^2+p^2)
--R
--R a cos(p x)sinh(a x) + p cosh(a x)sin(p x)
@@ -825,7 +667,7 @@
--R Type: Expression Integer
--E
---S 65
+--S 51
cc:=aa-bb
--R
--R (3)
@@ -840,7 +682,7 @@
--R Type: Expression Integer
--E
---S 66
+--S 52
coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2)
--R
--R 2 cosh(2x) + 1
@@ -849,7 +691,7 @@
--R Type: RewriteRule(Integer,Integer,Expression Integer)
--E
---S 67
+--S 53
dd:=coshsqrrule cc
--R
--R (5)
@@ -863,7 +705,7 @@
--R Type: Expression Integer
--E
---S 68
+--S 54
sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2)
--R
--R 2 cosh(2x) - 1
@@ -872,7 +714,7 @@
--R Type: RewriteRule(Integer,Integer,Expression Integer)
--E
---S 69 14:574 Schaums and Axiom agree
+--S 55 14:574 Schaums and Axiom agree
ee:=sinhsqrrule dd
--R
--R (7) 0
@@ -888,7 +730,7 @@
<<*>>=
)clear all
---S 70
+--S 56
aa:=integrate(1/(cosh(a*x)+1),x)
--R
--R
@@ -898,7 +740,7 @@
--R Type: Union(Expression Integer,...)
--E
---S 71
+--S 57
bb:=1/a*tanh((a*x)/2)
--R
--R a x
@@ -909,7 +751,7 @@
--R Type: Expression Integer
--E
---S 72
+--S 58
cc:=aa-bb
--R
--R a x
@@ -920,7 +762,7 @@
--R Type: Expression Integer
--E
---S 73
+--S 59
tanhrule:=rule(tanh(x) == sinh(x)/cosh(x))
--R
--R sinh(x)
@@ -929,7 +771,7 @@
--R Type: RewriteRule(Integer,Integer,Expression Integer)
--E
---S 74
+--S 60
dd:=tanhrule cc
--R
--R a x a x a x
@@ -942,7 +784,7 @@
--R Type: Expression Integer
--E
---S 75
+--S 61
sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y)))
--R
--I %B sinh(y + x) - %B sinh(y - x)
@@ -951,7 +793,7 @@
--R Type: RewriteRule(Integer,Integer,Expression Integer)
--E
---S 76
+--S 62
ee:=sinhcoshrule dd
--R
--R 3a x a x a x a x
@@ -964,7 +806,7 @@
--R Type: Expression Integer
--E
---S 77
+--S 63
sinhsinhrule:=rule(sinh(x)*sinh(y)==1/2*(cosh(x+y)-cosh(x-y)))
--R
--I %B cosh(y + x) - %B cosh(y - x)
@@ -973,7 +815,7 @@
--R Type: RewriteRule(Integer,Integer,Expression Integer)
--E
---S 78
+--S 64
ff:=sinhsinhrule ee
--R
--R 3a x a x 3a x a x
@@ -986,7 +828,7 @@
--R Type: Expression Integer
--E
---S 79
+--S 65
coshcoshrule:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y)))
--R
--I %BC cosh(y + x) + %BC cosh(y - x)
@@ -995,7 +837,7 @@
--R Type: RewriteRule(Integer,Integer,Expression Integer)
--E
---S 80 14:575 Schaums and Axiom differ by a constant
+--S 66 14:575 Schaums and Axiom differ by a constant
gg:=coshcoshrule ff
--R
--R 1
@@ -1013,7 +855,7 @@
<<*>>=
)clear all
---S 81
+--S 67
aa:=integrate(1/(cosh(a*x)-1),x)
--R
--R
@@ -1023,7 +865,7 @@
--R Type: Union(Expression Integer,...)
--E
---S 82
+--S 68
bb:=-1/a*coth((a*x)/2)
--R
--R a x
@@ -1034,7 +876,7 @@
--R Type: Expression Integer
--E
---S 83
+--S 69
cc:=aa-bb
--R
--R a x a x
@@ -1045,7 +887,7 @@
--R Type: Expression Integer
--E
---S 84
+--S 70
cothrule:=rule(coth(x) == cosh(x)/sinh(x))
--R
--R cosh(x)
@@ -1054,7 +896,7 @@
--R Type: RewriteRule(Integer,Integer,Expression Integer)
--E
---S 85
+--S 71
dd:=cothrule cc
--R
--R a x a x a x a x
@@ -1067,7 +909,7 @@
--R Type: Expression Integer
--E
---S 86
+--S 72
sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y)))
--R
--I %BD sinh(y + x) - %BD sinh(y - x)
@@ -1076,7 +918,7 @@
--R Type: RewriteRule(Integer,Integer,Expression Integer)
--E
---S 87
+--S 73
ee:=sinhcoshrule dd
--R
--R 3a x a x a x a x
@@ -1089,7 +931,7 @@
--R Type: Expression Integer
--E
---S 88
+--S 74
sinhsinhrule:=rule(sinh(x)*sinh(y)==1/2*(cosh(x+y)-cosh(x-y)))
--R
--I %BE cosh(y + x) - %BE cosh(y - x)
@@ -1098,7 +940,7 @@
--R Type: RewriteRule(Integer,Integer,Expression Integer)
--E
---S 89
+--S 75
ff:=sinhsinhrule ee
--R
--R 3a x a x a x a x
@@ -1111,7 +953,7 @@
--R Type: Expression Integer
--E
---S 90
+--S 76
coshcoshrule:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y)))
--R
--I %BF cosh(y + x) + %BF cosh(y - x)
@@ -1120,7 +962,7 @@
--R Type: RewriteRule(Integer,Integer,Expression Integer)
--E
---S 91 14:576 Schaums and Axiom differ by a constant
+--S 77 14:576 Schaums and Axiom differ by a constant
gg:=coshcoshrule ff
--R
--R 1
@@ -1138,7 +980,7 @@
<<*>>=
)clear all
---S 92
+--S 78
aa:=integrate(x/(cosh(a*x)+1),x)
--R
--R
@@ -1152,7 +994,7 @@
--R Type: Union(Expression Integer,...)
--E
---S 93
+--S 79
bb:=x/a*tanh((a*x)/2)-2/a^2*log(cosh((a*x)/2))
--R
--R a x a x
@@ -1164,7 +1006,7 @@
--R Type: Expression Integer
--E
---S 94
+--S 80
cc:=aa-bb
--R
--R (3)
@@ -1185,7 +1027,7 @@
--R Type: Expression Integer
--E
---S 95
+--S 81
tanhrule:=rule(tanh(x) == sinh(x)/cosh(x))
--R
--R sinh(x)
@@ -1194,7 +1036,7 @@
--R Type: RewriteRule(Integer,Integer,Expression Integer)
--E
---S 96
+--S 82
dd:=tanhrule cc
--R
--R (5)
@@ -1222,7 +1064,7 @@
--R Type: Expression Integer
--E
---S 97
+--S 83
coshcoshrule:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y)))
--R
--I %BG cosh(y + x) + %BG cosh(y - x)
@@ -1231,7 +1073,7 @@
--R Type: RewriteRule(Integer,Integer,Expression Integer)
--E
---S 98
+--S 84
ee:=coshcoshrule dd
--R
--R (7)
@@ -1259,7 +1101,7 @@
--R Type: Expression Integer
--E
---S 99
+--S 85
sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y)))
--R
--I %BH sinh(y + x) - %BH sinh(y - x)
@@ -1268,7 +1110,7 @@
--R Type: RewriteRule(Integer,Integer,Expression Integer)
--E
---S 100
+--S 86
ff:=sinhcoshrule ee
--R
--R (9)
@@ -1296,7 +1138,7 @@
--R Type: Expression Integer
--E
---S 101
+--S 87
sinhsinhrule:=rule(sinh(x)*sinh(y)==1/2*(cosh(x+y)-cosh(x-y)))
--R
--I %BI cosh(y + x) - %BI cosh(y - x)
@@ -1305,7 +1147,7 @@
--R Type: RewriteRule(Integer,Integer,Expression Integer)
--E
---S 102
+--S 88
gg:=sinhsinhrule ff
--R
--R a x
@@ -1317,7 +1159,7 @@
--R Type: Expression Integer
--E
---S 103 14:577 Schaums and Axiom differ by a constant
+--S 89 14:577 Schaums and Axiom differ by a constant
complexNormalize gg
--R
--R 2log(2)
@@ -1336,7 +1178,7 @@
<<*>>=
)clear all
---S 104
+--S 90
aa:=integrate(x/(cosh(a*x)-1),x)
--R
--R
@@ -1350,7 +1192,7 @@
--R Type: Union(Expression Integer,...)
--E
---S 105
+--S 91
bb:=-x/a*coth((a*x)/2)+2/a^2*log(sinh((a*x)/2))
--R
--R a x a x
@@ -1362,7 +1204,7 @@
--R Type: Expression Integer
--E
---S 106
+--S 92
cc:=aa-bb
--R
--R (3)
@@ -1383,7 +1225,7 @@
--R Type: Expression Integer
--E
---S 107
+--S 93
cothrule:=rule(coth(x) == cosh(x)/sinh(x))
--R
--R cosh(x)
@@ -1392,7 +1234,7 @@
--R Type: RewriteRule(Integer,Integer,Expression Integer)
--E
---S 108
+--S 94
dd:=cothrule cc
--R
--R (5)
@@ -1420,7 +1262,7 @@
--R Type: Expression Integer
--E
---S 109
+--S 95
sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y)))
--R
--I %BJ sinh(y + x) - %BJ sinh(y - x)
@@ -1429,7 +1271,7 @@
--R Type: RewriteRule(Integer,Integer,Expression Integer)
--E
---S 110
+--S 96
ee:=sinhcoshrule dd
--R
--R (7)
@@ -1457,7 +1299,7 @@
--R Type: Expression Integer
--E
---S 111
+--S 97
sinhsinhrule:=rule(sinh(x)*sinh(y)==1/2*(cosh(x+y)-cosh(x-y)))
--R
--I %BK cosh(y + x) - %BK cosh(y - x)
@@ -1466,7 +1308,7 @@
--R Type: RewriteRule(Integer,Integer,Expression Integer)
--E
---S 112
+--S 98
ff:=sinhsinhrule ee
--R
--R (9)
@@ -1494,7 +1336,7 @@
--R Type: Expression Integer
--E
---S 113
+--S 99
coshcoshrule:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y)))
--R
--I %BL cosh(y + x) + %BL cosh(y - x)
@@ -1503,7 +1345,7 @@
--R Type: RewriteRule(Integer,Integer,Expression Integer)
--E
---S 114
+--S 100
gg:=coshcoshrule ff
--R
--R a x
@@ -1515,7 +1357,7 @@
--R Type: Expression Integer
--E
---S 115 14:578 Schaums and Axiom differ by a constant
+--S 101 14:578 Schaums and Axiom differ by a constant
hh:=complexNormalize gg
--R
--R 2log(2)
@@ -1534,7 +1376,7 @@
<<*>>=
)clear all
---S 116
+--S 102
aa:=integrate(1/(cosh(a*x)+1)^2,x)
--R
--R
@@ -1552,7 +1394,7 @@
--R Type: Union(Expression Integer,...)
--E
---S 117
+--S 103
bb:=1/(2*a)*tanh((a*x)/2)-1/(6*a)*tanh((a*x)/2)^3
--R
--R a x 3 a x
@@ -1563,7 +1405,7 @@
--R Type: Expression Integer
--E
---S 118 14:579 Axiom cannot compute this integral
+--S 104 14:579 Axiom cannot compute this integral
cc:=aa-bb
--R
--R (3)
@@ -1614,7 +1456,7 @@
<<*>>=
)clear all
---S 119
+--S 105
aa:=integrate(1/(cosh(a*x)-1)^2,x)
--R
--R
@@ -1632,7 +1474,7 @@
--R Type: Union(Expression Integer,...)
--E
---S 120
+--S 106
bb:=1/(2*a)*coth((a*x)/2)-1/(6*a)*coth((a*x)/2)^3
--R
--R a x 3 a x
@@ -1643,7 +1485,7 @@
--R Type: Expression Integer
--E
---S 121 14:580 Axiom cannot compute this integral
+--S 107 14:580 Axiom cannot compute this integral
cc:=aa-bb
--R
--R (3)
@@ -1702,7 +1544,7 @@
<<*>>=
)clear all
---S 122
+--S 108
aa:=integrate(1/(p+q*cosh(a*x)),x)
--R
--R
@@ -1744,7 +1586,7 @@
--R Type: Union(List Expression Integer,...)
--E
---S 123
+--S 109
bb1:=2/(a*sqrt(q^2-p^2))*atan((q*%e^(a*x)+p)/sqrt(q^2-p^2))
--R
--R a x
@@ -1760,7 +1602,7 @@
--R Type: Expression Integer
--E
---S 124
+--S 110
bb2:=1/(a*sqrt(p^2-q^2))*log((q*%e^(a*x)+p-sqrt(p^2-q^2))/(q*%e^(a*x)+p+sqrt(p^2-q^2)))
--R
--R +---------+
@@ -1777,7 +1619,7 @@
--R Type: Expression Integer
--E
---S 125
+--S 111
cc1:=aa.1-bb1
--R
--R (4)
@@ -1817,7 +1659,7 @@
--R Type: Expression Integer
--E
---S 126
+--S 112
cc2:=aa.2-bb1
--R
--R +-------+
@@ -1834,7 +1676,7 @@
--R Type: Expression Integer
--E
---S 127
+--S 113
cc3:=aa.1-bb2
--R
--R (6)
@@ -1871,7 +1713,7 @@
--R Type: Expression Integer
--E
---S 128 14:581 Axiom cannot simplify this expression
+--S 114 14:581 Axiom cannot simplify this expression
cc4:=aa.2-bb2
--R
--R (7)
@@ -1907,7 +1749,7 @@
<<*>>=
)clear all
---S 129
+--S 115
aa:=integrate(1/(p+q*cosh(a*x))^2,x)
--R
--R
@@ -1989,7 +1831,7 @@
--R Type: Union(List Expression Integer,...)
--E
---S 130
+--S 116
t1:=integrate(1/(p+q*cosh(a*x)),x)
--R
--R (2)
@@ -2030,7 +1872,7 @@
--R Type: Union(List Expression Integer,...)
--E
---S 131
+--S 117
bb1:=(q*sinh(a*x))/(a*(q^2-p^2)*(p+q*cosh(a*x)))-p/(q^2-p^2)*t1.1
--R
--R (3)
@@ -2066,7 +1908,7 @@
--R Type: Expression Integer
--E
---S 132
+--S 118
bb2:=(q*sinh(a*x))/(a*(q^2-p^2)*(p+q*cosh(a*x)))-p/(q^2-p^2)*t1.2
--R
--R (4)
@@ -2087,7 +1929,7 @@
--R Type: Expression Integer
--E
---S 133
+--S 119
cc1:=aa.1-bb1
--R
--R (5)
@@ -2183,7 +2025,7 @@
--R Type: Expression Integer
--E
---S 134
+--S 120
cc2:=aa.2-bb1
--R
--R (6)
@@ -2271,7 +2113,7 @@
--R Type: Expression Integer
--E
---S 135
+--S 121
cc3:=aa.1-bb2
--R
--R (7)
@@ -2359,7 +2201,7 @@
--R Type: Expression Integer
--E
---S 136 14:582 Axiom cannot simplify this expression
+--S 122 14:582 Axiom cannot simplify this expression
cc4:=aa.2-bb2
--R
--R (8)
@@ -2409,7 +2251,7 @@
<<*>>=
)clear all
---S 137
+--S 123
aa:=integrate(1/(p^2-q^2*cosh(a*x)^2),x)
--R
--R
@@ -2477,7 +2319,7 @@
--R Type: Union(List Expression Integer,...)
--E
---S 138
+--S 124
bb1:=1/(2*a*p*sqrt(p^2-q^2))*log((p*tanh(a*x)+sqrt(p^2-q^2))/(p*tanh(a*x)-sqrt(p^2-q^2)))
--R
--R +---------+
@@ -2494,7 +2336,7 @@
--R Type: Expression Integer
--E
---S 139
+--S 125
bb2:=-1/(a*p*sqrt(q^2-p^2))*atan((p*tanh(a*x))/sqrt(q^2-p^2))
--R
--R p tanh(a x)
@@ -2509,7 +2351,7 @@
--R Type: Expression Integer
--E
---S 140
+--S 126
cc1:=aa.1-bb1
--R
--R (4)
@@ -2562,7 +2404,7 @@
--R Type: Expression Integer
--E
---S 141
+--S 127
cc2:=aa.2-bb1
--R
--R (5)
@@ -2599,7 +2441,7 @@
--R Type: Expression Integer
--E
---S 142
+--S 128
cc3:=aa.1-bb2
--R
--R (6)
@@ -2658,7 +2500,7 @@
--R Type: Expression Integer
--E
---S 143 14:583 Axiom cannot simplify this expression
+--S 129 14:583 Axiom cannot simplify this expression
cc4:=aa.2-bb2
--R
--R (7)
@@ -2708,7 +2550,7 @@
<<*>>=
)clear all
---S 144
+--S 130
aa:=integrate(1/(p^2+q^2*cosh(a*x)^2),x)
--R
--R
@@ -2754,7 +2596,7 @@
--R Type: Union(Expression Integer,...)
--E
---S 145
+--S 131
bb1:=1/(2*a*p*sqrt(p^2+q^2))*log((p*tanh(a*x)+sqrt(p^2+q^2))/(p*tanh(a*x)-sqrt(p^2+q^2)))
--R
--R +-------+
@@ -2771,7 +2613,7 @@
--R Type: Expression Integer
--E
---S 146
+--S 132
bb2:=1/(a*p*sqrt(p^2+q^2))*atan((p*tanh(a*x))/sqrt(p^2+q^2))
--R
--R p tanh(a x)
@@ -2786,7 +2628,7 @@
--R Type: Expression Integer
--E
---S 147
+--S 133
cc1:=aa-bb1
--R
--R (4)
@@ -2839,7 +2681,7 @@
--R Type: Expression Integer
--E
---S 148 14:584 Axiom cannot compute this integral
+--S 134 14:584 Axiom cannot compute this integral
cc2:=aa-bb2
--R
--R (5)
@@ -2899,7 +2741,7 @@
<<*>>=
)clear all
---S 149 14:585 Axiom cannot compute this integral
+--S 135 14:585 Axiom cannot compute this integral
aa:=integrate(x^m*cosh(a*x),x)
--R
--R
@@ -2919,7 +2761,7 @@
<<*>>=
)clear all
---S 150 14:586 Axiom cannot compute this integral
+--S 136 14:586 Axiom cannot compute this integral
aa:=integrate(cosh(a*x)^n,x)
--R
--R
@@ -2940,7 +2782,7 @@
<<*>>=
)clear all
---S 151 14:587 Axiom cannot compute this integral
+--S 137 14:587 Axiom cannot compute this integral
aa:=integrate(cosh(a*x)/x^n,x)
--R
--R
@@ -2962,7 +2804,7 @@
<<*>>=
)clear all
---S 152 14:588 Axiom cannot compute this integral
+--S 138 14:588 Axiom cannot compute this integral
aa:=integrate(1/cosh(a*x)^n,x)
--R
--R
@@ -2985,7 +2827,7 @@
<<*>>=
)clear all
---S 153 14:589 Axiom cannot compute this integral
+--S 139 14:589 Axiom cannot compute this integral
aa:=integrate(1/cosh(a*x)^n,x)
--R
--R
Modified: trunk/axiom/src/input/schaum29.input.pamphlet
===================================================================
--- trunk/axiom/src/input/schaum29.input.pamphlet 2008-04-27 06:01:45 UTC (rev 865)
+++ trunk/axiom/src/input/schaum29.input.pamphlet 2008-04-28 06:16:48 UTC (rev 866)
@@ -18,7 +18,7 @@
)set message auto off
)clear all
---S 1 of 14
+--S 1
aa:=integrate(sinh(a*x)*cosh(a*x),x)
--R
--R
@@ -28,6 +28,63 @@
--R 4a
--R Type: Union(Expression Integer,...)
--E
+
+--S 2
+bb:=sinh(a*x)^2/(2*a)
+--R
+--R 2
+--R sinh(a x)
+--R (2) ----------
+--R 2a
+--R Type: Expression Integer
+--E
+
+--S 3
+cc:=aa-bb
+--R
+--R 2 2
+--R - sinh(a x) + cosh(a x)
+--R (3) -------------------------
+--R 4a
+--R Type: Expression Integer
+--E
+
+--S 4
+sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2)
+--R
+--R 2 cosh(2x) - 1
+--R (4) sinh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 5
+dd:=sinhsqrrule cc
+--R
+--R 2
+--R - cosh(2a x) + 2cosh(a x) + 1
+--R (5) ------------------------------
+--R 8a
+--R Type: Expression Integer
+--E
+
+--S 6
+coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2)
+--R
+--R 2 cosh(2x) + 1
+--R (6) cosh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 7 14:590 Schaums and Axiom agree
+ee:=coshsqrrule dd
+--R
+--R 1
+--R (7) --
+--R 4a
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.591~~~~~$\displaystyle
@@ -38,7 +95,7 @@
<<*>>=
)clear all
---S 2 of 14
+--S 8
aa:=integrate(sinh(p*x)*cosh(q*x),x)
--R
--R
@@ -48,6 +105,35 @@
--R (q - p )sinh(p x) + (- q + p )cosh(p x)
--R Type: Union(Expression Integer,...)
--E
+
+--S 9
+bb:=(cosh(p+q)*x)/(2*(p+q))+(cosh(p-q)*x)/(2*(p-q))
+--R
+--R (q - p)x cosh(q + p) + (- q - p)x cosh(q - p)
+--R (2) ---------------------------------------------
+--R 2 2
+--R 2q - 2p
+--R Type: Expression Integer
+--E
+
+--S 10 14:591 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R (3)
+--R - 2q sinh(p x)sinh(q x)
+--R +
+--R 2
+--R ((- q + p)x cosh(q + p) + (q + p)x cosh(q - p))sinh(p x)
+--R +
+--R 2p cosh(p x)cosh(q x)
+--R +
+--R 2
+--R ((q - p)x cosh(q + p) + (- q - p)x cosh(q - p))cosh(p x)
+--R /
+--R 2 2 2 2 2 2
+--R (2q - 2p )sinh(p x) + (- 2q + 2p )cosh(p x)
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.592~~~~~$\displaystyle
@@ -58,7 +144,7 @@
<<*>>=
)clear all
---S 3 of 14
+--S 11
aa:=integrate(sinh(a*x)^n*cosh(a*x),x)
--R
--R
@@ -68,6 +154,29 @@
--R (a n + a)sinh(a x) + (- a n - a)cosh(a x)
--R Type: Union(Expression Integer,...)
--E
+
+--S 12
+bb:=sinh(a*x)/((n+1)*a)
+--R
+--R sinh(a x)
+--R (2) ---------
+--R a n + a
+--R Type: Expression Integer
+--E
+
+--S 13 14:592 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R (3)
+--R - sinh(a x)sinh(n log(sinh(a x))) - sinh(a x)cosh(n log(sinh(a x)))
+--R +
+--R 3 2
+--R - sinh(a x) + cosh(a x) sinh(a x)
+--R /
+--R 2 2
+--R (a n + a)sinh(a x) + (- a n - a)cosh(a x)
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.593~~~~~$\displaystyle
@@ -78,7 +187,7 @@
<<*>>=
)clear all
---S 4 of 14
+--S 14
aa:=integrate(cosh(a*x)^n*sinh(a*x),x)
--R
--R
@@ -88,6 +197,29 @@
--R (a n + a)sinh(a x) + (- a n - a)cosh(a x)
--R Type: Union(Expression Integer,...)
--E
+
+--S 15
+bb:=cosh(a*x)^(n+1)/((n+1)*a)
+--R
+--R n + 1
+--R cosh(a x)
+--R (2) --------------
+--R a n + a
+--R Type: Expression Integer
+--E
+
+--S 16 14:593 Axiom cannot simplify this expression
+--R
+--R (3)
+--R - cosh(a x)sinh(n log(cosh(a x))) - cosh(a x)cosh(n log(cosh(a x)))
+--R +
+--R 2 2 n + 1
+--R (- sinh(a x) + cosh(a x) )cosh(a x)
+--R /
+--R 2 2
+--R (a n + a)sinh(a x) + (- a n - a)cosh(a x)
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.594~~~~~$\displaystyle
@@ -98,7 +230,7 @@
<<*>>=
)clear all
---S 5 of 14
+--S 17
aa:=integrate(sinh(a*x)^2*cosh(a*x)^2,x)
--R
--R
@@ -108,6 +240,22 @@
--R 8a
--R Type: Union(Expression Integer,...)
--E
+
+--S 18
+bb:=sinh(4*a*x)/(32*a)-x/8
+--R
+--R sinh(4a x) - 4a x
+--R (2) -----------------
+--R 32a
+--R Type: Expression Integer
+--E
+
+--S 19 14:594 Schaums and Axiom agree
+cc:=complexNormalize(aa-bb)
+--R
+--R (3) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.595~~~~~$\displaystyle
@@ -118,7 +266,7 @@
<<*>>=
)clear all
---S 6 of 14
+--S 20
aa:=integrate(1/(sinh(a*x)*cosh(a*x)),x)
--R
--R
@@ -129,6 +277,67 @@
--R a
--R Type: Union(Expression Integer,...)
--E
+
+--S 21
+bb:=1/a*log(tanh(a*x))
+--R
+--R log(tanh(a x))
+--R (2) --------------
+--R a
+--R Type: Expression Integer
+--E
+
+--S 22
+cc:=aa-bb
+--R
+--R (3)
+--R 2cosh(a x)
+--R - log(tanh(a x)) - log(- ---------------------)
+--R sinh(a x) - cosh(a x)
+--R +
+--R 2sinh(a x)
+--R log(- ---------------------)
+--R sinh(a x) - cosh(a x)
+--R /
+--R a
+--R Type: Expression Integer
+--E
+
+--S 23
+dd:=expandLog cc
+--R
+--R - log(tanh(a x)) + log(sinh(a x)) - log(cosh(a x))
+--R (4) --------------------------------------------------
+--R a
+--R Type: Expression Integer
+--E
+
+--S 24
+tanhrule:=rule(tanh(x) == sinh(x)/cosh(x))
+--R
+--R sinh(x)
+--R (5) tanh(x) == -------
+--R cosh(x)
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 25
+ee:=tanhrule dd
+--R
+--R sinh(a x)
+--R log(sinh(a x)) - log(---------) - log(cosh(a x))
+--R cosh(a x)
+--R (6) ------------------------------------------------
+--R a
+--R Type: Expression Integer
+--E
+
+--S 26 14:595 Schaums and Axiom agree
+ff:=expandLog ee
+--R
+--R (7) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.596~~~~~$\displaystyle
@@ -139,17 +348,52 @@
<<*>>=
)clear all
---S 7 of 14
-aa:=integrate(1/(sinh(a*x)^2*cos(a*x)),x)
---R
+--S 27
+aa:=integrate(1/(sinh(a*x)^2*cosh(a*x)),x)
--R
---R x
---R ++ 1
---I (1) | -------------------- d%R
---R ++ 2
---I cos(%R a)sinh(%R a)
+--R (1)
+--R 2 2
+--R (- 2sinh(a x) - 4cosh(a x)sinh(a x) - 2cosh(a x) + 2)
+--R *
+--R atan(sinh(a x) + cosh(a x))
+--R +
+--R - 2sinh(a x) - 2cosh(a x)
+--R /
+--R 2 2
+--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) - a
--R Type: Union(Expression Integer,...)
--E
+
+--S 28
+bb:=-1/a*atan(sinh(a*x)-csch(a*x))/a
+--R
+--R atan(sinh(a x) - csch(a x))
+--R (2) - ---------------------------
+--R 2
+--R a
+--R Type: Expression Integer
+--E
+
+--S 29 14:596 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R (3)
+--R 2 2
+--R (- 2a sinh(a x) - 4a cosh(a x)sinh(a x) - 2a cosh(a x) + 2a)
+--R *
+--R atan(sinh(a x) + cosh(a x))
+--R +
+--R 2 2
+--R (sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) - 1)
+--R *
+--R atan(sinh(a x) - csch(a x))
+--R +
+--R - 2a sinh(a x) - 2a cosh(a x)
+--R /
+--R 2 2 2 2 2 2
+--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) - a
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.597~~~~~$\displaystyle
@@ -160,7 +404,7 @@
<<*>>=
)clear all
---S 8 of 14
+--S 30
aa:=integrate(1/(sinh(a*x)*cosh(a*x)^2),x)
--R
--R
@@ -181,6 +425,272 @@
--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) + a
--R Type: Union(Expression Integer,...)
--E
+
+--S 31
+bb:=sech(a*x)/a+1/a*log(tanh((a*x)/2))
+--R
+--R a x
+--R log(tanh(---)) + sech(a x)
+--R 2
+--R (2) --------------------------
+--R a
+--R Type: Expression Integer
+--E
+
+--S 32
+cc:=aa-bb
+--R
+--R (3)
+--R 2 2 a x
+--R (- sinh(a x) - 2cosh(a x)sinh(a x) - cosh(a x) - 1)log(tanh(---))
+--R 2
+--R +
+--R 2 2
+--R (- sinh(a x) - 2cosh(a x)sinh(a x) - cosh(a x) - 1)
+--R *
+--R log(sinh(a x) + cosh(a x) + 1)
+--R +
+--R 2 2
+--R (sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) + 1)
+--R *
+--R log(sinh(a x) + cosh(a x) - 1)
+--R +
+--R 2
+--R - sech(a x)sinh(a x) + (- 2cosh(a x)sech(a x) + 2)sinh(a x)
+--R +
+--R 2
+--R (- cosh(a x) - 1)sech(a x) + 2cosh(a x)
+--R /
+--R 2 2
+--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) + a
+--R Type: Expression Integer
+--E
+
+--S 33
+sechrule:=rule(sech(x) == 1/cosh(x))
+--R
+--R 1
+--R (4) sech(x) == -------
+--R cosh(x)
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 34
+dd:=sechrule cc
+--R
+--R (5)
+--R 2 2 3
+--R (- cosh(a x)sinh(a x) - 2cosh(a x) sinh(a x) - cosh(a x) - cosh(a x))
+--R *
+--R a x
+--R log(tanh(---))
+--R 2
+--R +
+--R 2 2 3
+--R (- cosh(a x)sinh(a x) - 2cosh(a x) sinh(a x) - cosh(a x) - cosh(a x))
+--R *
+--R log(sinh(a x) + cosh(a x) + 1)
+--R +
+--R 2 2 3
+--R (cosh(a x)sinh(a x) + 2cosh(a x) sinh(a x) + cosh(a x) + cosh(a x))
+--R *
+--R log(sinh(a x) + cosh(a x) - 1)
+--R +
+--R 2 2
+--R - sinh(a x) + cosh(a x) - 1
+--R /
+--R 2 2 3
+--R a cosh(a x)sinh(a x) + 2a cosh(a x) sinh(a x) + a cosh(a x) + a cosh(a x)
+--R Type: Expression Integer
+--E
+
+--S 35
+tanhrule:=rule(tanh(x) == sinh(x)/cosh(x))
+--R
+--R sinh(x)
+--R (6) tanh(x) == -------
+--R cosh(x)
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 36
+ee:=tanhrule dd
+--R
+--R (7)
+--R 2 2 3
+--R (- cosh(a x)sinh(a x) - 2cosh(a x) sinh(a x) - cosh(a x) - cosh(a x))
+--R *
+--R log(sinh(a x) + cosh(a x) + 1)
+--R +
+--R 2 2 3
+--R (cosh(a x)sinh(a x) + 2cosh(a x) sinh(a x) + cosh(a x) + cosh(a x))
+--R *
+--R log(sinh(a x) + cosh(a x) - 1)
+--R +
+--R 2 2 3
+--R (- cosh(a x)sinh(a x) - 2cosh(a x) sinh(a x) - cosh(a x) - cosh(a x))
+--R *
+--R a x
+--R sinh(---)
+--R 2
+--R log(---------)
+--R a x
+--R cosh(---)
+--R 2
+--R +
+--R 2 2
+--R - sinh(a x) + cosh(a x) - 1
+--R /
+--R 2 2 3
+--R a cosh(a x)sinh(a x) + 2a cosh(a x) sinh(a x) + a cosh(a x) + a cosh(a x)
+--R Type: Expression Integer
+--E
+
+--S 37
+coshcuberule:=rule(cosh(x)^3 == 1/4*cosh(3*x)-3/4*cosh(x))
+--R
+--R 3 cosh(3x) - 3cosh(x)
+--R (8) cosh(x) == -------------------
+--R 4
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 38
+ff:=coshcuberule ee
+--R
+--R (9)
+--R 2 2
+--R - 4cosh(a x)sinh(a x) - 8cosh(a x) sinh(a x) - cosh(3a x)
+--R +
+--R - cosh(a x)
+--R *
+--R log(sinh(a x) + cosh(a x) + 1)
+--R +
+--R 2 2
+--R (4cosh(a x)sinh(a x) + 8cosh(a x) sinh(a x) + cosh(3a x) + cosh(a x))
+--R *
+--R log(sinh(a x) + cosh(a x) - 1)
+--R +
+--R 2 2
+--R - 4cosh(a x)sinh(a x) - 8cosh(a x) sinh(a x) - cosh(3a x)
+--R +
+--R - cosh(a x)
+--R *
+--R a x
+--R sinh(---)
+--R 2
+--R log(---------)
+--R a x
+--R cosh(---)
+--R 2
+--R +
+--R 2 2
+--R - 4sinh(a x) + 4cosh(a x) - 4
+--R /
+--R 2 2
+--R 4a cosh(a x)sinh(a x) + 8a cosh(a x) sinh(a x) + a cosh(3a x)
+--R +
+--R a cosh(a x)
+--R Type: Expression Integer
+--E
+
+--S 39
+coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2)
+--R
+--R 2 cosh(2x) + 1
+--R (10) cosh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 40
+gg:=coshsqrrule ff
+--R
+--R (11)
+--R 2
+--R - 4cosh(a x)sinh(a x) + (- 4cosh(2a x) - 4)sinh(a x) - cosh(3a x)
+--R +
+--R - cosh(a x)
+--R *
+--R log(sinh(a x) + cosh(a x) + 1)
+--R +
+--R 2
+--R 4cosh(a x)sinh(a x) + (4cosh(2a x) + 4)sinh(a x) + cosh(3a x)
+--R +
+--R cosh(a x)
+--R *
+--R log(sinh(a x) + cosh(a x) - 1)
+--R +
+--R 2
+--R - 4cosh(a x)sinh(a x) + (- 4cosh(2a x) - 4)sinh(a x) - cosh(3a x)
+--R +
+--R - cosh(a x)
+--R *
+--R a x
+--R sinh(---)
+--R 2
+--R log(---------)
+--R a x
+--R cosh(---)
+--R 2
+--R +
+--R 2
+--R - 4sinh(a x) + 2cosh(2a x) - 2
+--R /
+--R 2
+--R 4a cosh(a x)sinh(a x) + (4a cosh(2a x) + 4a)sinh(a x) + a cosh(3a x)
+--R +
+--R a cosh(a x)
+--R Type: Expression Integer
+--E
+
+--S 41
+sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2)
+--R
+--R 2 cosh(2x) - 1
+--R (12) sinh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 42
+hh:=sinhsqrrule gg
+--R
+--R (13)
+--R - log(sinh(a x) + cosh(a x) + 1) + log(sinh(a x) + cosh(a x) - 1)
+--R +
+--R a x
+--R sinh(---)
+--R 2
+--R - log(---------)
+--R a x
+--R cosh(---)
+--R 2
+--R /
+--R a
+--R Type: Expression Integer
+--E
+
+--S 43
+ii:=expandLog hh
+--R
+--R (14)
+--R - log(sinh(a x) + cosh(a x) + 1) + log(sinh(a x) + cosh(a x) - 1)
+--R +
+--R a x a x
+--R - log(sinh(---)) + log(cosh(---))
+--R 2 2
+--R /
+--R a
+--R Type: Expression Integer
+--E
+
+--S 44 14:597 Schaums and Axiom agree
+jj:=complexNormalize ii
+--R
+--R (15) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.598~~~~~$\displaystyle
@@ -191,7 +701,7 @@
<<*>>=
)clear all
---S 9 of 14
+--S 45
aa:=integrate(1/(sinh(a*x)^2*cosh(a*x)^2),x)
--R
--R
@@ -206,6 +716,35 @@
--R 4a cosh(a x) sinh(a x) + a cosh(a x) - a
--R Type: Union(Expression Integer,...)
--E
+
+--S 46
+bb:=-(2*coth(2*a*x))/a
+--R
+--R 2coth(2a x)
+--R (2) - -----------
+--R a
+--R Type: Expression Integer
+--E
+
+--S 47 14:598 Axiom cannot simplify this expression
+--R
+--R (3)
+--R 4 3
+--R 2coth(2a x)sinh(a x) + 8cosh(a x)coth(2a x)sinh(a x)
+--R +
+--R 2 2 3
+--R 12cosh(a x) coth(2a x)sinh(a x) + 8cosh(a x) coth(2a x)sinh(a x)
+--R +
+--R 4
+--R (2cosh(a x) - 2)coth(2a x) - 4
+--R /
+--R 4 3 2 2
+--R a sinh(a x) + 4a cosh(a x)sinh(a x) + 6a cosh(a x) sinh(a x)
+--R +
+--R 3 4
+--R 4a cosh(a x) sinh(a x) + a cosh(a x) - a
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.599~~~~~$\displaystyle
@@ -216,7 +755,7 @@
<<*>>=
)clear all
---S 10 of 14
+--S 48
aa:=integrate(sinh(a*x)^2/cosh(a*x),x)
--R
--R
@@ -230,6 +769,28 @@
--R 2a sinh(a x) + 2a cosh(a x)
--R Type: Union(Expression Integer,...)
--E
+
+--S 49
+bb:=sinh(a*x)/a-1/a*atan(sinh(a*x))
+--R
+--R - atan(sinh(a x)) + sinh(a x)
+--R (2) -----------------------------
+--R a
+--R Type: Expression Integer
+--E
+
+--S 50 14:599 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R (3)
+--R (- 4sinh(a x) - 4cosh(a x))atan(sinh(a x) + cosh(a x))
+--R +
+--R 2 2
+--R (2sinh(a x) + 2cosh(a x))atan(sinh(a x)) - sinh(a x) + cosh(a x) - 1
+--R /
+--R 2a sinh(a x) + 2a cosh(a x)
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.600~~~~~$\displaystyle
@@ -240,7 +801,7 @@
<<*>>=
)clear all
---S 11 of 14
+--S 51
aa:=integrate(cosh(a*x)^2/sinh(a*x),x)
--R
--R
@@ -256,6 +817,37 @@
--R 2a sinh(a x) + 2a cosh(a x)
--R Type: Union(Expression Integer,...)
--E
+
+--S 52
+bb:=cosh(a*x)/a+1/a*log(tanh((a*x)/2))
+--R
+--R a x
+--R log(tanh(---)) + cosh(a x)
+--R 2
+--R (2) --------------------------
+--R a
+--R Type: Expression Integer
+--E
+
+--S 53 14:600 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R (3)
+--R a x
+--R (- 2sinh(a x) - 2cosh(a x))log(tanh(---))
+--R 2
+--R +
+--R (- 2sinh(a x) - 2cosh(a x))log(sinh(a x) + cosh(a x) + 1)
+--R +
+--R 2
+--R (2sinh(a x) + 2cosh(a x))log(sinh(a x) + cosh(a x) - 1) + sinh(a x)
+--R +
+--R 2
+--R - cosh(a x) + 1
+--R /
+--R 2a sinh(a x) + 2a cosh(a x)
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.601~~~~~$\displaystyle
@@ -267,7 +859,7 @@
<<*>>=
)clear all
---S 12 of 14
+--S 54
aa:=integrate(1/(cosh(a*x)*(1+sinh(a*x))),x)
--R
--R
@@ -281,6 +873,87 @@
--R 2a
--R Type: Union(Expression Integer,...)
--E
+
+--S 55
+bb:=1/(2*a)*log((1+sinh(a*x))/cosh(a*x))+1/a*atan(%e^(a*x))
+--R
+--R sinh(a x) + 1 a x
+--R log(-------------) + 2atan(%e )
+--R cosh(a x)
+--R (2) ---------------------------------
+--R 2a
+--R Type: Expression Integer
+--E
+
+--S 56
+cc:=aa-bb
+--R
+--R (3)
+--R sinh(a x) + 1 2cosh(a x)
+--R - log(-------------) - log(- ---------------------)
+--R cosh(a x) sinh(a x) - cosh(a x)
+--R +
+--R - 2sinh(a x) - 2 a x
+--R log(---------------------) + 2atan(sinh(a x) + cosh(a x)) - 2atan(%e )
+--R sinh(a x) - cosh(a x)
+--R /
+--R 2a
+--R Type: Expression Integer
+--E
+
+--S 57
+dd:=expandLog cc
+--R
+--R a x
+--R atan(sinh(a x) + cosh(a x)) - atan(%e )
+--R (4) -----------------------------------------
+--R a
+--R Type: Expression Integer
+--E
+
+--S 58
+atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x)))
+--R
+--R - x + %i
+--R %i log(--------)
+--R x + %i
+--R (5) atan(x) == - ----------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer)
+--E
+
+--S 59
+ee:=atanrule dd
+--R
+--R a x
+--R - %e + %i - sinh(a x) - cosh(a x) + %i
+--R %i log(------------) - %i log(----------------------------)
+--R a x sinh(a x) + cosh(a x) + %i
+--R %e + %i
+--R (6) -----------------------------------------------------------
+--R 2a
+--R Type: Expression Complex Integer
+--E
+
+--S 60
+ff:=expandLog ee
+--R
+--R (7)
+--R %i log(sinh(a x) + cosh(a x) + %i) - %i log(sinh(a x) + cosh(a x) - %i)
+--R +
+--R a x a x
+--R - %i log(%e + %i) + %i log(%e - %i)
+--R /
+--R 2a
+--R Type: Expression Complex Integer
+--E
+
+--S 61 14:601 Schaums and Axiom agree
+gg:=complexNormalize ff
+--R
+--R (8) 0
+--R Type: Expression Complex Integer
+--E
@
\section{\cite{1}:14.602~~~~~$\displaystyle
@@ -291,7 +964,7 @@
<<*>>=
)clear all
---S 13 of 14
+--S 62
aa:=integrate(1/(sinh(a*x)*(cosh(a*x)+1)),x)
--R
--R
@@ -316,6 +989,204 @@
--R 4a cosh(a x) + 2a
--R Type: Union(Expression Integer,...)
--E
+
+--S 63
+bb:=1/(2*a)*log(tanh((a*x)/2))+1/(2*a*(cosh(a*x)+1))
+--R
+--R a x
+--R (cosh(a x) + 1)log(tanh(---)) + 1
+--R 2
+--R (2) ---------------------------------
+--R 2a cosh(a x) + 2a
+--R Type: Expression Integer
+--E
+
+--S 64
+cc:=aa-bb
+--R
+--R (3)
+--R 2
+--R (- cosh(a x) - 1)sinh(a x)
+--R +
+--R 2 3 2
+--R (- 2cosh(a x) - 4cosh(a x) - 2)sinh(a x) - cosh(a x) - 3cosh(a x)
+--R +
+--R - 3cosh(a x) - 1
+--R *
+--R a x
+--R log(tanh(---))
+--R 2
+--R +
+--R 2
+--R (- cosh(a x) - 1)sinh(a x)
+--R +
+--R 2 3 2
+--R (- 2cosh(a x) - 4cosh(a x) - 2)sinh(a x) - cosh(a x) - 3cosh(a x)
+--R +
+--R - 3cosh(a x) - 1
+--R *
+--R log(sinh(a x) + cosh(a x) + 1)
+--R +
+--R 2 2
+--R (cosh(a x) + 1)sinh(a x) + (2cosh(a x) + 4cosh(a x) + 2)sinh(a x)
+--R +
+--R 3 2
+--R cosh(a x) + 3cosh(a x) + 3cosh(a x) + 1
+--R *
+--R log(sinh(a x) + cosh(a x) - 1)
+--R +
+--R 2 2
+--R - sinh(a x) + cosh(a x) - 1
+--R /
+--R 2
+--R (2a cosh(a x) + 2a)sinh(a x)
+--R +
+--R 2 3
+--R (4a cosh(a x) + 8a cosh(a x) + 4a)sinh(a x) + 2a cosh(a x)
+--R +
+--R 2
+--R 6a cosh(a x) + 6a cosh(a x) + 2a
+--R Type: Expression Integer
+--E
+
+--S 65
+coshcuberule:=rule(cosh(x)^3 == 1/4*cosh(3*x)-3/4*cosh(x))
+--R
+--R 3 cosh(3x) - 3cosh(x)
+--R (4) cosh(x) == -------------------
+--R 4
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 66
+dd:=coshcuberule cc
+--R
+--R (5)
+--R 2
+--R (- 4cosh(a x) - 4)sinh(a x)
+--R +
+--R 2
+--R (- 8cosh(a x) - 16cosh(a x) - 8)sinh(a x) - cosh(3a x)
+--R +
+--R 2
+--R - 12cosh(a x) - 9cosh(a x) - 4
+--R *
+--R a x
+--R log(tanh(---))
+--R 2
+--R +
+--R 2
+--R (- 4cosh(a x) - 4)sinh(a x)
+--R +
+--R 2
+--R (- 8cosh(a x) - 16cosh(a x) - 8)sinh(a x) - cosh(3a x)
+--R +
+--R 2
+--R - 12cosh(a x) - 9cosh(a x) - 4
+--R *
+--R log(sinh(a x) + cosh(a x) + 1)
+--R +
+--R 2 2
+--R (4cosh(a x) + 4)sinh(a x) + (8cosh(a x) + 16cosh(a x) + 8)sinh(a x)
+--R +
+--R 2
+--R cosh(3a x) + 12cosh(a x) + 9cosh(a x) + 4
+--R *
+--R log(sinh(a x) + cosh(a x) - 1)
+--R +
+--R 2 2
+--R - 4sinh(a x) + 4cosh(a x) - 4
+--R /
+--R 2
+--R (8a cosh(a x) + 8a)sinh(a x)
+...
[truncated message content] |
|
From: <gi...@ax...> - 2008-04-28 06:15:01
|
src/input/schaum28.input.pamphlet | 426 +++-------
src/input/schaum29.input.pamphlet | 1106 ++++++++++++++++++++++++-
src/input/schaum30.input.pamphlet | 480 +++++++++++-
src/input/schaum31.input.pamphlet | 471 ++++++++++-
src/input/schaum32.input.pamphlet | 725 ++++++++++++++++-
src/input/schaum33.input.pamphlet | 749 +++++++++++++++++-
src/input/schaum34.input.pamphlet | 1659 ++++++++++++++++++++++++++++++++++++-
7 files changed, 5229 insertions(+), 387 deletions(-)
New commits:
commit 9ad9a6e88e29ce63934ad9251ef39179c2cc2691
Author: Tim Daly <root@localhost.localdomain>
Date: Fri Apr 25 20:58:54 2008 -0400
20080427 tpd src/input/schaum34.input show Schaums-Axiom equivalence
20080427 tpd src/input/schaum33.input show Schaums-Axiom equivalence
20080427 tpd src/input/schaum32.input show Schaums-Axiom equivalence
20080427 tpd src/input/schaum31.input show Schaums-Axiom equivalence
20080427 tpd src/input/schaum30.input show Schaums-Axiom equivalence
20080427 tpd src/input/schaum29.input show Schaums-Axiom equivalence
20080427 tpd src/input/schaum28.input fix Schaums-Axiom equivalence
|
|
From: <da...@us...> - 2008-04-27 06:01:38
|
Revision: 865
http://axiom.svn.sourceforge.net/axiom/?rev=865&view=rev
Author: daly
Date: 2008-04-26 23:01:45 -0700 (Sat, 26 Apr 2008)
Log Message:
-----------
20080426 tpd src/input/schaum28.input show Schaums-Axiom equivalence
20080426 tpd src/input/schaum27.input show Schaums-Axiom equivalence
20080426 tpd src/input/schaum26.input show Schaums-Axiom equivalence
Modified Paths:
--------------
trunk/axiom/changelog
trunk/axiom/src/input/schaum26.input.pamphlet
trunk/axiom/src/input/schaum27.input.pamphlet
trunk/axiom/src/input/schaum28.input.pamphlet
Modified: trunk/axiom/changelog
===================================================================
--- trunk/axiom/changelog 2008-04-26 05:30:18 UTC (rev 864)
+++ trunk/axiom/changelog 2008-04-27 06:01:45 UTC (rev 865)
@@ -1,3 +1,6 @@
+20080426 tpd src/input/schaum28.input show Schaums-Axiom equivalence
+20080426 tpd src/input/schaum27.input show Schaums-Axiom equivalence
+20080426 tpd src/input/schaum26.input show Schaums-Axiom equivalence
20080425 tpd src/input/schaum25.input show Schaums-Axiom equivalence
20080425 tpd src/input/schaum24.input show Schaums-Axiom equivalence
20080424 tpd src/input/schaum23.input show Schaums-Axiom equivalence
Modified: trunk/axiom/src/input/schaum26.input.pamphlet
===================================================================
--- trunk/axiom/src/input/schaum26.input.pamphlet 2008-04-26 05:30:18 UTC (rev 864)
+++ trunk/axiom/src/input/schaum26.input.pamphlet 2008-04-27 06:01:45 UTC (rev 865)
@@ -18,13 +18,27 @@
)set message auto off
)clear all
---S 1 of 16
+--S 1
aa:=integrate(log(x),x)
--R
--R
--R (1) x log(x) - x
--R Type: Union(Expression Integer,...)
--E
+
+--S 2
+bb:=x*log(x)-x
+--R
+--R (2) x log(x) - x
+--R Type: Expression Integer
+--E
+
+--S 3 14:525 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.526~~~~~$\displaystyle
@@ -35,7 +49,7 @@
<<*>>=
)clear all
---S 2 of 16
+--S 4
aa:=integrate(x*log(x),x)
--R
--R
@@ -44,7 +58,24 @@
--R (1) --------------
--R 4
--R Type: Union(Expression Integer,...)
+--E
+
+--S 5
+bb:=x^2/2*(log(x)-1/2)
+--R
+--R 2 2
+--R 2x log(x) - x
+--R (2) --------------
+--R 4
+--R Type: Expression Integer
--E
+
+--S 6 14:526 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.527~~~~~$\displaystyle
@@ -55,7 +86,7 @@
<<*>>=
)clear all
---S 3 of 16
+--S 7
aa:=integrate(x^m*log(x),x)
--R
--R
@@ -65,7 +96,55 @@
--R 2
--R m + 2m + 1
--R Type: Union(Expression Integer,...)
---E
+--E
+
+--S 8
+bb:=x^(m+1)/(m+1)*(log(x)-1/(m+1))
+--R
+--R m + 1
+--R ((m + 1)log(x) - 1)x
+--R (2) -------------------------
+--R 2
+--R m + 2m + 1
+--R Type: Expression Integer
+--E
+
+--S 9
+cc:=aa-bb
+--R
+--R m log(x) m + 1
+--R ((m + 1)x log(x) - x)%e + ((- m - 1)log(x) + 1)x
+--R (3) -------------------------------------------------------------
+--R 2
+--R m + 2m + 1
+--R Type: Expression Integer
+--E
+
+--S 10
+explog:=rule(%e^(n*log(x)) == x^n)
+--R
+--R n log(x) n
+--R (4) %e == x
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 11
+dd:=explog cc
+--R
+--R m + 1 m
+--R ((- m - 1)log(x) + 1)x + ((m + 1)x log(x) - x)x
+--R (5) -----------------------------------------------------
+--R 2
+--R m + 2m + 1
+--R Type: Expression Integer
+--E
+
+--S 12 14:527 Schaums and Axiom agree
+ee:=complexNormalize dd
+--R
+--R (6) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.528~~~~~$\displaystyle
@@ -76,7 +155,7 @@
<<*>>=
)clear all
---S 4 of 16
+--S 13
aa:=integrate(log(x)/x,x)
--R
--R
@@ -85,7 +164,24 @@
--R (1) -------
--R 2
--R Type: Union(Expression Integer,...)
+--E
+
+--S 14
+bb:=1/2*log(x)^2
+--R
+--R 2
+--R log(x)
+--R (2) -------
+--R 2
+--R Type: Expression Integer
--E
+
+--S 15 14:528 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.529~~~~~$\displaystyle
@@ -96,7 +192,7 @@
<<*>>=
)clear all
---S 5 of 16
+--S 16
aa:=integrate(log(x)/x^2,x)
--R
--R
@@ -105,6 +201,22 @@
--R x
--R Type: Union(Expression Integer,...)
--E
+
+--S 17
+bb:=-log(x)/x-1/x
+--R
+--R - log(x) - 1
+--R (2) ------------
+--R x
+--R Type: Expression Integer
+--E
+
+--S 18 14:529 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.530~~~~~$\displaystyle
@@ -115,14 +227,29 @@
<<*>>=
)clear all
---S 6 of 16
+--S 19
aa:=integrate(log(x)^2,x)
--R
--R
--R 2
--R (1) x log(x) - 2x log(x) + 2x
--R Type: Union(Expression Integer,...)
+--E
+
+--S 20
+bb:=x*log(x)^2-2*x*log(x)+2*x
+--R
+--R 2
+--R (2) x log(x) - 2x log(x) + 2x
+--R Type: Expression Integer
--E
+
+--S 21 14:530 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.531~~~~~$\displaystyle
@@ -133,7 +260,7 @@
<<*>>=
)clear all
---S 7 of 16
+--S 22
aa:=integrate(log(x)^n/x,x)
--R
--R
@@ -142,7 +269,52 @@
--R (1) ---------------------
--R n + 1
--R Type: Union(Expression Integer,...)
+--E
+
+--S 23
+bb:=log(x)^(n+1)/(n+1)
+--R
+--R n + 1
+--R log(x)
+--R (2) -----------
+--R n + 1
+--R Type: Expression Integer
--E
+
+--S 24
+cc:=aa-bb
+--R
+--R n log(log(x)) n + 1
+--R log(x)%e - log(x)
+--R (3) -----------------------------------
+--R n + 1
+--R Type: Expression Integer
+--E
+
+--S 25
+explog:=rule(%e^(n*log(x)) == x^n)
+--R
+--R n log(x) n
+--R (4) %e == x
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 26
+dd:=explog cc
+--R
+--R n + 1 n
+--R - log(x) + log(x)log(x)
+--R (5) -----------------------------
+--R n + 1
+--R Type: Expression Integer
+--E
+
+--S 27 14:531 Schaums and Axiom agree
+ee:=complexNormalize dd
+--R
+--R (6) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.532~~~~~$\displaystyle
@@ -153,13 +325,28 @@
<<*>>=
)clear all
---S 8 of 16
+--S 28
aa:=integrate(1/(x*log(x)),x)
--R
--R
--R (1) log(log(x))
--R Type: Union(Expression Integer,...)
--E
+
+--S 29
+bb:=log(log(x))
+--R
+--R (2) log(log(x))
+--R Type: Expression Integer
+--E
+
+--S 30 14:532 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression Integer
+--E
+
@
\section{\cite{1}:14.533~~~~~$\displaystyle
@@ -171,7 +358,7 @@
<<*>>=
)clear all
---S 9 of 16
+--S 31 14:533 Schaums and Axiom agree by definition
aa:=integrate(1/log(x),x)
--R
--R
@@ -189,7 +376,7 @@
<<*>>=
)clear all
---S 10 of 16
+--S 32 14:534 Axiom cannot compute this integral
aa:=integrate(x^m/log(x),x)
--R
--R
@@ -209,7 +396,7 @@
<<*>>=
)clear all
---S 11 of 16
+--S 33 14:535 Axiom cannot compute this integral
aa:=integrate(log(x)^n,x)
--R
--R
@@ -229,7 +416,7 @@
<<*>>=
)clear all
---S 12 of 16
+--S 34 14:536 Axiom cannot compute this integral
aa:=integrate(x^m*log(x)^n,x)
--R
--R
@@ -249,7 +436,7 @@
<<*>>=
)clear all
---S 13 of 16
+--S 35
aa:=integrate(log(x^2+a^2),x)
--R
--R
@@ -258,6 +445,22 @@
--R a
--R Type: Union(Expression Integer,...)
--E
+
+--S 36
+bb:=x*log(x^2+a^2)-2*x+2*a*atan(x/a)
+--R
+--R 2 2 x
+--R (2) x log(x + a ) + 2a atan(-) - 2x
+--R a
+--R Type: Expression Integer
+--E
+
+--S 37 14:537 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.538~~~~~$\displaystyle
@@ -268,7 +471,7 @@
<<*>>=
)clear all
---S 14 of 16
+--S 38
aa:=integrate(log(x^2-a^2),x)
--R
--R
@@ -276,6 +479,31 @@
--R (1) x log(x - a ) + a log(x + a) - a log(x - a) - 2x
--R Type: Union(Expression Integer,...)
--E
+
+--S 39
+bb:=x*log(x^2-a^2)-2*x+a*log((x+a)/(x-a))
+--R
+--R 2 2 x + a
+--R (2) x log(x - a ) + a log(-----) - 2x
+--R x - a
+--R Type: Expression Integer
+--E
+
+--S 40
+cc:=aa-bb
+--R
+--R x + a
+--R (3) a log(x + a) - a log(x - a) - a log(-----)
+--R x - a
+--R Type: Expression Integer
+--E
+
+--S 41 14:538 Schaums and Axiom agree
+dd:=expandLog cc
+--R
+--R (4) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.539~~~~~$\displaystyle
@@ -287,7 +515,7 @@
<<*>>=
)clear all
---S 15a of 16
+--S 42
aa:=integrate(x^m*log(x^2+a^2),x)
--R
--R
@@ -296,11 +524,11 @@
--I (1) | log(a + %I )%I d%I
--R ++
--R Type: Union(Expression Integer,...)
---E
+--E
)clear all
---S 15b of 16
+--S 43 14:539 Axiom cannot compute this integral
aa:=integrate(x^m*log(x^2-a^2),x)
--R
--R
Modified: trunk/axiom/src/input/schaum27.input.pamphlet
===================================================================
--- trunk/axiom/src/input/schaum27.input.pamphlet 2008-04-26 05:30:18 UTC (rev 864)
+++ trunk/axiom/src/input/schaum27.input.pamphlet 2008-04-27 06:01:45 UTC (rev 865)
@@ -18,7 +18,7 @@
)set message auto off
)clear all
---S 1 of 22
+--S 1
aa:=integrate(sinh(a*x),x)
--R
--R cosh(a x)
@@ -26,6 +26,22 @@
--R a
--R Type: Union(Expression Integer,...)
--E
+
+--S 2
+bb:=cosh(a*x)/a
+--R
+--R cosh(a x)
+--R (2) ---------
+--R a
+--R Type: Expression Integer
+--E
+
+--S 3 14:540 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.541~~~~~$\displaystyle
@@ -36,7 +52,7 @@
<<*>>=
)clear all
---S 2 of 22
+--S 4
aa:=integrate(x*sinh(a*x),x)
--R
--R
@@ -46,6 +62,23 @@
--R a
--R Type: Union(Expression Integer,...)
--E
+
+--S 5
+bb:=(x*cosh(a*x))/a-sinh(a*x)/a^2
+--R
+--R - sinh(a x) + a x cosh(a x)
+--R (2) ---------------------------
+--R 2
+--R a
+--R Type: Expression Integer
+--E
+
+--S 6 14:541 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.542~~~~~$\displaystyle
@@ -56,7 +89,7 @@
<<*>>=
)clear all
---S 3 of 22
+--S 7
aa:=integrate(x^2*sinh(a*x),x)
--R
--R
@@ -67,6 +100,24 @@
--R a
--R Type: Union(Expression Integer,...)
--E
+
+--S 8
+bb:=(x^2/a+2/a^3)*cosh(a*x)-(2*x)/a^2*sinh(a*x)
+--R
+--R 2 2
+--R - 2a x sinh(a x) + (a x + 2)cosh(a x)
+--R (2) --------------------------------------
+--R 3
+--R a
+--R Type: Expression Integer
+--E
+
+--S 9 14:542 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.543~~~~~$\displaystyle
@@ -77,7 +128,7 @@
<<*>>=
)clear all
---S 4 of 22
+--S 10 14:543 Axiom cannot compute this integral
aa:=integrate(sinh(a*x)/x,x)
--R
--R
@@ -97,7 +148,7 @@
<<*>>=
)clear all
---S 5 of 22
+--S 11 14:544 Axiom cannot compute this integral
aa:=integrate(sinh(a*x)/x^2,x)
--R
--R
@@ -118,7 +169,7 @@
<<*>>=
)clear all
---S 6 of 22
+--S 12
aa:=integrate(1/sinh(a*x),x)
--R
--R
@@ -127,6 +178,31 @@
--R a
--R Type: Union(Expression Integer,...)
--E
+
+--S 13
+bb:=1/a*log(tanh(a*x)/2)
+--R
+--R tanh(a x)
+--R log(---------)
+--R 2
+--R (2) --------------
+--R a
+--R Type: Expression Integer
+--E
+
+--S 14 14:545 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R (3)
+--R tanh(a x)
+--R - log(---------) - log(sinh(a x) + cosh(a x) + 1)
+--R 2
+--R +
+--R log(sinh(a x) + cosh(a x) - 1)
+--R /
+--R a
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.546~~~~~$\displaystyle
@@ -138,7 +214,7 @@
<<*>>=
)clear all
---S 7 of 22
+--S 15 14:546 Axiom cannot compute this integral
aa:=integrate(x/sinh(a*x),x)
--R
--R
@@ -158,7 +234,7 @@
<<*>>=
)clear all
---S 8 of 22
+--S 16
aa:=integrate(sinh(a*x)^2,x)
--R
--R
@@ -167,6 +243,22 @@
--R 2a
--R Type: Union(Expression Integer,...)
--E
+
+--S 17
+bb:=(sinh(a*x)*cosh(a*x))/(2*a)-x/2
+--R
+--R cosh(a x)sinh(a x) - a x
+--R (2) ------------------------
+--R 2a
+--R Type: Expression Integer
+--E
+
+--S 18 14:547 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.548~~~~~$\displaystyle
@@ -177,7 +269,7 @@
<<*>>=
)clear all
---S 9 of 22
+--S 19
aa:=integrate(x*sinh(a*x)^2,x)
--R
--R
@@ -188,6 +280,87 @@
--R 8a
--R Type: Union(Expression Integer,...)
--E
+
+--S 20
+bb:=(x*sinh(2*a*x))/(4*a)-cosh(2*a*x)/(8*a^2)-x^2/4
+--R
+--R 2 2
+--R 2a x sinh(2a x) - cosh(2a x) - 2a x
+--R (2) ------------------------------------
+--R 2
+--R 8a
+--R Type: Expression Integer
+--E
+
+--S 21
+cc:=aa-bb
+--R
+--R (3)
+--R 2
+--R - 2a x sinh(2a x) - sinh(a x) + 4a x cosh(a x)sinh(a x) + cosh(2a x)
+--R +
+--R 2
+--R - cosh(a x)
+--R /
+--R 2
+--R 8a
+--R Type: Expression Integer
+--E
+
+--S 22
+sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2)
+--R
+--R 2 cosh(2x) - 1
+--R (4) sinh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 23
+dd:=sinhsqrrule cc
+--R
+--R (5)
+--R 2
+--R - 4a x sinh(2a x) + 8a x cosh(a x)sinh(a x) + cosh(2a x) - 2cosh(a x) + 1
+--R --------------------------------------------------------------------------
+--R 2
+--R 16a
+--R Type: Expression Integer
+--E
+
+--S 24
+coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2)
+--R
+--R 2 cosh(2x) + 1
+--R (6) cosh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 25
+ee:=coshsqrrule dd
+--R
+--R - x sinh(2a x) + 2x cosh(a x)sinh(a x)
+--R (7) --------------------------------------
+--R 4a
+--R Type: Expression Integer
+--E
+
+--S 26
+sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y)))
+--R
+--I %K sinh(y + x) - %K sinh(y - x)
+--I (8) %K cosh(y)sinh(x) == -------------------------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 27 14:548 Axiom cannot simplify this expression
+ff:=sinhcoshrule ee
+--R
+--R (9) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.549~~~~~$\displaystyle
@@ -198,7 +371,7 @@
<<*>>=
)clear all
---S 10 of 22
+--S 28
aa:=integrate(1/sinh(a*x)^2,x)
--R
--R
@@ -208,6 +381,158 @@
--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) - a
--R Type: Union(Expression Integer,...)
--E
+
+--S 29
+bb:=-coth(a*x)/a
+--R
+--R coth(a x)
+--R (2) - ---------
+--R a
+--R Type: Expression Integer
+--E
+
+--S 30
+cc:=aa-bb
+--R
+--R (3)
+--R 2
+--R coth(a x)sinh(a x) + 2cosh(a x)coth(a x)sinh(a x)
+--R +
+--R 2
+--R (cosh(a x) - 1)coth(a x) - 2
+--R /
+--R 2 2
+--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) - a
+--R Type: Expression Integer
+--E
+
+--S 31
+sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2)
+--R
+--R 2 cosh(2x) - 1
+--R (4) sinh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 32
+dd:=sinhsqrrule cc
+--R
+--R (5)
+--R 2
+--R 4cosh(a x)coth(a x)sinh(a x) + (cosh(2a x) + 2cosh(a x) - 3)coth(a x) - 4
+--R --------------------------------------------------------------------------
+--R 2
+--R 4a cosh(a x)sinh(a x) + a cosh(2a x) + 2a cosh(a x) - 3a
+--R Type: Expression Integer
+--E
+
+--S 33
+coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2)
+--R
+--R 2 cosh(2x) + 1
+--R (6) cosh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 34
+ee:=coshsqrrule dd
+--R
+--R 2cosh(a x)coth(a x)sinh(a x) + (cosh(2a x) - 1)coth(a x) - 2
+--R (7) ------------------------------------------------------------
+--R 2a cosh(a x)sinh(a x) + a cosh(2a x) - a
+--R Type: Expression Integer
+--E
+
+--S 35
+sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y)))
+--I %B sinh(y + x) - %B sinh(y - x)
+--I (8) %B cosh(y)sinh(x) == -------------------------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 36
+ff:=sinhcoshrule ee
+--R
+--R coth(a x)sinh(2a x) + (cosh(2a x) - 1)coth(a x) - 2
+--R (9) ---------------------------------------------------
+--R a sinh(2a x) + a cosh(2a x) - a
+--R Type: Expression Integer
+--E
+
+--S 37
+cothrule:=rule(coth(x) == cosh(x)/sinh(x))
+--R
+--R cosh(x)
+--R (10) coth(x) == -------
+--R sinh(x)
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 38
+gg:=cothrule ff
+--R
+--R cosh(a x)sinh(2a x) - 2sinh(a x) + cosh(a x)cosh(2a x) - cosh(a x)
+--R (11) ------------------------------------------------------------------
+--R a sinh(a x)sinh(2a x) + (a cosh(2a x) - a)sinh(a x)
+--R Type: Expression Integer
+--E
+
+--S 39
+hh:=sinhcoshrule gg
+--R
+--R sinh(3a x) - 3sinh(a x) + 2cosh(a x)cosh(2a x) - 2cosh(a x)
+--R (12) -----------------------------------------------------------
+--R a sinh(3a x) + 2a sinh(a x)sinh(2a x) - 3a sinh(a x)
+--R Type: Expression Integer
+--E
+
+--S 40
+sinhsinhrule:=rule(sinh(x)*sinh(y)==1/2*(cosh(x+y)-cosh(x-y)))
+--R
+--I %M cosh(y + x) - %M cosh(y - x)
+--I (13) %M sinh(x)sinh(y) == -------------------------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 41
+ii:=sinhsinhrule gg
+--R
+--R 2cosh(a x)sinh(2a x) - 4sinh(a x) + 2cosh(a x)cosh(2a x) - 2cosh(a x)
+--R (14) ---------------------------------------------------------------------
+--R (2a cosh(2a x) - 2a)sinh(a x) + a cosh(3a x) - a cosh(a x)
+--R Type: Expression Integer
+--E
+
+--S 42
+coshcoshrule:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y)))
+--R
+--I %N cosh(y + x) + %N cosh(y - x)
+--I (15) %N cosh(x)cosh(y) == -------------------------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 43
+jj:=coshcoshrule ii
+--R
+--R 2cosh(a x)sinh(2a x) - 4sinh(a x) + cosh(3a x) - cosh(a x)
+--R (16) ----------------------------------------------------------
+--R (2a cosh(2a x) - 2a)sinh(a x) + a cosh(3a x) - a cosh(a x)
+--R Type: Expression Integer
+--E
+
+--S 44 14:549 Schaums and Axiom differ by a constant
+kk:=sinhcoshrule jj
+--R
+--R 1
+--R (17) -
+--R a
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.550~~~~~$\displaystyle
@@ -218,7 +543,7 @@
<<*>>=
)clear all
---S 11 of 22
+--S 45
aa:=integrate(sinh(a*x)*sinh(p*x),x)
--R
--R
@@ -228,6 +553,36 @@
--R (p - a )sinh(a x) + (- p + a )cosh(a x)
--R Type: Union(Expression Integer,...)
--E
+
+--S 46
+bb:=(sinh(a+p)*x)/(2*(a+p))-(sinh(a-p)*x)/(2*(a-p))
+--R
+--R (p - a)x sinh(p + a) + (- p - a)x sinh(p - a)
+--R (2) ---------------------------------------------
+--R 2 2
+--R 2p - 2a
+--R Type: Expression Integer
+--E
+
+--S 47 14:550 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R (3)
+--R 2a cosh(a x)sinh(p x)
+--R +
+--R 2
+--R ((- p + a)x sinh(p + a) + (p + a)x sinh(p - a))sinh(a x)
+--R +
+--R 2
+--R - 2p cosh(p x)sinh(a x) + (p - a)x cosh(a x) sinh(p + a)
+--R +
+--R 2
+--R (- p - a)x cosh(a x) sinh(p - a)
+--R /
+--R 2 2 2 2 2 2
+--R (2p - 2a )sinh(a x) + (- 2p + 2a )cosh(a x)
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.551~~~~~$\displaystyle
@@ -238,7 +593,7 @@
<<*>>=
)clear all
---S 12 of 22
+--S 48
aa:=integrate(sinh(a*x)*sin(p*x),x)
--R
--R
@@ -255,6 +610,71 @@
--R (2p + 2a )sinh(a x) + (2p + 2a )cosh(a x)
--R Type: Union(Expression Integer,...)
--E
+
+--S 49
+bb:=(a*cosh(a*x)*sin(p*x)-p*sinh(a*x)*cos(p*x))/(a^2+p^2)
+--R
+--R - p cos(p x)sinh(a x) + a cosh(a x)sin(p x)
+--R (2) -------------------------------------------
+--R 2 2
+--R p + a
+--R Type: Expression Integer
+--E
+
+--S 50
+cc:=aa-bb
+--R
+--R (3)
+--R 2 2
+--R (a sin(p x) + p cos(p x))sinh(a x) + (- a cosh(a x) + a)sin(p x)
+--R +
+--R 2
+--R - p cos(p x)cosh(a x) + p cos(p x)
+--R /
+--R 2 2 2 2
+--R (2p + 2a )sinh(a x) + (2p + 2a )cosh(a x)
+--R Type: Expression Integer
+--E
+
+--S 51
+sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2)
+--R
+--R 2 cosh(2x) - 1
+--R (4) sinh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 52
+dd:=sinhsqrrule cc
+--R
+--R (5)
+--R 2
+--R (a cosh(2a x) - 2a cosh(a x) + a)sin(p x) + p cos(p x)cosh(2a x)
+--R +
+--R 2
+--R - 2p cos(p x)cosh(a x) + p cos(p x)
+--R /
+--R 2 2 2 2
+--R (4p + 4a )sinh(a x) + (4p + 4a )cosh(a x)
+--R Type: Expression Integer
+--E
+
+--S 53
+coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2)
+--R
+--R 2 cosh(2x) + 1
+--R (6) cosh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 54 14:551 Schaums and Axiom agree
+ee:=coshsqrrule dd
+--R
+--R (7) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.552~~~~~$\displaystyle
@@ -265,7 +685,7 @@
<<*>>=
)clear all
---S 13 of 22
+--S 55
aa:=integrate(sinh(a*x)*cos(p*x),x)
--R
--R
@@ -282,6 +702,71 @@
--R (2p + 2a )sinh(a x) + (2p + 2a )cosh(a x)
--R Type: Union(Expression Integer,...)
--E
+
+--S 56
+bb:=(a*cosh(a*x)*cos(p*x)+p*sinh(a*x)*sin(p*x))/(a^2+p^2)
+--R
+--R p sin(p x)sinh(a x) + a cos(p x)cosh(a x)
+--R (2) -----------------------------------------
+--R 2 2
+--R p + a
+--R Type: Expression Integer
+--E
+
+--S 57
+cc:=aa-bb
+--R
+--R (3)
+--R 2 2
+--R (- p sin(p x) + a cos(p x))sinh(a x) + (p cosh(a x) - p)sin(p x)
+--R +
+--R 2
+--R - a cos(p x)cosh(a x) + a cos(p x)
+--R /
+--R 2 2 2 2
+--R (2p + 2a )sinh(a x) + (2p + 2a )cosh(a x)
+--R Type: Expression Integer
+--E
+
+--S 58
+sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2)
+--R
+--R 2 cosh(2x) - 1
+--R (4) sinh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 59
+dd:=sinhsqrrule cc
+--R
+--R (5)
+--R 2
+--R (- p cosh(2a x) + 2p cosh(a x) - p)sin(p x) + a cos(p x)cosh(2a x)
+--R +
+--R 2
+--R - 2a cos(p x)cosh(a x) + a cos(p x)
+--R /
+--R 2 2 2 2
+--R (4p + 4a )sinh(a x) + (4p + 4a )cosh(a x)
+--R Type: Expression Integer
+--E
+
+--S 60
+coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2)
+--R
+--R 2 cosh(2x) + 1
+--R (6) cosh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 61 14:552 Schaums and Axiom agree
+ee:=coshsqrrule dd
+--R
+--R (7) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.553~~~~~$\displaystyle
@@ -293,7 +778,7 @@
<<*>>=
)clear all
---S 14 of 22
+--S 62
aa:=integrate(1/(p+q*sinh(a*x)),x)
--R
--R
@@ -322,6 +807,60 @@
--R a\|q + p
--R Type: Union(Expression Integer,...)
--E
+
+--S 63
+bb:=1/(a*sqrt(p^2+q^2))*log((q*%e^(a*x)+p-sqrt(p^2+q^2))/(q*%e^(a*x)+p+sqrt(p^2+q^2)))
+--R
+--R +-------+
+--R | 2 2 a x
+--R - \|q + p + q %e + p
+--R log(--------------------------)
+--R +-------+
+--R | 2 2 a x
+--R \|q + p + q %e + p
+--R (2) -------------------------------
+--R +-------+
+--R | 2 2
+--R a\|q + p
+--R Type: Expression Integer
+--E
+
+--S 64 14:553 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R (3)
+--R log
+--R 2 2 2 2 2
+--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x)
+--R +
+--R 2 2
+--R 2p q cosh(a x) + q + 2p
+--R *
+--R +-------+
+--R | 2 2
+--R \|q + p
+--R +
+--R 3 2 3 2 2 3
+--R (- 2q - 2p q)sinh(a x) + (- 2q - 2p q)cosh(a x) - 2p q - 2p
+--R /
+--R 2 2
+--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x)
+--R +
+--R 2p cosh(a x) - q
+--R +
+--R +-------+
+--R | 2 2 a x
+--R - \|q + p + q %e + p
+--R - log(--------------------------)
+--R +-------+
+--R | 2 2 a x
+--R \|q + p + q %e + p
+--R /
+--R +-------+
+--R | 2 2
+--R a\|q + p
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.554~~~~~$\displaystyle
@@ -333,7 +872,7 @@
<<*>>=
)clear all
---S 15 of 22
+--S 65
aa:=integrate(1/(p*q*sinh(a*x))^2,x)
--R
--R
@@ -344,6 +883,137 @@
--R a p q sinh(a x) + 2a p q cosh(a x)sinh(a x) + a p q cosh(a x) - a p q
--R Type: Union(Expression Integer,...)
--E
+
+--S 66
+t1:=integrate(1/(p+q*sinh(a*x)),x)
+--R
+--R (2)
+--R log
+--R 2 2 2 2 2
+--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x)
+--R +
+--R 2 2
+--R 2p q cosh(a x) + q + 2p
+--R *
+--R +-------+
+--R | 2 2
+--R \|q + p
+--R +
+--R 3 2 3 2 2 3
+--R (- 2q - 2p q)sinh(a x) + (- 2q - 2p q)cosh(a x) - 2p q - 2p
+--R /
+--R 2 2
+--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x)
+--R +
+--R 2p cosh(a x) - q
+--R /
+--R +-------+
+--R | 2 2
+--R a\|q + p
+--R Type: Union(Expression Integer,...)
+--E
+
+--S 67
+bb:=(-q*cosh(a*x))/(a*(p^2+q^2)*(p+q*sinh(a*x)))+p/(p^2+q^2)*t1
+--R
+--R (3)
+--R 2
+--R (p q sinh(a x) + p )
+--R *
+--R log
+--R 2 2 2 2 2
+--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x)
+--R +
+--R 2 2
+--R 2p q cosh(a x) + q + 2p
+--R *
+--R +-------+
+--R | 2 2
+--R \|q + p
+--R +
+--R 3 2 3 2 2 3
+--R (- 2q - 2p q)sinh(a x) + (- 2q - 2p q)cosh(a x) - 2p q - 2p
+--R /
+--R 2 2
+--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x)
+--R +
+--R 2p cosh(a x) - q
+--R +
+--R +-------+
+--R | 2 2
+--R - q cosh(a x)\|q + p
+--R /
+--R +-------+
+--R 3 2 2 3 | 2 2
+--R ((a q + a p q)sinh(a x) + a p q + a p )\|q + p
+--R Type: Expression Integer
+--E
+
+--S 68 14:554 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R (4)
+--R 3 3 3 3 3 4 2 2
+--R - p q sinh(a x) + (- 2p q cosh(a x) - p q )sinh(a x)
+--R +
+--R 3 3 2 4 2 3 3 4 2 2
+--R (- p q cosh(a x) - 2p q cosh(a x) + p q )sinh(a x) - p q cosh(a x)
+--R +
+--R 4 2
+--R p q
+--R *
+--R log
+--R 2 2 2 2 2
+--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x)
+--R +
+--R 2 2
+--R 2p q cosh(a x) + q + 2p
+--R *
+--R +-------+
+--R | 2 2
+--R \|q + p
+--R +
+--R 3 2 3 2 2 3
+--R (- 2q - 2p q)sinh(a x) + (- 2q - 2p q)cosh(a x) - 2p q - 2p
+--R /
+--R 2 2
+--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x)
+--R +
+--R 2p cosh(a x) - q
+--R +
+--R 2 3 2 2 3 2 3 2
+--R p q cosh(a x)sinh(a x) + (2p q cosh(a x) - 2q - 2p q)sinh(a x)
+--R +
+--R 2 3 3 2 3 2 3
+--R p q cosh(a x) - p q cosh(a x) - 2p q - 2p
+--R *
+--R +-------+
+--R | 2 2
+--R \|q + p
+--R /
+--R 2 5 4 3 3
+--R (a p q + a p q )sinh(a x)
+--R +
+--R 2 5 4 3 3 4 5 2 2
+--R ((2a p q + 2a p q )cosh(a x) + a p q + a p q )sinh(a x)
+--R +
+--R 2 5 4 3 2 3 4 5 2 2 5
+--R (a p q + a p q )cosh(a x) + (2a p q + 2a p q )cosh(a x) - a p q
+--R +
+--R 4 3
+--R - a p q
+--R *
+--R sinh(a x)
+--R +
+--R 3 4 5 2 2 3 4 5 2
+--R (a p q + a p q )cosh(a x) - a p q - a p q
+--R *
+--R +-------+
+--R | 2 2
+--R \|q + p
+--R Type: Expression Integer
+--E
+
@
\section{\cite{1}:14.555~~~~~$\displaystyle
@@ -363,7 +1033,7 @@
<<*>>=
)clear all
---S 16 of 22
+--S 69
aa:=integrate(1/(p^2+q^2*sinh(a*x)^2),x)
--R
--R
@@ -426,6 +1096,180 @@
--R ]
--R Type: Union(List Expression Integer,...)
--E
+
+--S 70
+bb1:=1/(a*p*sqrt(q^2-p^2))*atan((sqrt(q^2-p^2)*tanh(a*x))/p)
+--R
+--R +-------+
+--R | 2 2
+--R tanh(a x)\|q - p
+--R atan(-------------------)
+--R p
+--R (2) -------------------------
+--R +-------+
+--R | 2 2
+--R a p\|q - p
+--R Type: Expression Integer
+--E
+
+--S 71
+bb2:=1/(2*a*p*sqrt(p^2-q^2))*log((p+sqrt(p^2-q^2)*tanh(a*x))/(p-sqrt(p^2-q^2)*tanh(a*x)))
+--R
+--R +---------+
+--R | 2 2
+--R - tanh(a x)\|- q + p - p
+--R log(---------------------------)
+--R +---------+
+--R | 2 2
+--R tanh(a x)\|- q + p - p
+--R (3) --------------------------------
+--R +---------+
+--R | 2 2
+--R 2a p\|- q + p
+--R Type: Expression Integer
+--E
+
+--S 72
+cc1:=aa.1-bb1
+--R
+--R (4)
+--R +-------+
+--R | 2 2
+--R \|q - p
+--R *
+--R log
+--R 4 4 4 3
+--R q sinh(a x) + 4q cosh(a x)sinh(a x)
+--R +
+--R 4 2 4 2 2 2
+--R (6q cosh(a x) - 2q + 4p q )sinh(a x)
+--R +
+--R 4 3 4 2 2
+--R (4q cosh(a x) + (- 4q + 8p q )cosh(a x))sinh(a x)
+--R +
+--R 4 4 4 2 2 2 4 2 2 4
+--R q cosh(a x) + (- 2q + 4p q )cosh(a x) + q - 8p q + 8p
+--R *
+--R +---------+
+--R | 2 2
+--R \|- q + p
+--R +
+--R 4 3 2 2 4 3 2
+--R (4p q - 4p q )sinh(a x) + (8p q - 8p q )cosh(a x)sinh(a x)
+--R +
+--R 4 3 2 2 4 3 2 5
+--R (4p q - 4p q )cosh(a x) - 4p q + 12p q - 8p
+--R /
+--R 2 4 2 3
+--R q sinh(a x) + 4q cosh(a x)sinh(a x)
+--R +
+--R 2 2 2 2 2
+--R (6q cosh(a x) - 2q + 4p )sinh(a x)
+--R +
+--R 2 3 2 2 2 4
+--R (4q cosh(a x) + (- 4q + 8p )cosh(a x))sinh(a x) + q cosh(a x)
+--R +
+--R 2 2 2 2
+--R (- 2q + 4p )cosh(a x) + q
+--R +
+--R +-------+
+--R +---------+ | 2 2
+--R | 2 2 tanh(a x)\|q - p
+--R - 2\|- q + p atan(-------------------)
+--R p
+--R /
+--R +---------+ +-------+
+--R | 2 2 | 2 2
+--R 2a p\|- q + p \|q - p
+--R Type: Expression Integer
+--E
+
+--S 73
+cc2:=aa.2-bb1
+--R
+--R (5)
+--R +-------+
+--R | 2 2
+--R tanh(a x)\|q - p
+--R - atan(-------------------)
+--R p
+--R +
+--R atan
+--R 2 2 2 2 2 2 2
+--R (q sinh(a x) + 2q cosh(a x)sinh(a x) + q cosh(a x) - q + 2p )
+--R *
+--R +-------+
+--R | 2 2
+--R \|q - p
+--R /
+--R 2 3
+--R 2p q - 2p
+--R /
+--R +-------+
+--R | 2 2
+--R a p\|q - p
+--R Type: Expression Integer
+--E
+
+--S 74
+cc3:=aa.2-bb1
+--R
+--R (6)
+--R +-------+
+--R | 2 2
+--R tanh(a x)\|q - p
+--R - atan(-------------------)
+--R p
+--R +
+--R atan
+--R 2 2 2 2 2 2 2
+--R (q sinh(a x) + 2q cosh(a x)sinh(a x) + q cosh(a x) - q + 2p )
+--R *
+--R +-------+
+--R | 2 2
+--R \|q - p
+--R /
+--R 2 3
+--R 2p q - 2p
+--R /
+--R +-------+
+--R | 2 2
+--R a p\|q - p
+--R Type: Expression Integer
+--E
+
+--S 75 14:555 Axiom cannot simplify this expression
+cc4:=aa.2-bb2
+--R
+--R (7)
+--R +---------+
+--R +-------+ | 2 2
+--R | 2 2 - tanh(a x)\|- q + p - p
+--R - \|q - p log(---------------------------)
+--R +---------+
+--R | 2 2
+--R tanh(a x)\|- q + p - p
+--R +
+--R +---------+
+--R | 2 2
+--R 2\|- q + p
+--R *
+--R atan
+--R 2 2 2 2 2 2 2
+--R (q sinh(a x) + 2q cosh(a x)sinh(a x) + q cosh(a x) - q + 2p )
+--R *
+--R +-------+
+--R | 2 2
+--R \|q - p
+--R /
+--R 2 3
+--R 2p q - 2p
+--R /
+--R +---------+ +-------+
+--R | 2 2 | 2 2
+--R 2a p\|- q + p \|q - p
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.556~~~~~$\displaystyle
@@ -437,7 +1281,7 @@
<<*>>=
)clear all
---S 17 of 22
+--S 76
aa:=integrate(1/(p^2+q^2*sinh(a*x)^2),x)
--R
--R
@@ -500,6 +1344,113 @@
--R ]
--R Type: Union(List Expression Integer,...)
--E
+
+--S 77
+bb:=1/(2*a*p*sqrt(p^2+q^2))*log((p+sqrt(p^2+q^2)*tanh(a*x))/(p-sqrt(p^2+q^2)*tanh(a*x)))
+--R
+--R +-------+
+--R | 2 2
+--R - tanh(a x)\|q + p - p
+--R log(-------------------------)
+--R +-------+
+--R | 2 2
+--R tanh(a x)\|q + p - p
+--R (2) ------------------------------
+--R +-------+
+--R | 2 2
+--R 2a p\|q + p
+--R Type: Expression Integer
+--E
+
+--S 78
+cc1:=aa.1-bb
+--R
+--R (3)
+--R +-------+
+--R | 2 2
+--R \|q + p
+--R *
+--R log
+--R 4 4 4 3
+--R q sinh(a x) + 4q cosh(a x)sinh(a x)
+--R +
+--R 4 2 4 2 2 2
+--R (6q cosh(a x) - 2q + 4p q )sinh(a x)
+--R +
+--R 4 3 4 2 2
+--R (4q cosh(a x) + (- 4q + 8p q )cosh(a x))sinh(a x)
+--R +
+--R 4 4 4 2 2 2 4 2 2 4
+--R q cosh(a x) + (- 2q + 4p q )cosh(a x) + q - 8p q + 8p
+--R *
+--R +---------+
+--R | 2 2
+--R \|- q + p
+--R +
+--R 4 3 2 2 4 3 2
+--R (4p q - 4p q )sinh(a x) + (8p q - 8p q )cosh(a x)sinh(a x)
+--R +
+--R 4 3 2 2 4 3 2 5
+--R (4p q - 4p q )cosh(a x) - 4p q + 12p q - 8p
+--R /
+--R 2 4 2 3
+--R q sinh(a x) + 4q cosh(a x)sinh(a x)
+--R +
+--R 2 2 2 2 2
+--R (6q cosh(a x) - 2q + 4p )sinh(a x)
+--R +
+--R 2 3 2 2 2 4
+--R (4q cosh(a x) + (- 4q + 8p )cosh(a x))sinh(a x) + q cosh(a x)
+--R +
+--R 2 2 2 2
+--R (- 2q + 4p )cosh(a x) + q
+--R +
+--R +-------+
+--R +---------+ | 2 2
+--R | 2 2 - tanh(a x)\|q + p - p
+--R - \|- q + p log(-------------------------)
+--R +-------+
+--R | 2 2
+--R tanh(a x)\|q + p - p
+--R /
+--R +---------+ +-------+
+--R | 2 2 | 2 2
+--R 2a p\|- q + p \|q + p
+--R Type: Expression Integer
+--E
+
+--S 79 14:556 Axiom cannot simplify this expression
+cc2:=aa.2-bb
+--R
+--R (4)
+--R +-------+
+--R +-------+ | 2 2
+--R | 2 2 - tanh(a x)\|q + p - p
+--R - \|q - p log(-------------------------)
+--R +-------+
+--R | 2 2
+--R tanh(a x)\|q + p - p
+--R +
+--R +-------+
+--R | 2 2
+--R 2\|q + p
+--R *
+--R atan
+--R 2 2 2 2 2 2 2
+--R (q sinh(a x) + 2q cosh(a x)sinh(a x) + q cosh(a x) - q + 2p )
+--R *
+--R +-------+
+--R | 2 2
+--R \|q - p
+--R /
+--R 2 3
+--R 2p q - 2p
+--R /
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R 2a p\|q - p \|q + p
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.557~~~~~$\displaystyle
@@ -510,7 +1461,7 @@
<<*>>=
)clear all
---S 18 of 22
+--S 80 14:557 Axiom cannot compute this integral
aa:=integrate(x^m*sinh(a*x),x)
--R
--R
@@ -530,7 +1481,7 @@
<<*>>=
)clear all
---S 19 of 22
+--S 81 14:558 Axiom cannot compute this integral
aa:=integrate(sinh(a*x)^n,x)
--R
--R
@@ -550,16 +1501,17 @@
<<*>>=
)clear all
---S 20 of 22
-aa:=integrate(sinh(a*x)/a^n,x)
+--S 82 14:559 Axiom cannot compute this integral
+aa:=integrate(sinh(a*x)/x^n,x)
--R
---R 2 2
---R sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) + 1
---R (1) -------------------------------------------------
---R n
---R (2a sinh(a x) + 2a cosh(a x))a
+--R x
+--I ++ sinh(%T a)
+--I (3) | ---------- d%T
+--R ++ n
+--I %T
--R Type: Union(Expression Integer,...)
--E
+
@
\section{\cite{1}:14.560~~~~~$\displaystyle
@@ -571,7 +1523,7 @@
<<*>>=
)clear all
---S 21 of 22
+--S 83 14:560 Axiom cannot compute this integral
aa:=integrate(1/sinh(a*x)^n,x)
--R
--R
@@ -594,7 +1546,7 @@
<<*>>=
)clear all
---S 22 of 22
+--S 84 14:561 Axiom cannot compute this integral
aa:=integrate(x/sinh(a*x)^n,x)
--R
--R
Modified: trunk/axiom/src/input/schaum28.input.pamphlet
===================================================================
--- trunk/axiom/src/input/schaum28.input.pamphlet 2008-04-26 05:30:18 UTC (rev 864)
+++ trunk/axiom/src/input/schaum28.input.pamphlet 2008-04-27 06:01:45 UTC (rev 865)
@@ -18,7 +18,7 @@
)set message auto off
)clear all
---S 1 of 28
+--S 1
aa:=integrate(cosh(a*x),x)
--R
--R
@@ -27,6 +27,22 @@
--R a
--R Type: Union(Expression Integer,...)
--E
+
+--S 2
+bb:=sinh(a*x)/a
+--R
+--R sinh(a x)
+--R (2) ---------
+--R a
+--R Type: Expression Integer
+--E
+
+--S 3 14:562 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.563~~~~~$\displaystyle
@@ -37,7 +53,7 @@
<<*>>=
)clear all
---S 2 of 28
+--S 4
aa:=integrate(x*cosh(a*x),x)
--R
--R
@@ -47,6 +63,23 @@
--R a
--R Type: Union(Expression Integer,...)
--E
+
+--S 5
+bb:=(x*sinh(a*x))/a-cosh(a*x)/a^2
+--R
+--R a x sinh(a x) - cosh(a x)
+--R (2) -------------------------
+--R 2
+--R a
+--R Type: Expression Integer
+--E
+
+--S 6 14:563 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.564~~~~~$\displaystyle
@@ -57,7 +90,7 @@
<<*>>=
)clear all
---S 3 of 28
+--S 7
aa:=integrate(x^2*cosh(a*x),x)
--R
--R
@@ -68,6 +101,24 @@
--R a
--R Type: Union(Expression Integer,...)
--E
+
+--S 8
+bb:=-(2*x*cosh(a*x))/a^2+(x^2/a+2/a^3)*sinh(a*x)
+--R
+--R 2 2
+--R (a x + 2)sinh(a x) - 2a x cosh(a x)
+--R (2) ------------------------------------
+--R 3
+--R a
+--R Type: Expression Integer
+--E
+
+--S 9 14:564 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.565~~~~~$\displaystyle
@@ -80,7 +131,7 @@
<<*>>=
)clear all
---S 4 of 28
+--S 10 14:565 Axiom cannot compute this integral
aa:=integrate(cosh(a*x)/x,x)
--R
--R
@@ -100,7 +151,7 @@
<<*>>=
)clear all
---S 5 of 28
+--S 11 14:566 Axiom cannot compute this integral
aa:=integrate(cosh(a*x)/x^2,x)
--R
--R
@@ -121,7 +172,7 @@
<<*>>=
)clear all
---S 6 of 28
+--S 12
aa:=integrate(1/cosh(a*x),x)
--R
--R
@@ -130,6 +181,33 @@
--R a
--R Type: Union(Expression Integer,...)
--E
+
+--S 13
+bb:=2/a*atan(%e^(a*x))
+--R
+--R a x
+--R 2atan(%e )
+--R (2) ------------
+--R a
+--R Type: Expression Integer
+--E
+
+--S 14
+cc:=aa-bb
+--R
+--R a x
+--R 2atan(sinh(a x) + cosh(a x)) - 2atan(%e )
+--R (3) -------------------------------------------
+--R a
+--R Type: Expression Integer
+--E
+
+--S 15 14:567 Schaums and Axiom agree
+dd:=complexNormalize cc
+--R
+--R (4) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.568~~~~~$\displaystyle
@@ -141,7 +219,7 @@
<<*>>=
)clear all
---S 7 of 28
+--S 16 14:568 Axiom cannot compute this integral
aa:=integrate(x/cosh(a*x),x)
--R
--R
@@ -161,7 +239,7 @@
<<*>>=
)clear all
---S 8 of 28
+--S 17
aa:=integrate(cosh(a*x)^2,x)
--R
--R
@@ -170,6 +248,42 @@
--R 2a
--R Type: Union(Expression Integer,...)
--E
+
+--S 18
+bb:=x/2+(sinh(a*x)*cosh(a*x))/2
+--R
+--R cosh(a x)sinh(a x) + x
+--R (2) ----------------------
+--R 2
+--R Type: Expression Integer
+--E
+
+--S 19
+cc:=aa-bb
+--R
+--R (- a + 1)cosh(a x)sinh(a x)
+--R (3) ---------------------------
+--R 2a
+--R Type: Expression Integer
+--E
+
+--S 20
+sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y)))
+--R
+--I %B sinh(y + x) - %B sinh(y - x)
+--I (4) %B cosh(y)sinh(x) == -------------------------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 21 14:569 SCHAUMS AND AXIOM DISAGREE?
+dd:=sinhcoshrule cc
+--R
+--R (- a + 1)sinh(2a x)
+--R (5) -------------------
+--R 4a
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.570~~~~~$\displaystyle
@@ -180,7 +294,7 @@
<<*>>=
)clear all
---S 9 of 28
+--S 22
aa:=integrate(x*cosh(a*x)^2,x)
--R
--R
@@ -191,6 +305,87 @@
--R 8a
--R Type: Union(Expression Integer,...)
--E
+
+--S 23
+bb:=x^2/4+(x*sinh(2*a*x))/(4*a)-cosh(2*a*x)/(8*a^2)
+--R
+--R 2 2
+--R 2a x sinh(2a x) - cosh(2a x) + 2a x
+--R (2) ------------------------------------
+--R 2
+--R 8a
+--R Type: Expression Integer
+--E
+
+--S 24
+cc:=aa-bb
+--R
+--R (3)
+--R 2
+--R - 2a x sinh(2a x) - sinh(a x) + 4a x cosh(a x)sinh(a x) + cosh(2a x)
+--R +
+--R 2
+--R - cosh(a x)
+--R /
+--R 2
+--R 8a
+--R Type: Expression Integer
+--E
+
+--S 25
+sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2)
+--R
+--R 2 cosh(2x) - 1
+--R (4) sinh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 26
+dd:=sinhsqrrule cc
+--R
+--R (5)
+--R 2
+--R - 4a x sinh(2a x) + 8a x cosh(a x)sinh(a x) + cosh(2a x) - 2cosh(a x) + 1
+--R --------------------------------------------------------------------------
+--R 2
+--R 16a
+--R Type: Expression Integer
+--E
+
+--S 27
+coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2)
+--R
+--R 2 cosh(2x) + 1
+--R (6) cosh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 28
+ee:=coshsqrrule dd
+--R
+--R - x sinh(2a x) + 2x cosh(a x)sinh(a x)
+--R (7) --------------------------------------
+--R 4a
+--R Type: Expression Integer
+--E
+
+--S 29
+sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y)))
+--R
+--I %S sinh(y + x) - %S sinh(y - x)
+--I (8) %S cosh(y)sinh(x) == -------------------------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 30 14:570 Schaums and Axiom agree
+ff:=sinhcoshrule ee
+--R
+--R (9) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.571~~~~~$\displaystyle
@@ -201,7 +396,7 @@
<<*>>=
)clear all
---S 10 of 28
+--S 31
aa:=integrate(1/cosh(a*x)^2,x)
--R
--R
@@ -211,6 +406,176 @@
--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) + a
--R Type: Union(Expression Integer,...)
--E
+
+--S 32
+bb:=tanh(a*x)/a
+--R
+--R tanh(a x)
+--R (2) ---------
+--R a
+--R Type: Expression Integer
+--E
+
+--S 33
+cc:=aa-bb
+--R
+--R 2 2
+--R (- sinh(a x) - 2cosh(a x)sinh(a x) - cosh(a x) - 1)tanh(a x) - 2
+--R (3) ------------------------------------------------------------------
+--R 2 2
+--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) + a
+--R Type: Expression Integer
+--E
+
+--S 34
+tanhrule:=rule(tanh(x) == sinh(x)/cosh(x))
+--R
+--R sinh(x)
+--R (4) tanh(x) == -------
+--R cosh(x)
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 35
+dd:=tanhrule cc
+--R
+--R (5)
+--R 3 2 2
+--R - sinh(a x) - 2cosh(a x)sinh(a x) + (- cosh(a x) - 1)sinh(a x)
+--R +
+--R - 2cosh(a x)
+--R /
+--R 2 2 3
+--R a cosh(a x)sinh(a x) + 2a cosh(a x) sinh(a x) + a cosh(a x) + a cosh(a x)
+--R Type: Expression Integer
+--E
+
+--S 36
+sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2)
+--R
+--R 2 cosh(2x) - 1
+--R (6) sinh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 37
+ee:=sinhsqrrule dd
+--R
+--R (7)
+--R 3 2
+--R - 2sinh(a x) + (- 2cosh(a x) - 2)sinh(a x) - 2cosh(a x)cosh(2a x)
+--R +
+--R - 2cosh(a x)
+--R /
+--R 2 3
+--R 4a cosh(a x) sinh(a x) + a cosh(a x)cosh(2a x) + 2a cosh(a x)
+--R +
+--R a cosh(a x)
+--R Type: Expression Integer
+--E
+
+--S 38
+coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2)
+--R
+--R 2 cosh(2x) + 1
+--R (8) cosh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 39
+ff:=coshsqrrule ee
+--R
+--R (9)
+--R 3
+--R - 2sinh(a x) + (- cosh(2a x) - 3)sinh(a x) - 2cosh(a x)cosh(2a x)
+--R +
+--R - 2cosh(a x)
+--R /
+--R 3
+--R (2a cosh(2a x) + 2a)sinh(a x) + a cosh(a x)cosh(2a x) + 2a cosh(a x)
+--R +
+--R a cosh(a x)
+--R Type: Expression Integer
+--E
+
+--...
[truncated message content] |
|
From: <gi...@ax...> - 2008-04-27 06:00:48
|
changelog | 3 +
src/input/schaum26.input.pamphlet | 264 ++++-
src/input/schaum27.input.pamphlet | 1008 ++++++++++++++++-
src/input/schaum28.input.pamphlet | 2220 ++++++++++++++++++++++++++++++++++++-
4 files changed, 3421 insertions(+), 74 deletions(-)
New commits:
commit 865ff69fea526dfa87674fabd28510914cd80738
Author: Tim Daly <root@localhost.localdomain>
Date: Fri Apr 25 08:40:23 2008 -0400
20080426 tpd src/input/schaum28.input show Schaums-Axiom equivalence
20080426 tpd src/input/schaum27.input show Schaums-Axiom equivalence
20080426 tpd src/input/schaum26.input show Schaums-Axiom equivalence
|
|
From: <da...@us...> - 2008-04-26 05:30:11
|
Revision: 864
http://axiom.svn.sourceforge.net/axiom/?rev=864&view=rev
Author: daly
Date: 2008-04-25 22:30:18 -0700 (Fri, 25 Apr 2008)
Log Message:
-----------
20080425 tpd src/input/schaum25.input show Schaums-Axiom equivalence
20080425 tpd src/input/schaum24.input show Schaums-Axiom equivalence
Modified Paths:
--------------
trunk/axiom/changelog
trunk/axiom/src/input/schaum24.input.pamphlet
trunk/axiom/src/input/schaum25.input.pamphlet
Modified: trunk/axiom/changelog
===================================================================
--- trunk/axiom/changelog 2008-04-25 06:20:21 UTC (rev 863)
+++ trunk/axiom/changelog 2008-04-26 05:30:18 UTC (rev 864)
@@ -1,3 +1,5 @@
+20080425 tpd src/input/schaum25.input show Schaums-Axiom equivalence
+20080425 tpd src/input/schaum24.input show Schaums-Axiom equivalence
20080424 tpd src/input/schaum23.input show Schaums-Axiom equivalence
20080423 tpd src/input/schaum22.input show Schaums-Axiom equivalence
20080423 tpd src/input/schaum21.input show Schaums-Axiom equivalence
Modified: trunk/axiom/src/input/schaum24.input.pamphlet
===================================================================
--- trunk/axiom/src/input/schaum24.input.pamphlet 2008-04-25 06:20:21 UTC (rev 863)
+++ trunk/axiom/src/input/schaum24.input.pamphlet 2008-04-26 05:30:18 UTC (rev 864)
@@ -18,7 +18,7 @@
)set message auto off
)clear all
---S 1 of 38
+--S 1
aa:=integrate(asin(x/a),x)
--R
--R
@@ -32,6 +32,31 @@
--R 2
--R Type: Union(Expression Integer,...)
--E
+
+--S 2
+bb:=s+asin(x/a)+sqrt(a^2-x^2)
+--R
+--R +---------+
+--R | 2 2 x
+--R (2) \|- x + a + asin(-) + s
+--R a
+--R Type: Expression Integer
+--E
+
+--S 3 14:471 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +---------+
+--R | 2 2
+--R 2x\|- x + a x
+--R - x atan(--------------) - 2asin(-) - 2s
+--R 2 2 a
+--R 2x - a
+--R (3) ----------------------------------------
+--R 2
+--R Type: Expression Integer
+--E
+
@
\section{\cite{1}:14.472~~~~~$\displaystyle
@@ -43,7 +68,7 @@
<<*>>=
)clear all
---S 2 of 38
+--S 4
aa:=integrate(x*asin(x/a),x)
--R
--R
@@ -57,6 +82,32 @@
--R 8
--R Type: Union(Expression Integer,...)
--E
+
+--S 5
+bb:=(x^2/2-a^2/4)*asin(x/a)+(x*sqrt(a^2-x^2))/4
+--R
+--R +---------+
+--R | 2 2 2 2 x
+--R x\|- x + a + (2x - a )asin(-)
+--R a
+--R (2) ---------------------------------
+--R 4
+--R Type: Expression Integer
+--E
+
+--S 6 14:472 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +---------+
+--R | 2 2
+--R 2 2 2x\|- x + a 2 2 x
+--R (- 2x + a )atan(--------------) + (- 4x + 2a )asin(-)
+--R 2 2 a
+--R 2x - a
+--R (3) -------------------------------------------------------
+--R 8
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.473~~~~~$\displaystyle
@@ -67,7 +118,7 @@
<<*>>=
)clear all
---S 3 of 38
+--S 7
aa:=integrate(x^2*asin(x/a),x)
--R
--R
@@ -81,6 +132,32 @@
--R 18
--R Type: Union(Expression Integer,...)
--E
+
+--S 8
+bb:=x^3/3*asin(x/a)+((x^2+2*a^2)*sqrt(a^2-x^2))/9
+--R
+--R +---------+
+--R 2 2 | 2 2 3 x
+--R (x + 2a )\|- x + a + 3x asin(-)
+--R a
+--R (2) -----------------------------------
+--R 9
+--R Type: Expression Integer
+--E
+
+--S 9 14:473 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +---------+
+--R | 2 2
+--R 3 2x\|- x + a 3 x
+--R - x atan(--------------) - 2x asin(-)
+--R 2 2 a
+--R 2x - a
+--R (3) -------------------------------------
+--R 6
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.474~~~~~$\displaystyle
@@ -93,7 +170,7 @@
<<*>>=
)clear all
---S 4 of 38
+--S 10 14:474 Axiom cannot compute this integral
aa:=integrate(asin(x/a)/x,x)
--R
--R
@@ -115,7 +192,7 @@
<<*>>=
)clear all
---S 5 of 38
+--S 11
aa:=integrate(asin(x/a)/x^2,x)
--R
--R
@@ -130,6 +207,38 @@
--R 2a x
--R Type: Union(Expression Integer,...)
--E
+
+--S 12
+bb:=-asin(x/a)/x-1/a*log((a+sqrt(a^2-x^2))/x)
+--R
+--R +---------+
+--R | 2 2
+--R \|- x + a + a x
+--R - x log(----------------) - a asin(-)
+--R x a
+--R (2) -------------------------------------
+--R a x
+--R Type: Expression Integer
+--E
+
+--S 13 14:475 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R (3)
+--R +---------+ +---------+
+--R | 2 2 | 2 2
+--R - x log(\|- x + a + a) + x log(\|- x + a - a)
+--R +
+--R +---------+ +---------+
+--R | 2 2 | 2 2
+--R \|- x + a + a 2x\|- x + a x
+--R 2x log(----------------) + a atan(--------------) + 2a asin(-)
+--R x 2 2 a
+--R 2x - a
+--R /
+--R 2a x
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.476~~~~~$\displaystyle
@@ -140,7 +249,7 @@
<<*>>=
)clear all
---S 6 of 38
+--S 14
aa:=integrate(asin(x/a)^2,x)
--R
--R
@@ -154,6 +263,36 @@
--R 4
--R Type: Union(Expression Integer,...)
--E
+
+--S 15
+bb:=x*asin(x/a)^2-2*x+2*sqrt(a^2-x^2)*asin(x/a)
+--R
+--R +---------+
+--R x | 2 2 x 2
+--R (2) 2asin(-)\|- x + a + x asin(-) - 2x
+--R a a
+--R Type: Expression Integer
+--E
+
+--S 16 14:476 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R (3)
+--R +---------+ 2 +---------+
+--R | 2 2 +---------+ | 2 2
+--R 2x\|- x + a | 2 2 2x\|- x + a
+--R x atan(--------------) - 4\|- x + a atan(--------------)
+--R 2 2 2 2
+--R 2x - a 2x - a
+--R +
+--R +---------+
+--R x | 2 2 x 2
+--R - 8asin(-)\|- x + a - 4x asin(-)
+--R a a
+--R /
+--R 4
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.477~~~~~$\displaystyle
@@ -164,7 +303,7 @@
<<*>>=
)clear all
---S 7 of 38
+--S 17
aa:=integrate(acos(x/a),x)
--R
--R
@@ -178,6 +317,30 @@
--R 2
--R Type: Union(Expression Integer,...)
--E
+
+--S 18
+bb:=x*acos(x/a)-sqrt(a^2-x^2)
+--R
+--R +---------+
+--R | 2 2 x
+--R (2) - \|- x + a + x acos(-)
+--R a
+--R Type: Expression Integer
+--E
+
+--S 19 14:477 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +---------+
+--R | 2 2
+--R 2x\|- x + a x
+--R x atan(--------------) - 2x acos(-)
+--R 2 2 a
+--R 2x - a
+--R (3) -----------------------------------
+--R 2
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.478~~~~~$\displaystyle
@@ -189,7 +352,7 @@
<<*>>=
)clear all
---S 8 of 38
+--S 20
aa:=integrate(x*acos(x/a),x)
--R
--R
@@ -203,6 +366,32 @@
--R 8
--R Type: Union(Expression Integer,...)
--E
+
+--S 21
+bb:=(x^2/2-a^2/4)*acos(x/a)-(x*sqrt(a^2-x^2))/4
+--R
+--R +---------+
+--R | 2 2 2 2 x
+--R - x\|- x + a + (2x - a )acos(-)
+--R a
+--R (2) -----------------------------------
+--R 4
+--R Type: Expression Integer
+--E
+
+--S 22 14:478 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +---------+
+--R | 2 2
+--R 2 2 2x\|- x + a 2 2 x
+--R (2x - a )atan(--------------) + (- 4x + 2a )acos(-)
+--R 2 2 a
+--R 2x - a
+--R (3) -----------------------------------------------------
+--R 8
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.479~~~~~$\displaystyle
@@ -213,7 +402,7 @@
<<*>>=
)clear all
---S 9 of 38
+--S 23
aa:=integrate(x^2*acos(x/a),x)
--R
--R
@@ -227,6 +416,32 @@
--R 18
--R Type: Union(Expression Integer,...)
--E
+
+--S 24
+bb:=x^3/3*acos(x/a)-((x^2+2*a^2)*sqrt(a^2-x^2))/9
+--R
+--R +---------+
+--R 2 2 | 2 2 3 x
+--R (- x - 2a )\|- x + a + 3x acos(-)
+--R a
+--R (2) -------------------------------------
+--R 9
+--R Type: Expression Integer
+--E
+
+--S 25 14:479 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +---------+
+--R | 2 2
+--R 3 2x\|- x + a 3 x
+--R x atan(--------------) - 2x acos(-)
+--R 2 2 a
+--R 2x - a
+--R (3) -----------------------------------
+--R 6
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.480~~~~~$\displaystyle
@@ -237,7 +452,7 @@
<<*>>=
)clear all
---S 10 of 38
+--S 26 14:480 Axiom cannot compute this integral
aa:=integrate(acos(x/a)/x,x)
--R
--R
@@ -258,7 +473,7 @@
<<*>>=
)clear all
---S 11 of 38
+--S 27
aa:=integrate(acos(x/a)/x^2,x)
--R
--R
@@ -273,6 +488,38 @@
--R 2a x
--R Type: Union(Expression Integer,...)
--E
+
+--S 28
+bb:=-acos(x/a)/x+1/a*log((a+sqrt(a^2-x^2))/x)
+--R
+--R +---------+
+--R | 2 2
+--R \|- x + a + a x
+--R x log(----------------) - a acos(-)
+--R x a
+--R (2) -----------------------------------
+--R a x
+--R Type: Expression Integer
+--E
+
+--S 29 14:481 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R (3)
+--R +---------+ +---------+
+--R | 2 2 | 2 2
+--R x log(\|- x + a + a) - x log(\|- x + a - a)
+--R +
+--R +---------+ +---------+
+--R | 2 2 | 2 2
+--R \|- x + a + a 2x\|- x + a x
+--R - 2x log(----------------) - a atan(--------------) + 2a acos(-)
+--R x 2 2 a
+--R 2x - a
+--R /
+--R 2a x
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.482~~~~~$\displaystyle
@@ -283,7 +530,7 @@
<<*>>=
)clear all
---S 12 of 38
+--S 30
aa:=integrate(acos(x/a)^2,x)
--R
--R
@@ -296,7 +543,37 @@
--R (1) ----------------------------------------------------------------
--R 4
--R Type: Union(Expression Integer,...)
---E
+--E
+
+--S 31
+bb:=x*acos(x/a)^2-2*x-2*sqrt(a^2-x^2)*acos(x/a)
+--R
+--R +---------+
+--R x | 2 2 x 2
+--R (2) - 2acos(-)\|- x + a + x acos(-) - 2x
+--R a a
+--R Type: Expression Integer
+--E
+
+--S 32 14:482 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R (3)
+--R +---------+ 2 +---------+
+--R | 2 2 +---------+ | 2 2
+--R 2x\|- x + a | 2 2 2x\|- x + a
+--R x atan(--------------) - 4\|- x + a atan(--------------)
+--R 2 2 2 2
+--R 2x - a 2x - a
+--R +
+--R +---------+
+--R x | 2 2 x 2
+--R 8acos(-)\|- x + a - 4x acos(-)
+--R a a
+--R /
+--R 4
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.483~~~~~$\displaystyle
@@ -307,7 +584,7 @@
<<*>>=
)clear all
---S 13 of 38
+--S 33
aa:=integrate(atan(x/a),x)
--R
--R
@@ -319,6 +596,62 @@
--R 2
--R Type: Union(Expression Integer,...)
--E
+
+--S 34
+bb:=x*atan(x/a)-a/2*log(x^2+a^2)
+--R
+--R 2 2 x
+--R - a log(x + a ) + 2x atan(-)
+--R a
+--R (2) -----------------------------
+--R 2
+--R Type: Expression Integer
+--E
+
+--S 35
+cc:=aa-bb
+--R
+--R x 2a x
+--R - 2x atan(-) - x atan(-------)
+--R a 2 2
+--R x - a
+--R (3) ------------------------------
+--R 2
+--R Type: Expression Integer
+--E
+
+--S 36
+atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x)))
+--R
+--R - x + %i
+--R %i log(--------)
+--R x + %i
+--R (4) atan(x) == - ----------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer)
+--E
+
+--S 37
+dd:=atanrule cc
+--R
+--R 2 2
+--R x + 2%i a x - a - x + %i a
+--R %i x log(-----------------) + 2%i x log(----------)
+--R 2 2 x + %i a
+--R x - 2%i a x - a
+--R (5) ---------------------------------------------------
+--R 4
+--R Type: Expression Complex Integer
+--E
+
+--S 38 14:483 SCHAUMS AND AXIOM DIFFER? (BRANCH CUTS?)
+ee:=expandLog dd
+--R
+--R %i x log(- 1)
+--R (6) -------------
+--R 2
+--R Type: Expression Complex Integer
+--E
@
\section{\cite{1}:14.484~~~~~$\displaystyle
@@ -329,7 +662,7 @@
<<*>>=
)clear all
---S 14 of 38
+--S 39 14:484 Axiom cannot compute this integral
aa:=integrate(x*tan(x/a),x)
--R
--R
@@ -349,7 +682,7 @@
<<*>>=
)clear all
---S 15 of 38
+--S 40
aa:=integrate(x^2*atan(x/a),x)
--R
--R
@@ -361,6 +694,29 @@
--R 6
--R Type: Union(Expression Integer,...)
--E
+
+--S 41
+bb:=x^3/2*atan(x/a)-(a*x^2)/6+a^3/6*log(x^2+a^2)
+--R
+--R 3 2 2 3 x 2
+--R a log(x + a ) + 3x atan(-) - a x
+--R a
+--R (2) ----------------------------------
+--R 6
+--R Type: Expression Integer
+--E
+
+--S 42 14:485 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R 3 x 3 2a x
+--R - 3x atan(-) - x atan(-------)
+--R a 2 2
+--R x - a
+--R (3) ------------------------------
+--R 6
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.486~~~~~$\displaystyle
@@ -371,7 +727,7 @@
<<*>>=
)clear all
---S 16 of 38
+--S 43 14:486 Axiom cannot compute this integral
aa:=integrate(atan(x/a)/x,x)
--R
--R
@@ -393,7 +749,7 @@
<<*>>=
)clear all
---S 17 of 38
+--S 44
aa:=integrate(atan(x/a)/x^2,x)
--R
--R
@@ -405,6 +761,73 @@
--R 2a x
--R Type: Union(Expression Integer,...)
--E
+
+--S 45
+bb:=-1/x*atan(x/a)-1/(2*a)*log((x^2+a^2)/x^2)
+--R
+--R 2 2
+--R x + a x
+--R - x log(-------) - 2a atan(-)
+--R 2 a
+--R x
+--R (2) -----------------------------
+--R 2a x
+--R Type: Expression Integer
+--E
+
+--S 46
+cc:=aa-bb
+--R
+--R (3)
+--R 2 2
+--R 2 2 x + a x 2a x
+--R - x log(x + a ) + 2x log(x) + x log(-------) + 2a atan(-) + a atan(-------)
+--R 2 a 2 2
+--R x x - a
+--R ----------------------------------------------------------------------------
+--R 2a x
+--R Type: Expression Integer
+--E
+
+--S 47
+atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x)))
+--R
+--R - x + %i
+--R %i log(--------)
+--R x + %i
+--R (4) atan(x) == - ----------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer)
+--E
+
+--S 48
+dd:=atanrule cc
+--R
+--R (5)
+--R 2 2
+--R 2 2 x + 2%i a x - a
+--R - 2x log(x + a ) + 4x log(x) - %i a log(-----------------)
+--R 2 2
+--R x - 2%i a x - a
+--R +
+--R 2 2
+--R x + a - x + %i a
+--R 2x log(-------) - 2%i a log(----------)
+--R 2 x + %i a
+--R x
+--R /
+--R 4a x
+--R Type: Expression Complex Integer
+--E
+
+--S 49 14:487 SCHAUMS AND AXIOM DIFFER? (branch cuts?)
+ee:=expandLog dd
+--R
+--R %i log(- 1)
+--R (6) - -----------
+--R 2x
+--R Type: Expression Complex Integer
+--E
@
\section{\cite{1}:14.488~~~~~$\displaystyle
@@ -415,7 +838,7 @@
<<*>>=
)clear all
---S 18 of 38
+--S 50
aa:=integrate(acot(x/a),x)
--R
--R
@@ -426,7 +849,85 @@
--R (1) --------------------------------
--R 2
--R Type: Union(Expression Integer,...)
+--E
+
+--S 51
+bb:=x*acot(x/a)+a/2*log(x^2+a^2)
+--R
+--R 2 2 x
+--R a log(x + a ) + 2x acot(-)
+--R a
+--R (2) ---------------------------
+--R 2
+--R Type: Expression Integer
--E
+
+--S 52
+cc:=aa-bb
+--R
+--R 2a x x
+--R x atan(-------) - 2x acot(-)
+--R 2 2 a
+--R x - a
+--R (3) ----------------------------
+--R 2
+--R Type: Expression Integer
+--E
+
+--S 53
+atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x)))
+--R
+--R - x + %i
+--R %i log(--------)
+--R x + %i
+--R (4) atan(x) == - ----------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer)
+--E
+
+--S 54
+dd:=atanrule cc
+--R
+--R 2 2
+--R x + 2%i a x - a x
+--R - %i x log(-----------------) - 4x acot(-)
+--R 2 2 a
+--R x - 2%i a x - a
+--R (5) ------------------------------------------
+--R 4
+--R Type: Expression Complex Integer
+--E
+
+--S 55
+acotrule:=rule(acot(x) == -%i/2*log((%i*x-1)/(%i*x+1)))
+--R
+--R x + %i
+--R %i log(------)
+--R x - %i
+--R (6) acot(x) == - --------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer)
+--E
+
+--S 56
+ee:=acotrule dd
+--R
+--R 2 2
+--R x + 2%i a x - a x + %i a
+--R - %i x log(-----------------) + 2%i x log(--------)
+--R 2 2 x - %i a
+--R x - 2%i a x - a
+--R (7) ---------------------------------------------------
+--R 4
+--R Type: Expression Complex Integer
+--E
+
+--S 57 14:488 Axiom and Schaums agree
+ff:=expandLog %
+--R
+--R (8) 0
+--R Type: Expression Complex Integer
+--E
@
\section{\cite{1}:14.489~~~~~$\displaystyle
@@ -437,7 +938,7 @@
<<*>>=
)clear all
---S 19 of 38
+--S 58
aa:=integrate(x*acot(x/a),x)
--R
--R
@@ -449,6 +950,84 @@
--R 4
--R Type: Union(Expression Integer,...)
--E
+
+--S 59
+bb:=1/2*(x^2+a^2)*acot(x/a)+(a*x)/2
+--R
+--R 2 2 x
+--R (x + a )acot(-) + a x
+--R a
+--R (2) ----------------------
+--R 2
+--R Type: Expression Integer
+--E
+
+--S 60
+cc:=aa-bb
+--R
+--R 2 2 2a x 2 2 x
+--R (x + a )atan(-------) + (- 2x - 2a )acot(-)
+--R 2 2 a
+--R x - a
+--R (3) ---------------------------------------------
+--R 4
+--R Type: Expression Integer
+--E
+
+--S 61
+acotrule:=rule(acot(x) == -%i/2*log((%i*x-1)/(%i*x+1)))
+--R
+--R x + %i
+--R %i log(------)
+--R x - %i
+--R (4) acot(x) == - --------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer)
+--E
+
+--S 62
+dd:=acotrule cc
+--R
+--R 2 2 x + %i a 2 2 2a x
+--R (%i x + %i a )log(--------) + (x + a )atan(-------)
+--R x - %i a 2 2
+--R x - a
+--R (5) -----------------------------------------------------
+--R 4
+--R Type: Expression Complex Integer
+--E
+
+--S 63
+atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x)))
+--R
+--R - x + %i
+--R %i log(--------)
+--R x + %i
+--R (6) atan(x) == - ----------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer)
+--E
+
+--S 64
+ee:=atanrule dd
+--R
+--R (7)
+--R 2 2
+--R 2 2 x + 2%i a x - a 2 2 x + %i a
+--R (- %i x - %i a )log(-----------------) + (2%i x + 2%i a )log(--------)
+--R 2 2 x - %i a
+--R x - 2%i a x - a
+--R ------------------------------------------------------------------------
+--R 8
+--R Type: Expression Complex Integer
+--E
+
+--S 65 14:489 Axiom and Schaums agree
+ff:=expandLog ee
+--R
+--R (8) 0
+--R Type: Expression Complex Integer
+--E
@
\section{\cite{1}:14.490~~~~~$\displaystyle
@@ -459,7 +1038,7 @@
<<*>>=
)clear all
---S 20 of 38
+--S 66
aa:=integrate(x^2*acot(x/a),x)
--R
--R
@@ -470,7 +1049,85 @@
--R (1) -----------------------------------------
--R 6
--R Type: Union(Expression Integer,...)
+--E
+
+--S 67
+bb:=x^3/3*acot(x/a)+(a*x^2)/6-a^3/6*log(x^2+a^2)
+--R
+--R 3 2 2 3 x 2
+--R - a log(x + a ) + 2x acot(-) + a x
+--R a
+--R (2) ------------------------------------
+--R 6
+--R Type: Expression Integer
--E
+
+--S 68
+cc:=aa-bb
+--R
+--R 3 2a x 3 x
+--R x atan(-------) - 2x acot(-)
+--R 2 2 a
+--R x - a
+--R (3) ----------------------------
+--R 6
+--R Type: Expression Integer
+--E
+
+--S 69
+atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x)))
+--R
+--R - x + %i
+--R %i log(--------)
+--R x + %i
+--R (4) atan(x) == - ----------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer)
+--E
+
+--S 70
+dd:=atanrule cc
+--R
+--R 2 2
+--R 3 x + 2%i a x - a 3 x
+--R - %i x log(-----------------) - 4x acot(-)
+--R 2 2 a
+--R x - 2%i a x - a
+--R (5) ------------------------------------------
+--R 12
+--R Type: Expression Complex Integer
+--E
+
+--S 71
+acotrule:=rule(acot(x) == -%i/2*log((%i*x-1)/(%i*x+1)))
+--R
+--R x + %i
+--R %i log(------)
+--R x - %i
+--R (6) acot(x) == - --------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer)
+--E
+
+--S 72
+ee:=acotrule dd
+--R
+--R 2 2
+--R 3 x + 2%i a x - a 3 x + %i a
+--R - %i x log(-----------------) + 2%i x log(--------)
+--R 2 2 x - %i a
+--R x - 2%i a x - a
+--R (7) ---------------------------------------------------
+--R 12
+--R Type: Expression Complex Integer
+--E
+
+--S 73 14:490 Axiom and Schaums agree
+ff:=expandLog ee
+--R
+--R (8) 0
+--R Type: Expression Complex Integer
+--E
@
\section{\cite{1}:14.491~~~~~$\displaystyle
@@ -481,7 +1138,7 @@
<<*>>=
)clear all
---S 21 of 38
+--S 74 14:491 Axiom cannot compute this integral
aa:=integrate(acot(x/a)/x,x)
--R
--R
@@ -502,7 +1159,7 @@
<<*>>=
)clear all
---S 22 of 38
+--S 75
aa:=integrate(acot(x/a)/x^2,x)
--R
--R
@@ -514,6 +1171,101 @@
--R 2a x
--R Type: Union(Expression Integer,...)
--E
+
+--S 76
+bb:=-acot(x/a)/x+1/(2*a)*log((x^2+a^2)/x^2)
+--R
+--R 2 2
+--R x + a x
+--R x log(-------) - 2a acot(-)
+--R 2 a
+--R x
+--R (2) ---------------------------
+--R 2a x
+--R Type: Expression Integer
+--E
+
+--S 77
+cc:=aa-bb
+--R
+--R (3)
+--R 2 2
+--R 2 2 x + a 2a x x
+--R x log(x + a ) - 2x log(x) - x log(-------) - a atan(-------) + 2a acot(-)
+--R 2 2 2 a
+--R x x - a
+--R --------------------------------------------------------------------------
+--R 2a x
+--R Type: Expression Integer
+--E
+
+--S 78
+acotrule:=rule(acot(x) == -%i/2*log((%i*x-1)/(%i*x+1)))
+--R
+--R x + %i
+--R %i log(------)
+--R x - %i
+--R (4) acot(x) == - --------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer)
+--E
+
+--S 79
+dd:=acotrule cc
+--R
+--R (5)
+--R 2 2
+--R 2 2 x + %i a x + a
+--R x log(x + a ) - 2x log(x) - %i a log(--------) - x log(-------)
+--R x - %i a 2
+--R x
+--R +
+--R 2a x
+--R - a atan(-------)
+--R 2 2
+--R x - a
+--R /
+--R 2a x
+--R Type: Expression Complex Integer
+--E
+
+--S 80
+atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x)))
+--R
+--R - x + %i
+--R %i log(--------)
+--R x + %i
+--R (6) atan(x) == - ----------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer)
+--E
+
+--S 81
+ee:=atanrule dd
+--R
+--R (7)
+--R 2 2
+--R 2 2 x + 2%i a x - a
+--R 2x log(x + a ) - 4x log(x) + %i a log(-----------------)
+--R 2 2
+--R x - 2%i a x - a
+--R +
+--R 2 2
+--R x + %i a x + a
+--R - 2%i a log(--------) - 2x log(-------)
+--R x - %i a 2
+--R x
+--R /
+--R 4a x
+--R Type: Expression Complex Integer
+--E
+
+--S 82 14:492 Schaums and Axiom agree
+ff:=expandLog ee
+--R
+--R (8) 0
+--R Type: Expression Complex Integer
+--E
@
\section{\cite{1}:14.493~~~~~$\displaystyle
@@ -534,7 +1286,7 @@
<<*>>=
)clear all
---S 23 of 38
+--S 83
aa:=integrate(asec(x/a),x)
--R
--R
@@ -555,6 +1307,72 @@
--R 2
--R Type: Union(Expression Integer,...)
--E
+
+--S 84
+bb1:=x*asec(x/a)-a*log(x+sqrt(x^2-a^2))
+--R
+--R +-------+
+--R | 2 2 x
+--R (2) - a log(\|x - a + x) + x asec(-)
+--R a
+--R Type: Expression Integer
+--E
+
+--S 85
+bb2:=x*asec(x/a)+a*log(x+sqrt(x^2-a^2))
+--R
+--R +-------+
+--R | 2 2 x
+--R (3) a log(\|x - a + x) + x asec(-)
+--R a
+--R Type: Expression Integer
+--E
+
+--S 86
+cc1:=aa-bb1
+--R
+--R (4)
+--R +---------+
+--R +-------+ +-+ | 2 2
+--R | 2 2 +-+ 2x\|2 \|- x + a
+--R 2a log(\|x - a + x) - a\|2 atan(------------------)
+--R 2 2
+--R 3x - 2a
+--R +
+--R +---------+
+--R | 2 2
+--R 2a\|- x + a x x
+--R x atan(--------------) - 2a atan(------------) - 2x asec(-)
+--R 2 +---------+ a
+--R x | 2 2
+--R \|- x + a
+--R /
+--R 2
+--R Type: Expression Integer
+--E
+
+--S 87 14:493 Axiom cannot simplify these expressions
+cc2:=aa-bb2
+--R
+--R (5)
+--R +---------+
+--R +-------+ +-+ | 2 2
+--R | 2 2 +-+ 2x\|2 \|- x + a
+--R - 2a log(\|x - a + x) - a\|2 atan(------------------)
+--R 2 2
+--R 3x - 2a
+--R +
+--R +---------+
+--R | 2 2
+--R 2a\|- x + a x x
+--R x atan(--------------) - 2a atan(------------) - 2x asec(-)
+--R 2 +---------+ a
+--R x | 2 2
+--R \|- x + a
+--R /
+--R 2
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.494~~~~~$\displaystyle
@@ -575,7 +1393,7 @@
<<*>>=
)clear all
---S 24 of 38
+--S 88
aa:=integrate(x*asec(x/a),x)
--R
--R
@@ -589,6 +1407,60 @@
--R 4
--R Type: Union(Expression Integer,...)
--E
+
+--S 89
+bb1:=x^2/2*asec(x/a)-(a*sqrt(x^2-a^2))/2
+--R
+--R +-------+
+--R | 2 2 2 x
+--R - a\|x - a + x asec(-)
+--R a
+--R (2) -------------------------
+--R 2
+--R Type: Expression Integer
+--E
+
+--S 90
+bb2:=x^2/2*asec(x/a)+(a*sqrt(x^2-a^2))/2
+--R
+--R +-------+
+--R | 2 2 2 x
+--R a\|x - a + x asec(-)
+--R a
+--R (3) -----------------------
+--R 2
+--R Type: Expression Integer
+--E
+
+--S 91
+cc1:=aa-bb1
+--R
+--R (4)
+--R +---------+
+--R | 2 2 +-------+ +---------+
+--R 2 2 2a\|- x + a | 2 2 | 2 2 2 x
+--R (x - 2a )atan(--------------) + 2a\|x - a + 2a\|- x + a - 2x asec(-)
+--R 2 a
+--R x
+--R ---------------------------------------------------------------------------
+--R 4
+--R Type: Expression Integer
+--E
+
+--S 92 14:494 Axiom cannot simplify these expressions
+cc2:=aa-bb2
+--R
+--R (5)
+--R +---------+
+--R | 2 2 +-------+ +---------+
+--R 2 2 2a\|- x + a | 2 2 | 2 2 2 x
+--R (x - 2a )atan(--------------) - 2a\|x - a + 2a\|- x + a - 2x asec(-)
+--R 2 a
+--R x
+--R ---------------------------------------------------------------------------
+--R 4
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.495~~~~~$\displaystyle
@@ -616,7 +1488,7 @@
<<*>>=
)clear all
---S 25 of 38
+--S 93
aa:=integrate(x^2*asec(x/a),x)
--R
--R
@@ -638,6 +1510,87 @@
--R 6
--R Type: Union(Expression Integer,...)
--E
+
+--S 94
+bb1:=x^3/3*asec(x/a)-(a*x*sqrt(x^2-a^2))/6-a^3/6*log(x+sqrt(x^2-a^2))
+--R
+--R +-------+ +-------+
+--R 3 | 2 2 | 2 2 3 x
+--R - a log(\|x - a + x) - a x\|x - a + 2x asec(-)
+--R a
+--R (2) ----------------------------------------------------
+--R 6
+--R Type: Expression Integer
+--E
+
+--S 95
+bb2:=x^3/3*asec(x/a)+(a*x*sqrt(x^2-a^2))/6+a^3/6*log(x+sqrt(x^2-a^2))
+--R
+--R +-------+ +-------+
+--R 3 | 2 2 | 2 2 3 x
+--R a log(\|x - a + x) + a x\|x - a + 2x asec(-)
+--R a
+--R (3) --------------------------------------------------
+--R 6
+--R Type: Expression Integer
+--E
+
+--S 96
+cc1:=aa-bb1
+--R
+--R (4)
+--R +---------+
+--R +-------+ +-+ | 2 2
+--R 3 | 2 2 3 +-+ 2x\|2 \|- x + a
+--R a log(\|x - a + x) - 2a \|2 atan(------------------)
+--R 2 2
+--R 3x - 2a
+--R +
+--R +---------+
+--R | 2 2 +-------+
+--R 3 2a\|- x + a 3 x | 2 2
+--R x atan(--------------) - 5a atan(------------) + a x\|x - a
+--R 2 +---------+
+--R x | 2 2
+--R \|- x + a
+--R +
+--R +---------+
+--R | 2 2 3 x
+--R a x\|- x + a - 2x asec(-)
+--R a
+--R /
+--R 6
+--R Type: Expression Integer
+--E
+
+--S 97 14:495 Axiom cannot simplify these expressions
+cc2:=aa-bb2
+--R
+--R (5)
+--R +---------+
+--R +-------+ +-+ | 2 2
+--R 3 | 2 2 3 +-+ 2x\|2 \|- x + a
+--R - a log(\|x - a + x) - 2a \|2 atan(------------------)
+--R 2 2
+--R 3x - 2a
+--R +
+--R +---------+
+--R | 2 2 +-------+
+--R 3 2a\|- x + a 3 x | 2 2
+--R x atan(--------------) - 5a atan(------------) - a x\|x - a
+--R 2 +---------+
+--R x | 2 2
+--R \|- x + a
+--R +
+--R +---------+
+--R | 2 2 3 x
+--R a x\|- x + a - 2x asec(-)
+--R a
+--R /
+--R 6
+--R Type: Expression Integer
+--E
+
@
\section{\cite{1}:14.496~~~~~$\displaystyle
@@ -650,7 +1603,7 @@
<<*>>=
)clear all
---S 26 of 38
+--S 98 14:496 Axiom cannot compute this integral
aa:=integrate(asec(x/a)/x,x)
--R
--R
@@ -681,7 +1634,7 @@
<<*>>=
)clear all
---S 27 of 38
+--S 99
aa:=integrate(asec(x/a)/x^2,x)
--R
--R
@@ -696,6 +1649,70 @@
--R 2a x\|2
--R Type: Union(Expression Integer,...)
--E
+
+--S 100
+bb1:=-asec(x/a)/x+sqrt(x^2-a^2)/(a*x)
+--R
+--R +-------+
+--R | 2 2 x
+--R \|x - a - a asec(-)
+--R a
+--R (2) ----------------------
+--R a x
+--R Type: Expression Integer
+--E
+
+--S 101
+bb2:=-asec(x/a)/x-sqrt(x^2-a^2)/(a*x)
+--R
+--R +-------+
+--R | 2 2 x
+--R - \|x - a - a asec(-)
+--R a
+--R (3) ------------------------
+--R a x
+--R Type: Expression Integer
+--E
+
+--S 102
+cc1:=aa-bb1
+--R
+--R (4)
+--R +---------+ +---------+
+--R +-+ | 2 2 | 2 2 +-------+
+--R 2x\|2 \|- x + a +-+ 2a\|- x + a +-+ | 2 2
+--R x atan(------------------) - a\|2 atan(--------------) - 2\|2 \|x - a
+--R 2 2 2
+--R 3x - 2a x
+--R +
+--R +-+ x
+--R 2a\|2 asec(-)
+--R a
+--R /
+--R +-+
+--R 2a x\|2
+--R Type: Expression Integer
+--E
+
+--S 103 14:497 Axiom cannot simplify these expressions
+cc2:=aa-bb2
+--R
+--R (5)
+--R +---------+ +---------+
+--R +-+ | 2 2 | 2 2 +-------+
+--R 2x\|2 \|- x + a +-+ 2a\|- x + a +-+ | 2 2
+--R x atan(------------------) - a\|2 atan(--------------) + 2\|2 \|x - a
+--R 2 2 2
+--R 3x - 2a x
+--R +
+--R +-+ x
+--R 2a\|2 asec(-)
+--R a
+--R /
+--R +-+
+--R 2a x\|2
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.498~~~~~$\displaystyle
@@ -716,7 +1733,7 @@
<<*>>=
)clear all
---S 28 of 38
+--S 104
aa:=integrate(acsc(x/a),x)
--R
--R
@@ -737,6 +1754,73 @@
--R 2
--R Type: Union(Expression Integer,...)
--E
+
+--S 105
+bb1:=x*acsc(x/a)+a*log(x+sqrt(x^2-a^2))
+--R
+--R +-------+
+--R | 2 2 x
+--R (2) a log(\|x - a + x) + x acsc(-)
+--R a
+--R Type: Expression Integer
+--E
+
+--S 106
+bb2:=x*acsc(x/a)-a*log(x+sqrt(x^2-a^2))
+--R
+--R +-------+
+--R | 2 2 x
+--R (3) - a log(\|x - a + x) + x acsc(-)
+--R a
+--R Type: Expression Integer
+--E
+
+--S 107
+cc1:=aa-bb1
+--R
+--R (4)
+--R +---------+
+--R +-------+ +-+ | 2 2
+--R | 2 2 +-+ 2x\|2 \|- x + a
+--R - 2a log(\|x - a + x) + a\|2 atan(------------------)
+--R 2 2
+--R 3x - 2a
+--R +
+--R +---------+
+--R | 2 2
+--R 2a\|- x + a x x
+--R - x atan(--------------) + 2a atan(------------) - 2x acsc(-)
+--R 2 +---------+ a
+--R x | 2 2
+--R \|- x + a
+--R /
+--R 2
+--R Type: Expression Integer
+--E
+
+--S 108 14:498 Axiom cannot simplify these expressions
+cc2:=aa-bb2
+--R
+--R (5)
+--R +---------+
+--R +-------+ +-+ | 2 2
+--R | 2 2 +-+ 2x\|2 \|- x + a
+--R 2a log(\|x - a + x) + a\|2 atan(------------------)
+--R 2 2
+--R 3x - 2a
+--R +
+--R +---------+
+--R | 2 2
+--R 2a\|- x + a x x
+--R - x atan(--------------) + 2a atan(------------) - 2x acsc(-)
+--R 2 +---------+ a
+--R x | 2 2
+--R \|- x + a
+--R /
+--R 2
+--R Type: Expression Integer
+--E
+
@
\section{\cite{1}:14.499~~~~~$\displaystyle
@@ -757,7 +1841,7 @@
<<*>>=
)clear all
---S 29 of 38
+--S 109
aa:=integrate(x*acsc(x/a),x)
--R
--R
@@ -771,6 +1855,60 @@
--R 4
--R Type: Union(Expression Integer,...)
--E
+
+--S 110
+bb1:=x^2/2*acsc(x/a)+(a*sqrt(x^2-a^2))/2
+--R
+--R +-------+
+--R | 2 2 2 x
+--R a\|x - a + x acsc(-)
+--R a
+--R (2) -----------------------
+--R 2
+--R Type: Expression Integer
+--E
+
+--S 111
+bb2:=x^2/2*acsc(x/a)-(a*sqrt(x^2-a^2))/2
+--R
+--R +-------+
+--R | 2 2 2 x
+--R - a\|x - a + x acsc(-)
+--R a
+--R (3) -------------------------
+--R 2
+--R Type: Expression Integer
+--E
+
+--S 112
+cc1:=aa-bb1
+--R
+--R (4)
+--R +---------+
+--R | 2 2 +-------+ +---------+
+--R 2 2 2a\|- x + a | 2 2 | 2 2 2 x
+--R (- x + 2a )atan(--------------) - 2a\|x - a - 2a\|- x + a - 2x acsc(-)
+--R 2 a
+--R x
+--R -----------------------------------------------------------------------------
+--R 4
+--R Type: Expression Integer
+--E
+
+--S 113 14:499 Axiom cannot simplify these expressions
+cc2:=aa-bb2
+--R
+--R (5)
+--R +---------+
+--R | 2 2 +-------+ +---------+
+--R 2 2 2a\|- x + a | 2 2 | 2 2 2 x
+--R (- x + 2a )atan(--------------) + 2a\|x - a - 2a\|- x + a - 2x acsc(-)
+--R 2 a
+--R x
+--R -----------------------------------------------------------------------------
+--R 4
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.500~~~~~$\displaystyle
@@ -798,7 +1936,7 @@
<<*>>=
)clear all
---S 30 of 38
+--S 114
aa:=integrate(x^2*acsc(x/a),x)
--R
--R
@@ -820,6 +1958,86 @@
--R 6
--R Type: Union(Expression Integer,...)
--E
+
+--S 115
+bb1:=x^3/3*acsc(x/a)+(a*x*sqrt(x^2-a^2))/6+a^3/6*log(x+sqrt(x^2-a^2))
+--R
+--R +-------+ +-------+
+--R 3 | 2 2 | 2 2 3 x
+--R a log(\|x - a + x) + a x\|x - a + 2x acsc(-)
+--R a
+--R (2) --------------------------------------------------
+--R 6
+--R Type: Expression Integer
+--E
+
+--S 116
+bb2:=x^3/3*acsc(x/a)-(a*x*sqrt(x^2-a^2))/6-a^3/6*log(x+sqrt(x^2-a^2))
+--R
+--R +-------+ +-------+
+--R 3 | 2 2 | 2 2 3 x
+--R - a log(\|x - a + x) - a x\|x - a + 2x acsc(-)
+--R a
+--R (3) ----------------------------------------------------
+--R 6
+--R Type: Expression Integer
+--E
+
+--S 117
+cc1:=aa-bb1
+--R
+--R (4)
+--R +---------+
+--R +-------+ +-+ | 2 2
+--R 3 | 2 2 3 +-+ 2x\|2 \|- x + a
+--R - a log(\|x - a + x) + 2a \|2 atan(------------------)
+--R 2 2
+--R 3x - 2a
+--R +
+--R +---------+
+--R | 2 2 +-------+
+--R 3 2a\|- x + a 3 x | 2 2
+--R - x atan(--------------) + 5a atan(------------) - a x\|x - a
+--R 2 +---------+
+--R x | 2 2
+--R \|- x + a
+--R +
+--R +---------+
+--R | 2 2 3 x
+--R - a x\|- x + a - 2x acsc(-)
+--R a
+--R /
+--R 6
+--R Type: Expression Integer
+--E
+
+--S 118 14:500 Axiom cannot simplify this expression
+cc2:=aa-bb2
+--R
+--R (5)
+--R +---------+
+--R +-------+ +-+ | 2 2
+--R 3 | 2 2 3 +-+ 2x\|2 \|- x + a
+--R a log(\|x - a + x) + 2a \|2 atan(------------------)
+--R 2 2
+--R 3x - 2a
+--R +
+--R +---------+
+--R | 2 2 +-------+
+--R 3 2a\|- x + a 3 x | 2 2
+--R - x atan(--------------) + 5a atan(------------) + a x\|x - a
+--R 2 +---------+
+--R x | 2 2
+--R \|- x + a
+--R +
+--R +---------+
+--R | 2 2 3 x
+--R - a x\|- x + a - 2x acsc(-)
+--R a
+--R /
+--R 6
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.501~~~~~$\displaystyle
@@ -832,7 +2050,7 @@
<<*>>=
)clear all
---S 31 of 38
+--S 119 14:501 Axiom cannot compute this integral
aa:=integrate(acsc(x/a)/x,x)
--R
--R
@@ -863,7 +2081,7 @@
<<*>>=
)clear all
---S 32 of 38
+--S 120
aa:=integrate(acsc(x/a)/x^2,x)
--R
--R
@@ -878,6 +2096,72 @@
--R 2a x\|2
--R Type: Union(Expression Integer,...)
--E
+
+--S 121
+bb1:=-acsc(x/a)/x-sqrt(x^2-a^2)/(a*x)
+--R
+--R +-------+
+--R | 2 2 x
+--R - \|x - a - a acsc(-)
+--R a
+--R (2) ------------------------
+--R a x
+--R Type: Expression Integer
+--E
+
+--S 122
+bb2:=-acsc(x/a)/x+sqrt(x^2-a^2)/(a*x)
+--R
+--R +-------+
+--R | 2 2 x
+--R \|x - a - a acsc(-)
+--R a
+--R (3) ----------------------
+--R a x
+--R Type: Expression Integer
+--E
+
+--S 123
+cc1:=aa-bb1
+--R
+--R (4)
+--R +---------+ +---------+
+--R +-+ | 2 2 | 2 2
+--R 2x\|2 \|- x + a +-+ 2a\|- x + a
+--R - x atan(------------------) + a\|2 atan(--------------)
+--R 2 2 2
+--R 3x - 2a x
+--R +
+--R +-------+
+--R +-+ | 2 2 +-+ x
+--R 2\|2 \|x - a + 2a\|2 acsc(-)
+--R a
+--R /
+--R +-+
+--R 2a x\|2
+--R Type: Expression Integer
+--E
+
+--S 124 14:502 Axiom cannot simplify this expression
+cc2:=aa-bb2
+--R
+--R (5)
+--R +---------+ +---------+
+--R +-+ | 2 2 | 2 2
+--R 2x\|2 \|- x + a +-+ 2a\|- x + a
+--R - x atan(------------------) + a\|2 atan(--------------)
+--R 2 2 2
+--R 3x - 2a x
+--R +
+--R +-------+
+--R +-+ | 2 2 +-+ x
+--R - 2\|2 \|x - a + 2a\|2 acsc(-)
+--R a
+--R /
+--R +-+
+--R 2a x\|2
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.503~~~~~$\displaystyle
@@ -888,7 +2172,7 @@
<<*>>=
)clear all
---S 33 of 38
+--S 125 14:503 Axiom cannot compute this integral
aa:=integrate(x^m*asin(x/a),x)
--R
--R
@@ -908,7 +2192,7 @@
<<*>>=
)clear all
---S 34 of 38
+--S 126 14:504 Axiom cannot compute this integral
aa:=integrate(x^m*acos(x/a),x)
--R
--R
@@ -926,10 +2210,15 @@
\frac{x^{m_1}}{m+1}\tan^{-1}\frac{x}{a}
-\frac{a}{m+1}\int{\frac{x^{m+1}}{x^2+a^2}}
$$
+This appears to be an interesting integral. Axiom found a closed
+form solution to the problem. However, the t1 integral below does
+not have a closed form solution. Note that we did not return a
+result for the prior two integrals, nor for the next integral. They
+have the same form but are expressed in terms of asin, acos, and acot.
<<*>>=
)clear all
---S 35 of 38
+--S 127
aa:=integrate(x*m*atan(x/a),x)
--R
--R
@@ -941,8 +2230,99 @@
--R 4
--R Type: Union(Expression Integer,...)
--E
+
+--S 128
+t1:=integrate(x^(m+1)/(x^2+a^2),x)
+--E
@
+Since we cannot get a closed form version of the prior integral
+we proceed to try to prove that Axiom got a correct answer. We
+do this by computing the derivate of 'aa' above and finding the
+difference from the original formula.
+So first we generate the derivative:
+<<*>>=
+
+--S 129
+bb:=D(aa,x)
+--R
+--R 2a x
+--R m x atan(-------)
+--R 2 2
+--R x - a
+--R (3) - -----------------
+--R 2
+--R Type: Expression Integer
+--E
+@
+Then we input the original expression
+<<*>>=
+--S 130
+aa1:=x*m*atan(x/a)
+--R
+--R x
+--R (4) m x atan(-)
+--R a
+--R Type: Expression Integer
+--E
+@
+Now we take their difference
+<<*>>=
+--S 131
+dd:=aa1-bb
+--R
+--R x 2a x
+--R 2m x atan(-) + m x atan(-------)
+--R a 2 2
+--R x - a
+--R (5) --------------------------------
+--R 2
+--R Type: Expression Integer
+--E
+@
+Now we input the atan transformation
+<<*>>=
+--S 132
+atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x)))
+--R
+--R - x + %i
+--R %i log(--------)
+--R x + %i
+--R (6) atan(x) == - ----------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer)
+--E
+@
+And apply the transformation to the difference
+<<*>>=
+--S 133
+ee:=atanrule dd
+--R
+--R 2 2
+--R x + 2%i a x - a - x + %i a
+--R - %i m x log(-----------------) - 2%i m x log(----------)
+--R 2 2 x + %i a
+--R x - 2%i a x - a
+--R (7) ---------------------------------------------------------
+--R 4
+--R Type: Expression Complex Integer
+--E
+@
+And now we simplify
+<<*>>=
+--S 134 14:505 SCHAUMS AND AXIOM DISAGREE? (branch cuts?)
+ff:=expandLog ee
+--R
+--R %i m x log(- 1)
+--R (8) - ---------------
+--R 2
+--R Type: Expression Complex Integer
+--E
+And we get the surprising result that they are not equal.
+In fact, they differ by a complex value depending on x.
+Likely there is a branch-cut issue lurking somewhere.
+@
+
\section{\cite{1}:14.506~~~~~$\displaystyle
\int{x^m\cot^{-1}\frac{x}{a}}~dx$}
$$\int{x^m\cot^{-1}\frac{x}{a}}=
@@ -952,7 +2332,7 @@
<<*>>=
)clear all
---S 36 of 38
+--S 135 14:506 Axiom cannot compute this integral
aa:=integrate(x^m*acot(x/a),x)
--R
--R
@@ -982,7 +2362,7 @@
<<*>>=
)clear all
---S 37 of 38
+--S 136 14:507 Axiom cannot compute this integral
aa:=integrate(x^m*asec(x/a),x)
--R
--R
@@ -1012,7 +2392,7 @@
<<*>>=
)clear all
---S 38 of 38
+--S 137 14:508 Axiom cannot compute this integral
aa:=integrate(x^m*acsc(x/a),x)
--R
--R
Modified: trunk/axiom/src/input/schaum25.input.pamphlet
===================================================================
--- trunk/axiom/src/input/schaum25.input.pamphlet 2008-04-25 06:20:21 UTC (rev 863)
+++ trunk/axiom/src/input/schaum25.input.pamphlet 2008-04-26 05:30:18 UTC (rev 864)
@@ -18,16 +18,32 @@
)set message auto off
)clear all
---S 1 of 16
-aa:=integrate(e^(a*x),x)
---R
+--S 1
+aa:=integrate(%e^(a*x),x)
--R
---R a...
[truncated message content] |
|
From: <gi...@ax...> - 2008-04-26 05:29:03
|
changelog | 2 +
src/input/schaum24.input.pamphlet | 1458 ++++++++++++++++++++++++++++++++++++-
src/input/schaum25.input.pamphlet | 481 +++++++++----
3 files changed, 1779 insertions(+), 162 deletions(-)
New commits:
commit c0e73bf51bf809c83cfc6c80b00958e76457f436
Author: Tim Daly <root@localhost.localdomain>
Date: Thu Apr 24 22:21:05 2008 -0400
20080425 tpd src/input/schaum25.input show Schaums-Axiom equivalence
20080425 tpd src/input/schaum24.input show Schaums-Axiom equivalence
|
|
From: <da...@us...> - 2008-04-25 06:20:19
|
Revision: 863
http://axiom.svn.sourceforge.net/axiom/?rev=863&view=rev
Author: daly
Date: 2008-04-24 23:20:21 -0700 (Thu, 24 Apr 2008)
Log Message:
-----------
20080424 tpd src/input/schaum23.input show Schaums-Axiom equivalence
Modified Paths:
--------------
trunk/axiom/changelog
trunk/axiom/src/input/schaum23.input.pamphlet
Modified: trunk/axiom/changelog
===================================================================
--- trunk/axiom/changelog 2008-04-24 05:55:47 UTC (rev 862)
+++ trunk/axiom/changelog 2008-04-25 06:20:21 UTC (rev 863)
@@ -1,3 +1,4 @@
+20080424 tpd src/input/schaum23.input show Schaums-Axiom equivalence
20080423 tpd src/input/schaum22.input show Schaums-Axiom equivalence
20080423 tpd src/input/schaum21.input show Schaums-Axiom equivalence
20080423 tpd src/input/schaum20.input show Schaums-Axiom equivalence
Modified: trunk/axiom/src/input/schaum23.input.pamphlet
===================================================================
--- trunk/axiom/src/input/schaum23.input.pamphlet 2008-04-24 05:55:47 UTC (rev 862)
+++ trunk/axiom/src/input/schaum23.input.pamphlet 2008-04-25 06:20:21 UTC (rev 863)
@@ -19,7 +19,7 @@
)set message auto off
)clear all
---S 1 of 10
+--S 1
aa:=integrate(csc(a*x),x)
--R
--R
@@ -30,6 +30,148 @@
--R a
--R Type: Union(Expression Integer,...)
--E
+
+--S 2
+bb1:=1/a*log(csc(a*x)-cot(a*x))
+--R
+--R log(csc(a x) - cot(a x))
+--R (2) ------------------------
+--R a
+--R Type: Expression Integer
+--E
+
+--S 3
+bb2:=1/a*log(tan((a*x)/2))
+--R
+--R a x
+--R log(tan(---))
+--R 2
+--R (3) -------------
+--R a
+--R Type: Expression Integer
+--E
+
+--S 4
+cc1:=aa-bb1
+--R
+--R sin(a x)
+--R log(------------) - log(csc(a x) - cot(a x))
+--R cos(a x) + 1
+--R (4) --------------------------------------------
+--R a
+--R Type: Expression Integer
+--E
+
+--S 5
+cotrule:=rule(cot(a) == cos(a)/sin(a))
+--R
+--R cos(a)
+--R (5) cot(a) == ------
+--R sin(a)
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 6
+dd1:=cotrule cc1
+--R
+--R sin(a x) csc(a x)sin(a x) - cos(a x)
+--R log(------------) - log(---------------------------)
+--R cos(a x) + 1 sin(a x)
+--R (6) ----------------------------------------------------
+--R a
+--R Type: Expression Integer
+--E
+
+--S 7
+cscrule:=rule(csc(a) == 1/sin(a))
+--R
+--R 1
+--R (7) csc(a) == ------
+--R sin(a)
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 8
+ee1:=cscrule dd1
+--R
+--R sin(a x) - cos(a x) + 1
+--R log(------------) - log(--------------)
+--R cos(a x) + 1 sin(a x)
+--R (8) ---------------------------------------
+--R a
+--R Type: Expression Integer
+--E
+
+--S 9
+ff1:=expandLog ee1
+--R
+--R 2log(sin(a x)) - log(cos(a x) + 1) - log(cos(a x) - 1) - log(- 1)
+--R (9) -----------------------------------------------------------------
+--R a
+--R Type: Expression Integer
+--E
+
+--S 10
+gg1:=complexNormalize ff1
+--R
+--R 2log(- 1)
+--R (10) - ---------
+--R a
+--R Type: Expression Integer
+--E
+
+--S 11
+cc2:=aa-bb2
+--R
+--R a x sin(a x)
+--R - log(tan(---)) + log(------------)
+--R 2 cos(a x) + 1
+--R (11) -----------------------------------
+--R a
+--R Type: Expression Integer
+--E
+
+--S 12
+tanrule:=rule(tan(a) == sin(a)/cos(a))
+--R
+--R sin(a)
+--R (12) tan(a) == ------
+--R cos(a)
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 13
+dd2:=tanrule cc2
+--R
+--R a x
+--R sin(---)
+--R sin(a x) 2
+--R log(------------) - log(--------)
+--R cos(a x) + 1 a x
+--R cos(---)
+--R 2
+--R (13) ---------------------------------
+--R a
+--R Type: Expression Integer
+--E
+
+--S 14
+ee2:=expandLog dd2
+--R
+--R a x a x
+--R log(sin(a x)) - log(sin(---)) - log(cos(a x) + 1) + log(cos(---))
+--R 2 2
+--R (14) -----------------------------------------------------------------
+--R a
+--R Type: Expression Integer
+--E
+
+--S 15 14:461 Schaums and Axiom agree
+ff2:=complexNormalize ee2
+--R
+--R (15) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.462~~~~~$\displaystyle
@@ -40,7 +182,7 @@
<<*>>=
)clear all
---S 2 of 10
+--S 16
aa:=integrate(csc(a*x)^2,x)
--R
--R
@@ -49,6 +191,40 @@
--R a sin(a x)
--R Type: Union(Expression Integer,...)
--E
+
+--S 17
+bb:=-cot(a*x)/a
+--R
+--R cot(a x)
+--R (2) - --------
+--R a
+--R Type: Expression Integer
+--E
+
+--S 18
+cc:=aa-bb
+--R
+--R cot(a x)sin(a x) - cos(a x)
+--R (3) ---------------------------
+--R a sin(a x)
+--R Type: Expression Integer
+--E
+
+--S 19
+cotrule:=rule(cot(a) == cos(a)/sin(a))
+--R
+--R cos(a)
+--R (4) cot(a) == ------
+--R sin(a)
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 20 14:462 Schaums and Axiom agree
+dd:=cotrule cc
+--R
+--R (5) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.463~~~~~$\displaystyle
@@ -59,7 +235,7 @@
<<*>>=
)clear all
---S 3 of 10
+--S 21
aa:=integrate(csc(a*x)^3,x)
--R
--R
@@ -71,6 +247,159 @@
--R 2a cos(a x) - 2a
--R Type: Union(Expression Integer,...)
--E
+
+--S 22
+bb:=-(csc(a*x)*cot(a*x))/(2*a)+1/(2*a)*log(tan((a*x)/2))
+--R
+--R a x
+--R log(tan(---)) - cot(a x)csc(a x)
+--R 2
+--R (2) --------------------------------
+--R 2a
+--R Type: Expression Integer
+--E
+
+--S 23
+cc:=aa-bb
+--R
+--R (3)
+--R 2 a x 2 sin(a x)
+--R (- cos(a x) + 1)log(tan(---)) + (cos(a x) - 1)log(------------)
+--R 2 cos(a x) + 1
+--R +
+--R 2
+--R (cos(a x) - 1)cot(a x)csc(a x) + cos(a x)
+--R /
+--R 2
+--R 2a cos(a x) - 2a
+--R Type: Expression Integer
+--E
+
+--S 24
+cotrule:=rule(cot(a) == cos(a)/sin(a))
+--R
+--R cos(a)
+--R (4) cot(a) == ------
+--R sin(a)
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 25
+dd:=cotrule cc
+--R
+--R (5)
+--R 2 a x
+--R (- cos(a x) + 1)sin(a x)log(tan(---))
+--R 2
+--R +
+--R 2 sin(a x)
+--R (cos(a x) - 1)sin(a x)log(------------) + cos(a x)sin(a x)
+--R cos(a x) + 1
+--R +
+--R 3
+--R (cos(a x) - cos(a x))csc(a x)
+--R /
+--R 2
+--R (2a cos(a x) - 2a)sin(a x)
+--R Type: Expression Integer
+--E
+
+--S 26
+tanrule:=rule(tan(a) == sin(a)/cos(a))
+--R
+--R sin(a)
+--R (6) tan(a) == ------
+--R cos(a)
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 27
+ee:=tanrule dd
+--R
+--R (7)
+--R 2 sin(a x)
+--R (cos(a x) - 1)sin(a x)log(------------)
+--R cos(a x) + 1
+--R +
+--R a x
+--R sin(---)
+--R 2 2
+--R (- cos(a x) + 1)sin(a x)log(--------) + cos(a x)sin(a x)
+--R a x
+--R cos(---)
+--R 2
+--R +
+--R 3
+--R (cos(a x) - cos(a x))csc(a x)
+--R /
+--R 2
+--R (2a cos(a x) - 2a)sin(a x)
+--R Type: Expression Integer
+--E
+
+--S 28
+cscrule:=rule(csc(a) == 1/sin(a))
+--R
+--R 1
+--R (8) csc(a) == ------
+--R sin(a)
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 29
+ff:=cscrule ee
+--R
+--R (9)
+--R 2 2 sin(a x)
+--R (cos(a x) - 1)sin(a x) log(------------)
+--R cos(a x) + 1
+--R +
+--R a x
+--R sin(---)
+--R 2 2 2 2 3
+--R (- cos(a x) + 1)sin(a x) log(--------) + cos(a x)sin(a x) + cos(a x)
+--R a x
+--R cos(---)
+--R 2
+--R +
+--R - cos(a x)
+--R /
+--R 2 2
+--R (2a cos(a x) - 2a)sin(a x)
+--R Type: Expression Integer
+--E
+
+--S 30
+gg:=expandLog ff
+--R
+--R (10)
+--R 2 2
+--R (cos(a x) - 1)sin(a x) log(sin(a x))
+--R +
+--R 2 2 a x
+--R (- cos(a x) + 1)sin(a x) log(sin(---))
+--R 2
+--R +
+--R 2 2
+--R (- cos(a x) + 1)sin(a x) log(cos(a x) + 1)
+--R +
+--R 2 2 a x 2 3
+--R (cos(a x) - 1)sin(a x) log(cos(---)) + cos(a x)sin(a x) + cos(a x)
+--R 2
+--R +
+--R - cos(a x)
+--R /
+--R 2 2
+--R (2a cos(a x) - 2a)sin(a x)
+--R Type: Expression Integer
+--E
+
+--S 31 14:463 Schaums and Axiom agree
+hh:=complexNormalize gg
+--R
+--R (11) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.464~~~~~$\displaystyle
@@ -81,7 +410,7 @@
<<*>>=
)clear all
---S 4 of 10
+--S 32
aa:=integrate(csc(a*x)^n*cot(a*x),x)
--R
--R
@@ -96,6 +425,31 @@
--R a n
--R Type: Union(Expression Integer,...)
--E
+
+--S 33
+bb:=-csc(a*x)^n/(n*a)
+--R
+--R n
+--R csc(a x)
+--R (2) - ---------
+--R a n
+--R Type: Expression Integer
+--E
+
+--S 34 14:464 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R 1
+--R n log(- -------------)
+--R 2
+--R cos(a x) - 1
+--R ----------------------
+--R 2 n
+--R - %e + csc(a x)
+--R (3) --------------------------------------
+--R a n
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.465~~~~~$\displaystyle
@@ -106,7 +460,7 @@
<<*>>=
)clear all
---S 5 of 10
+--S 35
aa:=integrate(1/csc(a*x),x)
--R
--R
@@ -114,7 +468,23 @@
--R (1) - --------
--R a
--R Type: Union(Expression Integer,...)
+--E
+
+--S 36
+bb:=-cos(a*x)/a
+--R
+--R cos(a x)
+--R (2) - --------
+--R a
+--R Type: Expression Integer
--E
+
+--S 37 14:465 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.466~~~~~$\displaystyle
@@ -126,7 +496,7 @@
<<*>>=
)clear all
---S 6 of 10
+--S 38 14:466 Axiom cannot compute this integral
aa:=integrate(x*csc(a*x),x)
--R
--R
@@ -147,7 +517,7 @@
<<*>>=
)clear all
---S 7 of 10
+--S 39 14:467 Axiom cannot compute this integral
aa:=integrate(csc(a*x)/x,x)
--R
--R
@@ -167,7 +537,7 @@
<<*>>=
)clear all
---S 8 of 10
+--S 40
aa:=integrate(x*csc(a*x)^2,x)
--R
--R
@@ -179,6 +549,64 @@
--R a sin(a x)
--R Type: Union(Expression Integer,...)
--E
+
+--S 41
+bb:=-(x*cot(a*x))/a+1/a^2*log(sin(a*x))
+--R
+--R log(sin(a x)) - a x cot(a x)
+--R (2) ----------------------------
+--R 2
+--R a
+--R Type: Expression Integer
+--E
+
+--S 42
+cc:=aa-bb
+--R
+--R (3)
+--R sin(a x)
+--R - sin(a x)log(sin(a x)) + sin(a x)log(------------)
+--R cos(a x) + 1
+--R +
+--R 2
+--R - sin(a x)log(------------) + a x cot(a x)sin(a x) - a x cos(a x)
+--R cos(a x) + 1
+--R /
+--R 2
+--R a sin(a x)
+--R Type: Expression Integer
+--E
+
+--S 43
+cotrule:=rule(cot(a) == cos(a)/sin(a))
+--R
+--R cos(a)
+--R (4) cot(a) == ------
+--R sin(a)
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 44
+dd:=cotrule cc
+--R
+--R sin(a x) 2
+--R - log(sin(a x)) + log(------------) - log(------------)
+--R cos(a x) + 1 cos(a x) + 1
+--R (5) -------------------------------------------------------
+--R 2
+--R a
+--R Type: Expression Integer
+--E
+
+--S 45 14:468 Schaums and Axiom differ by a constant
+ee:=expandLog dd
+--R
+--R log(2)
+--R (6) - ------
+--R 2
+--R a
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.469~~~~~$\displaystyle
@@ -189,7 +617,7 @@
<<*>>=
)clear all
---S 9 of 10
+--S 46
aa:=integrate(1/(q+p*csc(a*x)),x)
--R
--R
@@ -227,6 +655,156 @@
--R a q\|- q + p
--R Type: Union(List Expression Integer,...)
--E
+
+--S 47
+t1:=integrate(1/(p+q*sin(a*x)),x)
+--E
+
+--S 48
+bb1:=x/q-p/q*t1.1
+--R
+--R (3)
+--R -
+--R p
+--R *
+--R log
+--R +-------+
+--R 2 2 2 | 2 2
+--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p
+--R +
+--R 2 3 3 2 3 2
+--R (- p q + p )sin(a x) + (- q + p q)cos(a x) - q + p q
+--R /
+--R q sin(a x) + p
+--R +
+--R +-------+
+--R | 2 2
+--R a x\|q - p
+--R /
+--R +-------+
+--R | 2 2
+--R a q\|q - p
+--R Type: Expression Integer
+--E
+
+--S 49
+bb2:=x/q-p/q*t1.2
+--R
+--R +---------+
+--R | 2 2 +---------+
+--R (p sin(a x) + q cos(a x) + q)\|- q + p | 2 2
+--R 2p atan(-----------------------------------------) + a x\|- q + p
+--R 2 2 2 2
+--R (q - p )cos(a x) + q - p
+--R (4) --------------------------------------------------------------------
+--R +---------+
+--R | 2 2
+--R a q\|- q + p
+--R Type: Expression Integer
+--E
+
+--S 50
+cc1:=aa.1-bb1
+--R
+--R (5)
+--R p
+--R *
+--R log
+--R +-------+
+--R 2 2 2 | 2 2
+--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p
+--R +
+--R 2 3 3 2 3 2
+--R (p q - p )sin(a x) + (q - p q)cos(a x) + q - p q
+--R /
+--R q sin(a x) + p
+--R +
+--R p
+--R *
+--R log
+--R +-------+
+--R 2 2 2 | 2 2
+--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p
+--R +
+--R 2 3 3 2 3 2
+--R (- p q + p )sin(a x) + (- q + p q)cos(a x) - q + p q
+--R /
+--R q sin(a x) + p
+--R /
+--R +-------+
+--R | 2 2
+--R a q\|q - p
+--R Type: Expression Integer
+--E
+
+--S 51
+cc2:=aa.2-bb1
+--R
+--R (6)
+--R +---------+
+--R | 2 2
+--R p\|- q + p
+--R *
+--R log
+--R +-------+
+--R 2 2 2 | 2 2
+--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p
+--R +
+--R 2 3 3 2 3 2
+--R (- p q + p )sin(a x) + (- q + p q)cos(a x) - q + p q
+--R /
+--R q sin(a x) + p
+--R +
+--R +---------+
+--R +-------+ | 2 2
+--R | 2 2 (p sin(a x) + q cos(a x) + q)\|- q + p
+--R 2p\|q - p atan(-----------------------------------------)
+--R 2 2 2 2
+--R (q - p )cos(a x) + q - p
+--R /
+--R +---------+ +-------+
+--R | 2 2 | 2 2
+--R a q\|- q + p \|q - p
+--R Type: Expression Integer
+--E
+
+--S 52
+cc3:=aa.1-bb2
+--R
+--R (7)
+--R +---------+
+--R | 2 2
+--R p\|- q + p
+--R *
+--R log
+--R +-------+
+--R 2 2 2 | 2 2
+--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p
+--R +
+--R 2 3 3 2 3 2
+--R (p q - p )sin(a x) + (q - p q)cos(a x) + q - p q
+--R /
+--R q sin(a x) + p
+--R +
+--R +---------+
+--R +-------+ | 2 2
+--R | 2 2 (p sin(a x) + q cos(a x) + q)\|- q + p
+--R - 2p\|q - p atan(-----------------------------------------)
+--R 2 2 2 2
+--R (q - p )cos(a x) + q - p
+--R /
+--R +---------+ +-------+
+--R | 2 2 | 2 2
+--R a q\|- q + p \|q - p
+--R Type: Expression Integer
+--E
+
+--S 53 14:469 Schaums and Axiom agree
+cc4:=aa.2-bb2
+--R
+--R (8) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.470~~~~~$\displaystyle
@@ -238,7 +816,7 @@
<<*>>=
)clear all
---S 10 of 10
+--S 54 14:470 Axiom cannot compute this integral
aa:=integrate(csc(a*x)^n,x)
--R
--R
This was sent by the SourceForge.net collaborative development platform, the world's largest Open Source development site.
|
|
From: <gi...@ax...> - 2008-04-25 06:18:29
|
changelog | 1 +
src/input/schaum23.input.pamphlet | 598 ++++++++++++++++++++++++++++++++++++-
2 files changed, 589 insertions(+), 10 deletions(-)
New commits:
commit e6cf073b0b9be112e0e43633d8201798d6a2c043
Author: Tim Daly <root@localhost.localdomain>
Date: Thu Apr 24 09:03:00 2008 -0400
20080424 tpd src/input/schaum23.input show Schaums-Axiom equivalence
|
|
From: <da...@us...> - 2008-04-24 05:55:40
|
Revision: 862
http://axiom.svn.sourceforge.net/axiom/?rev=862&view=rev
Author: daly
Date: 2008-04-23 22:55:47 -0700 (Wed, 23 Apr 2008)
Log Message:
-----------
20080423 tpd src/input/schaum22.input show Schaums-Axiom equivalence
20080423 tpd src/input/schaum21.input show Schaums-Axiom equivalence
20080423 tpd src/input/schaum20.input show Schaums-Axiom equivalence
20080423 tpd src/input/schaum19.input show Schaums-Axiom equivalence
20080423 tpd src/input/schaum18.input show Schaums-Axiom equivalence
20080423 tpd src/input/schaum17.input show Schaums-Axiom equivalence
Modified Paths:
--------------
trunk/axiom/changelog
trunk/axiom/src/input/schaum17.input.pamphlet
trunk/axiom/src/input/schaum18.input.pamphlet
trunk/axiom/src/input/schaum19.input.pamphlet
trunk/axiom/src/input/schaum20.input.pamphlet
trunk/axiom/src/input/schaum21.input.pamphlet
trunk/axiom/src/input/schaum22.input.pamphlet
Modified: trunk/axiom/changelog
===================================================================
--- trunk/axiom/changelog 2008-04-22 05:50:59 UTC (rev 861)
+++ trunk/axiom/changelog 2008-04-24 05:55:47 UTC (rev 862)
@@ -1,3 +1,9 @@
+20080423 tpd src/input/schaum22.input show Schaums-Axiom equivalence
+20080423 tpd src/input/schaum21.input show Schaums-Axiom equivalence
+20080423 tpd src/input/schaum20.input show Schaums-Axiom equivalence
+20080423 tpd src/input/schaum19.input show Schaums-Axiom equivalence
+20080423 tpd src/input/schaum18.input show Schaums-Axiom equivalence
+20080423 tpd src/input/schaum17.input show Schaums-Axiom equivalence
20080421 tpd src/input/schaum16.input show Schaums-Axiom equivalence
20080421 tpd src/input/schaum15.input show Schaums-Axiom equivalence
20080421 tpd src/input/schaum14.input show Schaums-Axiom equivalence
Modified: trunk/axiom/src/input/schaum17.input.pamphlet
===================================================================
--- trunk/axiom/src/input/schaum17.input.pamphlet 2008-04-22 05:50:59 UTC (rev 861)
+++ trunk/axiom/src/input/schaum17.input.pamphlet 2008-04-24 05:55:47 UTC (rev 862)
@@ -18,7 +18,7 @@
)set message auto off
)clear all
---S 1 of 30
+--S 1
aa:=integrate(sin(a*x),x)
--R
--R
@@ -27,6 +27,22 @@
--R a
--R Type: Union(Expression Integer,...)
--E
+
+--S 2
+bb:=-cos(a*x)/a
+--R
+--R cos(a x)
+--R (2) - --------
+--R a
+--R Type: Expression Integer
+--E
+
+--S 3 14:339 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.340~~~~~$\displaystyle
@@ -37,7 +53,7 @@
<<*>>=
)clear all
---S 2 of 30
+--S 4
aa:=integrate(x*sin(a*x),x)
--R
--R
@@ -47,6 +63,23 @@
--R a
--R Type: Union(Expression Integer,...)
--E
+
+--S 5
+bb:=sin(a*x)/a^2-(x*cos(a*x))/a
+--R
+--R sin(a x) - a x cos(a x)
+--R (2) -----------------------
+--R 2
+--R a
+--R Type: Expression Integer
+--E
+
+--S 6 14:340 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.341~~~~~$\displaystyle
@@ -57,7 +90,7 @@
<<*>>=
)clear all
---S 3 of 30
+--S 7
aa:=integrate(x^2*sin(a*x),x)
--R
--R
@@ -68,6 +101,24 @@
--R a
--R Type: Union(Expression Integer,...)
--E
+
+--S 8
+bb:=(2*x)/a^2*sin(a*x)+(2/a^3-x^2/a)*cos(a*x)
+--R
+--R 2 2
+--R 2a x sin(a x) + (- a x + 2)cos(a x)
+--R (2) ------------------------------------
+--R 3
+--R a
+--R Type: Expression Integer
+--E
+
+--S 9 14:341 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.342~~~~~$\displaystyle
@@ -79,7 +130,7 @@
<<*>>=
)clear all
---S 4 of 30
+--S 10
aa:=integrate(x^3*sin(a*x),x)
--R
--R
@@ -90,6 +141,24 @@
--R a
--R Type: Union(Expression Integer,...)
--E
+
+--S 11
+bb:=((3*x^2)/a^2-6/a^4)*sin(a*x)+(6*x/a^3-x^3/a)*cos(a*x)
+--R
+--R 2 2 3 3
+--R (3a x - 6)sin(a x) + (- a x + 6a x)cos(a x)
+--R (2) ---------------------------------------------
+--R 4
+--R a
+--R Type: Expression Integer
+--E
+
+--S 12 14:342 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.343~~~~~$\displaystyle
@@ -100,7 +169,7 @@
<<*>>=
)clear all
---S 5 of 30
+--S 13 14:343 Schaums and Axiom agree by definition
aa:=integrate(sin(x)/x,x)
--R
--R
@@ -117,7 +186,7 @@
<<*>>=
)clear all
---S 6 of 30
+--S 14 14:344 Axiom cannot compute this integral
aa:=integrate(sin(a*x)/x^2,x)
--R
--R
@@ -139,7 +208,7 @@
<<*>>=
)clear all
---S 7 of 30
+--S 15
aa:=integrate(1/sin(a*x),x)
--R
--R
@@ -150,6 +219,35 @@
--R a
--R Type: Union(Expression Integer,...)
--E
+
+--S 16
+bb:=1/a*log(tan((a*x)/2))
+--R
+--R a x
+--R log(tan(---))
+--R 2
+--R (2) -------------
+--R a
+--R Type: Expression Integer
+--E
+
+--S 17
+cc:=aa-bb
+--R
+--R a x sin(a x)
+--R - log(tan(---)) + log(------------)
+--R 2 cos(a x) + 1
+--R (3) -----------------------------------
+--R a
+--R Type: Expression Integer
+--E
+
+--S 18 14:345 Schaums and Axiom agree
+dd:=complexNormalize cc
+--R
+--R (4) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.346~~~~~$\displaystyle
@@ -162,7 +260,7 @@
<<*>>=
)clear all
---S 8 of 30
+--S 19 14:346 Axiom cannot compute this integral
aa:=integrate(x/sin(a*x),x)
--R
--R
@@ -182,7 +280,7 @@
<<*>>=
)clear all
---S 9 of 30
+--S 20
aa:=integrate(sin(a*x)^2,x)
--R
--R
@@ -191,6 +289,31 @@
--R 2a
--R Type: Union(Expression Integer,...)
--E
+
+--S 21
+bb:=x/2-sin(2*a*x)/(4*a)
+--R
+--R - sin(2a x) + 2a x
+--R (2) ------------------
+--R 4a
+--R Type: Expression Integer
+--E
+
+--S 22
+cc:=aa-bb
+--R
+--R sin(2a x) - 2cos(a x)sin(a x)
+--R (3) -----------------------------
+--R 4a
+--R Type: Expression Integer
+--E
+
+--S 23 14:347 Schaums and Axiom agreee
+dd:=complexNormalize cc
+--R
+--R (4) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.348~~~~~$\displaystyle
@@ -201,7 +324,7 @@
<<*>>=
)clear all
---S 10 of 30
+--S 24
aa:=integrate(x*sin(a*x)^2,x)
--R
--R
@@ -212,6 +335,38 @@
--R 4a
--R Type: Union(Expression Integer,...)
--E
+
+--S 25
+bb:=x^2/4-(x*sin(2*a*x))/(4*a)-cos(2*a*x)/(8*a^2)
+--R
+--R 2 2
+--R - 2a x sin(2a x) - cos(2a x) + 2a x
+--R (2) ------------------------------------
+--R 2
+--R 8a
+--R Type: Expression Integer
+--E
+
+--S 26
+cc:=aa-bb
+--R
+--R 2
+--R 2a x sin(2a x) - 4a x cos(a x)sin(a x) + cos(2a x) - 2cos(a x)
+--R (3) ---------------------------------------------------------------
+--R 2
+--R 8a
+--R Type: Expression Integer
+--E
+
+--S 27 14:348 Schaums and Axiom differ by a constant
+dd:=complexNormalize cc
+--R
+--R 1
+--R (4) - ---
+--R 2
+--R 8a
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.349~~~~~$\displaystyle
@@ -222,7 +377,7 @@
<<*>>=
)clear all
---S 11 of 30
+--S 28
aa:=integrate(sin(a*x)^3,x)
--R
--R
@@ -232,6 +387,23 @@
--R 3a
--R Type: Union(Expression Integer,...)
--E
+
+--S 29
+bb:=-cos(a*x)/a+cos(a*x)^3/(3*a)
+--R
+--R 3
+--R cos(a x) - 3cos(a x)
+--R (2) ---------------------
+--R 3a
+--R Type: Expression Integer
+--E
+
+--S 30 14:349 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.350~~~~~$\displaystyle
@@ -242,7 +414,7 @@
<<*>>=
)clear all
---S 12 of 30
+--S 31
aa:=integrate(sin(a*x)^4,x)
--R
--R
@@ -252,6 +424,32 @@
--R 8a
--R Type: Union(Expression Integer,...)
--E
+
+--S 32
+bb:=(3*x)/8-sin(2*a*x)/(4*a)+sin(4*a*x)/(32*a)
+--R
+--R sin(4a x) - 8sin(2a x) + 12a x
+--R (2) ------------------------------
+--R 32a
+--R Type: Expression Integer
+--E
+
+--S 33
+cc:=aa-bb
+--R
+--R 3
+--R - sin(4a x) + 8sin(2a x) + (8cos(a x) - 20cos(a x))sin(a x)
+--R (3) ------------------------------------------------------------
+--R 32a
+--R Type: Expression Integer
+--E
+
+--S 34 14:350 Schaums and Axiom agree
+dd:=complexNormalize cc
+--R
+--R (4) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.351~~~~~$\displaystyle
@@ -262,7 +460,7 @@
<<*>>=
)clear all
---S 13 of 30
+--S 35
aa:=integrate(1/sin(a*x)^2,x)
--R
--R
@@ -271,6 +469,31 @@
--R a sin(a x)
--R Type: Union(Expression Integer,...)
--E
+
+--S 36
+bb:=-1/a*cot(a*x)
+--R
+--R cot(a x)
+--R (2) - --------
+--R a
+--R Type: Expression Integer
+--E
+
+--S 37
+cc:=aa-bb
+--R
+--R cot(a x)sin(a x) - cos(a x)
+--R (3) ---------------------------
+--R a sin(a x)
+--R Type: Expression Integer
+--E
+
+--S 38 14:351 Schaums and Axiom agree
+dd:=complexNormalize cc
+--R
+--R (4) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.352~~~~~$\displaystyle
@@ -281,7 +504,7 @@
<<*>>=
)clear all
---S 14 of 30
+--S 39
aa:=integrate(1/sin(a*x)^3,x)
--R
--R
@@ -293,6 +516,66 @@
--R 2a cos(a x) - 2a
--R Type: Union(Expression Integer,...)
--E
+
+--S 40
+bb:=-cos(a*x)/(2*a*sin(a*x)^2)+1/(2*a)*log(tan((a*x)/2))
+--R
+--R 2 a x
+--R sin(a x) log(tan(---)) - cos(a x)
+--R 2
+--R (2) ---------------------------------
+--R 2
+--R 2a sin(a x)
+--R Type: Expression Integer
+--E
+
+--S 41
+cc:=aa-bb
+--R
+--R (3)
+--R 2 2 a x
+--R (- cos(a x) + 1)sin(a x) log(tan(---))
+--R 2
+--R +
+--R 2 2 sin(a x) 2 3
+--R (cos(a x) - 1)sin(a x) log(------------) + cos(a x)sin(a x) + cos(a x)
+--R cos(a x) + 1
+--R +
+--R - cos(a x)
+--R /
+--R 2 2
+--R (2a cos(a x) - 2a)sin(a x)
+--R Type: Expression Integer
+--E
+
+--S 42
+dd:=expandLog cc
+--R
+--R (4)
+--R 2 2 a x
+--R (- cos(a x) + 1)sin(a x) log(tan(---))
+--R 2
+--R +
+--R 2 2
+--R (cos(a x) - 1)sin(a x) log(sin(a x))
+--R +
+--R 2 2 2
+--R (- cos(a x) + 1)sin(a x) log(cos(a x) + 1) + cos(a x)sin(a x)
+--R +
+--R 3
+--R cos(a x) - cos(a x)
+--R /
+--R 2 2
+--R (2a cos(a x) - 2a)sin(a x)
+--R Type: Expression Integer
+--E
+
+--S 43 14:352 Schaums and Axiom agree
+ee:=complexNormalize dd
+--R
+--R (5) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.353~~~~~$\displaystyle
@@ -303,7 +586,7 @@
<<*>>=
)clear all
---S 15 of 30
+--S 44
aa:=integrate(sin(p*x)*sin(q*x),x)
--R
--R
@@ -312,7 +595,30 @@
--R 2 2
--R q - p
--R Type: Union(Expression Integer,...)
+--E
+
+--S 45
+bb:=(sin(p-q)*x)/(2*(p-q))-(sin(p+q)*x)/(2*(p+q))
+--R
+--R (- q + p)x sin(q + p) + (q + p)x sin(q - p)
+--R (2) -------------------------------------------
+--R 2 2
+--R 2q - 2p
+--R Type: Expression Integer
--E
+
+--S 46 14:353 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R (3)
+--R 2p cos(p x)sin(q x) - 2q cos(q x)sin(p x) + (q - p)x sin(q + p)
+--R +
+--R (- q - p)x sin(q - p)
+--R /
+--R 2 2
+--R 2q - 2p
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.354~~~~~$\displaystyle
@@ -323,7 +629,7 @@
<<*>>=
)clear all
---S 16 of 30
+--S 47
aa:=integrate(1/(1-sin(a*x)),x)
--R
--R
@@ -331,7 +637,38 @@
--R (1) ---------------------------
--R a sin(a x) - a cos(a x) - a
--R Type: Union(Expression Integer,...)
+--E
+
+--S 48
+bb:=1/a*tan(%pi/4+(a*x)/2)
+--R
+--R 2a x + %pi
+--R tan(----------)
+--R 4
+--R (2) ---------------
+--R a
+--R Type: Expression Integer
--E
+
+--S 49
+cc:=aa-bb
+--R
+--R 2a x + %pi
+--R (- sin(a x) + cos(a x) + 1)tan(----------) - 2cos(a x) - 2
+--R 4
+--R (3) ----------------------------------------------------------
+--R a sin(a x) - a cos(a x) - a
+--R Type: Expression Integer
+--E
+
+--S 50 14:354 Schaums and Axiom differ by a constant
+dd:=complexNormalize cc
+--R
+--R 1
+--R (4) -
+--R a
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.355~~~~~$\displaystyle
@@ -343,7 +680,7 @@
<<*>>=
)clear all
---S 17 of 30
+--S 51
aa:=integrate(x/(1-sin(ax)),x)
--R
--R
@@ -353,6 +690,35 @@
--R 2sin(ax) - 2
--R Type: Union(Expression Integer,...)
--E
+
+--S 52
+bb:=x/a*tan(%pi/4+(a*x)/2)+2/a^2*log(sin(%pi/4-(a*x)/2))
+--R
+--R 2a x - %pi 2a x + %pi
+--R 2log(- sin(----------)) + a x tan(----------)
+--R 4 4
+--R (2) ---------------------------------------------
+--R 2
+--R a
+--R Type: Expression Integer
+--E
+
+--S 53 14:355 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R (3)
+--R 2a x - %pi
+--R (- 4sin(ax) + 4)log(- sin(----------))
+--R 4
+--R +
+--R 2a x + %pi 2 2
+--R (- 2a x sin(ax) + 2a x)tan(----------) - a x
+--R 4
+--R /
+--R 2 2
+--R 2a sin(ax) - 2a
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.356~~~~~$\displaystyle
@@ -363,7 +729,7 @@
<<*>>=
)clear all
---S 18 of 30
+--S 54
aa:=integrate(1/(1+sin(ax)),x)
--R
--R
@@ -372,6 +738,46 @@
--R sin(ax) + 1
--R Type: Union(Expression Integer,...)
--E
+
+--S 55
+bb:=-1/a*tan(%pi/4-(a*x)/2)
+--R
+--R 2a x - %pi
+--R tan(----------)
+--R 4
+--R (2) ---------------
+--R a
+--R Type: Expression Integer
+--E
+
+--S 56
+cc:=aa-bb
+--R
+--R 2a x - %pi
+--R (- sin(ax) - 1)tan(----------) + a x
+--R 4
+--R (3) ------------------------------------
+--R a sin(ax) + a
+--R Type: Expression Integer
+--E
+
+--S 57
+tanrule:=rule(tan(a/b) == sin(a)/cos(b))
+--R
+--R a sin(a)
+--R (4) tan(-) == ------
+--R b cos(b)
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 58 14:356 Axiom cannot simplify this expression
+dd:=tanrule cc
+--R
+--R (- sin(ax) - 1)sin(2a x - %pi) + a x cos(4)
+--R (5) -------------------------------------------
+--R a cos(4)sin(ax) + a cos(4)
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.357~~~~~$\displaystyle
@@ -383,7 +789,7 @@
<<*>>=
)clear all
---S 19 of 30
+--S 59
aa:=integrate(x/(1+sin(a*x)),x)
--R
--R
@@ -402,6 +808,100 @@
--R a sin(a x) + a cos(a x) + a
--R Type: Union(Expression Integer,...)
--E
+
+--S 60
+bb:=-x/a*tan(%pi/4-(a*x)/2)+2/a^2*log(sin(%pi/4+(a*x)/2))
+--R
+--R 2a x + %pi 2a x - %pi
+--R 2log(sin(----------)) + a x tan(----------)
+--R 4 4
+--R (2) -------------------------------------------
+--R 2
+--R a
+--R Type: Expression Integer
+--E
+
+--S 61
+cc:=aa-bb
+--R
+--R (3)
+--R sin(a x) + cos(a x) + 1
+--R (2sin(a x) + 2cos(a x) + 2)log(-----------------------)
+--R cos(a x) + 1
+--R +
+--R 2a x + %pi
+--R (- 2sin(a x) - 2cos(a x) - 2)log(sin(----------))
+--R 4
+--R +
+--R 2
+--R (- sin(a x) - cos(a x) - 1)log(------------)
+--R cos(a x) + 1
+--R +
+--R 2a x - %pi
+--R (- a x sin(a x) - a x cos(a x) - a x)tan(----------) + a x sin(a x)
+--R 4
+--R +
+--R - a x cos(a x) - a x
+--R /
+--R 2 2 2
+--R a sin(a x) + a cos(a x) + a
+--R Type: Expression Integer
+--E
+
+--S 62
+dd:=expandLog cc
+--R
+--R (4)
+--R (2sin(a x) + 2cos(a x) + 2)log(sin(a x) + cos(a x) + 1)
+--R +
+--R 2a x + %pi
+--R (- 2sin(a x) - 2cos(a x) - 2)log(sin(----------))
+--R 4
+--R +
+--R (- sin(a x) - cos(a x) - 1)log(cos(a x) + 1)
+--R +
+--R 2a x - %pi
+--R (- a x sin(a x) - a x cos(a x) - a x)tan(----------)
+--R 4
+--R +
+--R (- log(2) + a x)sin(a x) + (- log(2) - a x)cos(a x) - log(2) - a x
+--R /
+--R 2 2 2
+--R a sin(a x) + a cos(a x) + a
+--R Type: Expression Integer
+--E
+
+--S 63
+tanrule:=rule(tan(a/b) == sin(a)/cos(b))
+--R
+--R a sin(a)
+--R (5) tan(-) == ------
+--R b cos(b)
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 64 14:357 Axiom cannot simplify this expression
+ee:=tanrule dd
+--R
+--R (6)
+--R (2cos(4)sin(a x) + 2cos(4)cos(a x) + 2cos(4))log(sin(a x) + cos(a x) + 1)
+--R +
+--R 2a x + %pi
+--R (- 2cos(4)sin(a x) - 2cos(4)cos(a x) - 2cos(4))log(sin(----------))
+--R 4
+--R +
+--R (- cos(4)sin(a x) - cos(4)cos(a x) - cos(4))log(cos(a x) + 1)
+--R +
+--R (- a x sin(a x) - a x cos(a x) - a x)sin(2a x - %pi)
+--R +
+--R (- cos(4)log(2) + a x cos(4))sin(a x)
+--R +
+--R (- cos(4)log(2) - a x cos(4))cos(a x) - cos(4)log(2) - a x cos(4)
+--R /
+--R 2 2 2
+--R a cos(4)sin(a x) + a cos(4)cos(a x) + a cos(4)
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.358~~~~~$\displaystyle
@@ -413,7 +913,7 @@
<<*>>=
)clear all
---S 20 of 30
+--S 65
aa:=integrate(1/(1-sin(a*x))^2,x)
--R
--R
@@ -423,7 +923,76 @@
--R 2
--R (3a cos(a x) + 6a)sin(a x) + 3a cos(a x) - 3a cos(a x) - 6a
--R Type: Union(Expression Integer,...)
+--E
+
+--S 66
+bb:=-1/(2*a)*tan(%pi/4+(a*x)/2)+1/(6*a)*tan(%pi/4+(a*x)/2)^3
+--R
+--R 2a x + %pi 3 2a x + %pi
+--R tan(----------) - 3tan(----------)
+--R 4 4
+--R (2) -----------------------------------
+--R 6a
+--R Type: Expression Integer
--E
+
+--S 67
+cc:=aa-bb
+--R
+--R (3)
+--R 2 2a x + %pi 3
+--R ((- cos(a x) - 2)sin(a x) - cos(a x) + cos(a x) + 2)tan(----------)
+--R 4
+--R +
+--R 2 2a x + %pi
+--R ((3cos(a x) + 6)sin(a x) + 3cos(a x) - 3cos(a x) - 6)tan(----------)
+--R 4
+--R +
+--R 2
+--R (6cos(a x) + 6)sin(a x) + 2cos(a x) - 8cos(a x) - 10
+--R /
+--R 2
+--R (6a cos(a x) + 12a)sin(a x) + 6a cos(a x) - 6a cos(a x) - 12a
+--R Type: Expression Integer
+--E
+
+--S 68
+tanrule:=rule(tan(a/b) == sin(a)/cos(b))
+--R
+--R a sin(a)
+--R (4) tan(-) == ------
+--R b cos(b)
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 69 14:358 Axiom cannot simplify this expression
+dd:=tanrule cc
+--R
+--R (5)
+--R 2 3
+--R ((- cos(a x) - 2)sin(a x) - cos(a x) + cos(a x) + 2)sin(2a x + %pi)
+--R +
+--R 2 2 2 2
+--R (3cos(4) cos(a x) + 6cos(4) )sin(a x) + 3cos(4) cos(a x)
+--R +
+--R 2 2
+--R - 3cos(4) cos(a x) - 6cos(4)
+--R *
+--R sin(2a x + %pi)
+--R +
+--R 3 3 3 2
+--R (6cos(4) cos(a x) + 6cos(4) )sin(a x) + 2cos(4) cos(a x)
+--R +
+--R 3 3
+--R - 8cos(4) cos(a x) - 10cos(4)
+--R /
+--R 3 3 3 2
+--R (6a cos(4) cos(a x) + 12a cos(4) )sin(a x) + 6a cos(4) cos(a x)
+--R +
+--R 3 3
+--R - 6a cos(4) cos(a x) - 12a cos(4)
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.359~~~~~$\displaystyle
@@ -435,7 +1004,7 @@
<<*>>=
)clear all
---S 21 of 30
+--S 70
aa:=integrate(1/(1+sin(a*x))^2,x)
--R
--R
@@ -445,7 +1014,38 @@
--R 2
--R (3a cos(a x) + 6a)sin(a x) - 3a cos(a x) + 3a cos(a x) + 6a
--R Type: Union(Expression Integer,...)
+--E
+
+--S 71
+bb:=-1/(2*a)*tan(%pi/4-(a*x)/2)-1/(6*a)*tan(%pi/4-(a*x)/2)^3
+--R
+--R 2a x - %pi 3 2a x - %pi
+--R tan(----------) + 3tan(----------)
+--R 4 4
+--R (2) -----------------------------------
+--R 6a
+--R Type: Expression Integer
--E
+
+--S 72 14:359 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R (3)
+--R 2 2a x - %pi 3
+--R ((- cos(a x) - 2)sin(a x) + cos(a x) - cos(a x) - 2)tan(----------)
+--R 4
+--R +
+--R 2 2a x - %pi
+--R ((- 3cos(a x) - 6)sin(a x) + 3cos(a x) - 3cos(a x) - 6)tan(----------)
+--R 4
+--R +
+--R 2
+--R (- 6cos(a x) - 6)sin(a x) + 2cos(a x) - 8cos(a x) - 10
+--R /
+--R 2
+--R (6a cos(a x) + 12a)sin(a x) - 6a cos(a x) + 6a cos(a x) + 12a
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.360~~~~~$\displaystyle
@@ -466,7 +1066,7 @@
<<*>>=
)clear all
---S 22 of 30
+--S 73
aa:=integrate(1/(p+q*sin(a*x)),x)
--R
--R
@@ -498,6 +1098,156 @@
--R a\|- q + p
--R Type: Union(List Expression Integer,...)
--E
+
+--S 74
+bb1:=2/(a*sqrt(p^2-q^2))*atan((p*tan((a*x)/2+q))/sqrt(p^2-q^2))
+--R
+--R a x + 2q
+--R p tan(--------)
+--R 2
+--R 2atan(---------------)
+--R +---------+
+--R | 2 2
+--R \|- q + p
+--R (2) ----------------------
+--R +---------+
+--R | 2 2
+--R a\|- q + p
+--R Type: Expression Integer
+--E
+
+--S 75
+bb2:=1/(a*sqrt(q^2-p^2))*log((p*tan((a*x)/2+q-sqrt(q^2-p^2)))/(p*tan((a*x)/2+q+sqrt(q^2-p^2))))
+--R
+--R +-------+
+--R | 2 2
+--R 2\|q - p - a x - 2q
+--R tan(----------------------)
+--R 2
+--R log(- ---------------------------)
+--R +-------+
+--R | 2 2
+--R 2\|q - p + a x + 2q
+--R tan(----------------------)
+--R 2
+--R (3) ----------------------------------
+--R +-------+
+--R | 2 2
+--R a\|q - p
+--R Type: Expression Integer
+--E
+
+--S 76
+cc1:=aa.1-bb1
+--R
+--R (4)
+--R +---------+
+--R | 2 2
+--R \|- q + p
+--R *
+--R log
+--R +-------+
+--R 2 2 2 | 2 2
+--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p
+--R +
+--R 2 3 3 2 3 2
+--R (- p q + p )sin(a x) + (- q + p q)cos(a x) - q + p q
+--R /
+--R q sin(a x) + p
+--R +
+--R a x + 2q
+--R +-------+ p tan(--------)
+--R | 2 2 2
+--R - 2\|q - p atan(---------------)
+--R +---------+
+--R | 2 2
+--R \|- q + p
+--R /
+--R +---------+ +-------+
+--R | 2 2 | 2 2
+--R a\|- q + p \|q - p
+--R Type: Expression Integer
+--E
+
+--S 77
+cc2:=aa.2-bb1
+--R
+--R (5)
+--R +---------+ a x + 2q
+--R | 2 2 p tan(--------)
+--R (p sin(a x) + q cos(a x) + q)\|- q + p 2
+--R - 2atan(-----------------------------------------) - 2atan(---------------)
+--R 2 2 2 2 +---------+
+--R (q - p )cos(a x) + q - p | 2 2
+--R \|- q + p
+--R ---------------------------------------------------------------------------
+--R +---------+
+--R | 2 2
+--R a\|- q + p
+--R Type: Expression Integer
+--E
+
+--S 78
+cc3:=aa.1-bb2
+--R
+--R (6)
+--R +-------+
+--R | 2 2
+--R 2\|q - p - a x - 2q
+--R tan(----------------------)
+--R 2
+--R - log(- ---------------------------)
+--R +-------+
+--R | 2 2
+--R 2\|q - p + a x + 2q
+--R tan(----------------------)
+--R 2
+--R +
+--R log
+--R +-------+
+--R 2 2 2 | 2 2
+--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p
+--R +
+--R 2 3 3 2 3 2
+--R (- p q + p )sin(a x) + (- q + p q)cos(a x) - q + p q
+--R /
+--R q sin(a x) + p
+--R /
+--R +-------+
+--R | 2 2
+--R a\|q - p
+--R Type: Expression Integer
+--E
+
+--S 79 14:360 Axiom cannot simplify these equations
+cc4:=aa.2-bb2
+--R
+--R (7)
+--R +-------+
+--R | 2 2
+--R 2\|q - p - a x - 2q
+--R +---------+ tan(----------------------)
+--R | 2 2 2
+--R - \|- q + p log(- ---------------------------)
+--R +-------+
+--R | 2 2
+--R 2\|q - p + a x + 2q
+--R tan(----------------------)
+--R 2
+--R +
+--R +---------+
+--R +-------+ | 2 2
+--R | 2 2 (p sin(a x) + q cos(a x) + q)\|- q + p
+--R - 2\|q - p atan(-----------------------------------------)
+--R 2 2 2 2
+--R (q - p )cos(a x) + q - p
+--R /
+--R +---------+ +-------+
+--R | 2 2 | 2 2
+--R a\|- q + p \|q - p
+--R Type: Expression Integer
+--E
+
@
\section{\cite{1}:14.361~~~~~$\displaystyle
@@ -509,7 +1259,7 @@
<<*>>=
)clear all
---S 23 of 30
+--S 80
aa:=integrate(1/(p+q*sin(a*x))^2,x)
--R
--R
@@ -554,6 +1304,206 @@
--R ]
--R Type: Union(List Expression Integer,...)
--E
+
+--S 81
+t1:=integrate(1/(p+q*sin(a*x)),x)
+--R
+--R (2)
+--R [
+--R log
+--R +-------+
+--R 2 2 2 | 2 2
+--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p
+--R +
+--R 2 3 3 2 3 2
+--R (- p q + p )sin(a x) + (- q + p q)cos(a x) - q + p q
+--R /
+--R q sin(a x) + p
+--R /
+--R +-------+
+--R | 2 2
+--R a\|q - p
+--R ,
+--R +---------+
+--R | 2 2
+--R (p sin(a x) + q cos(a x) + q)\|- q + p
+--R 2atan(-----------------------------------------)
+--R 2 2 2 2
+--R (q - p )cos(a x) + q - p
+--R - ------------------------------------------------]
+--R +---------+
+--R | 2 2
+--R a\|- q + p
+--R Type: Union(List Expression Integer,...)
+--E
+
+--S 82
+bb1:=(q*cos(a*x))/(a*(p^2-q^2)*(p+q*sin(a*x)))+p/(p^2-q^2)*t1.1
+--R
+--R (3)
+--R 2
+--R (- p q sin(a x) - p )
+--R *
+--R log
+--R +-------+
+--R 2 2 2 | 2 2
+--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p
+--R +
+--R 2 3 3 2 3 2
+--R (- p q + p )sin(a x) + (- q + p q)cos(a x) - q + p q
+--R /
+--R q sin(a x) + p
+--R +
+--R +-------+
+--R | 2 2
+--R - q cos(a x)\|q - p
+--R /
+--R +-------+
+--R 3 2 2 3 | 2 2
+--R ((a q - a p q)sin(a x) + a p q - a p )\|q - p
+--R Type: Expression Integer
+--E
+
+--S 83
+bb2:=(q*cos(a*x))/(a*(p^2-q^2)*(p+q*sin(a*x)))+p/(p^2-q^2)*t1.2
+--R
+--R (4)
+--R +---------+
+--R | 2 2
+--R 2 (p sin(a x) + q cos(a x) + q)\|- q + p
+--R (2p q sin(a x) + 2p )atan(-----------------------------------------)
+--R 2 2 2 2
+--R (q - p )cos(a x) + q - p
+--R +
+--R +---------+
+--R | 2 2
+--R - q cos(a x)\|- q + p
+--R /
+--R +---------+
+--R 3 2 2 3 | 2 2
+--R ((a q - a p q)sin(a x) + a p q - a p )\|- q + p
+--R Type: Expression Integer
+--E
+
+--S 84
+cc1:=aa.1-bb1
+--R
+--R (5)
+--R 2
+--R p
+--R *
+--R log
+--R +-------+
+--R 2 2 2 | 2 2
+--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p
+--R +
+--R 2 3 3 2 3 2
+--R (p q - p )sin(a x) + (q - p q)cos(a x) + q - p q
+--R /
+--R q sin(a x) + p
+--R +
+--R 2
+--R p
+--R *
+--R log
+--R +-------+
+--R 2 2 2 | 2 2
+--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p
+--R +
+--R 2 3 3 2 3 2
+--R (- p q + p )sin(a x) + (- q + p q)cos(a x) - q + p q
+--R /
+--R q sin(a x) + p
+--R +
+--R +-------+
+--R | 2 2
+--R - q\|q - p
+--R /
+--R +-------+
+--R 2 3 | 2 2
+--R (a p q - a p )\|q - p
+--R Type: Expression Integer
+--E
+
+--S 85
+cc2:=aa.2-bb1
+--R
+--R (6)
+--R +---------+
+--R 2 | 2 2
+--R p \|- q + p
+--R *
+--R log
+--R +-------+
+--R 2 2 2 | 2 2
+--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p
+--R +
+--R 2 3 3 2 3 2
+--R (- p q + p )sin(a x) + (- q + p q)cos(a x) - q + p q
+--R /
+--R q sin(a x) + p
+--R +
+--R +---------+
+--R +-------+ | 2 2
+--R 2 | 2 2 (p sin(a x) + q cos(a x) + q)\|- q + p
+--R 2p \|q - p atan(-----------------------------------------)
+--R 2 2 2 2
+--R (q - p )cos(a x) + q - p
+--R +
+--R +---------+ +-------+
+--R | 2 2 | 2 2
+--R - q\|- q + p \|q - p
+--R /
+--R +---------+ +-------+
+--R 2 3 | 2 2 | 2 2
+--R (a p q - a p )\|- q + p \|q - p
+--R Type: Expression Integer
+--E
+
+--S 86
+cc3:=aa.1-bb2
+--R
+--R (7)
+--R +---------+
+--R 2 | 2 2
+--R p \|- q + p
+--R *
+--R log
+--R +-------+
+--R 2 2 2 | 2 2
+--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p
+--R +
+--R 2 3 3 2 3 2
+--R (p q - p )sin(a x) + (q - p q)cos(a x) + q - p q
+--R /
+--R q sin(a x) + p
+--R +
+--R +---------+
+--R +-------+ | 2 2
+--R 2 | 2 2 (p sin(a x) + q cos(a x) + q)\|- q + p
+--R - 2p \|q - p atan(-----------------------------------------)
+--R 2 2 2 2
+--R (q - p )cos(a x) + q - p
+--R +
+--R +---------+ +-------+
+--R | 2 2 | 2 2
+--R - q\|- q + p \|q - p
+--R /
+--R +---------+ +-------+
+--R 2 3 | 2 2 | 2 2
+--R (a p q - a p )\|- q + p \|q - p
+--R Type: Expression Integer
+--E
+
+--S 87 14:361 Schaums and Axiom differ by a constant
+cc4:=aa.2-bb2
+--R
+--R q
+--R (8) - -------------
+--R 2 3
+--R a p q - a p
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.362~~~~~$\displaystyle
@@ -564,7 +1514,7 @@
<<*>>=
)clear all
---S 24 of 30
+--S 88
aa:=integrate(1/(p^2+a^2*sin(a*x)),x)
--R
--R
@@ -597,6 +1547,76 @@
--R a\|p - a
--R Type: Union(List Expression Integer,...)
--E
+
+--S 89
+bb:=1/(a*p*sqrt(p^2+q^2))*atan((sqrt(p^2+q^2)*tan(a*x))/p)
+--R
+--R +-------+
+--R | 2 2
+--R tan(a x)\|q + p
+--R atan(------------------)
+--R p
+--R (2) ------------------------
+--R +-------+
+--R | 2 2
+--R a p\|q + p
+--R Type: Expression Integer
+--E
+
+--S 90
+cc1:=aa.1-bb
+--R
+--R (3)
+--R +-------+
+--R | 2 2
+--R p\|q + p
+--R *
+--R log
+--R +---------+
+--R 2 2 4 4 4 | 4 4
+--R (a p sin(a x) + (- p + a )cos(a x) + a )\|- p + a
+--R +
+--R 6 4 2 2 4 6 2 4 6
+--R (p - a p )sin(a x) + (a p - a )cos(a x) + a p - a
+--R /
+--R 2 2
+--R a sin(a x) + p
+--R +
+--R +-------+
+--R +---------+ | 2 2
+--R | 4 4 tan(a x)\|q + p
+--R - \|- p + a atan(------------------)
+--R p
+--R /
+--R +---------+ +-------+
+--R | 4 4 | 2 2
+--R a p\|- p + a \|q + p
+--R Type: Expression Integer
+--E
+
+--S 91 14:362 Axiom cannot simplify this expression
+cc2:=aa.2-bb
+--R
+--R (4)
+--R +-------+
+--R +-------+ | 2 2
+--R | 4 4 tan(a x)\|q + p
+--R - \|p - a atan(------------------)
+--R p
+--R +
+--R +-------+
+--R +-------+ 2 2 2 | 4 4
+--R | 2 2 (p sin(a x) + a cos(a x) + a )\|p - a
+--R 2p\|q + p atan(----------------------------------------)
+--R 4 4 4 4
+--R (p - a )cos(a x) + p - a
+--R /
+--R +-------+ +-------+
+--R | 4 4 | 2 2
+--R a p\|p - a \|q + p
+--R Type: Expression Integer
+--E
+
@
\section{\cite{1}:14.363~~~~~$\displaystyle
@@ -616,7 +1636,7 @@
<<*>>=
)clear all
---S 25 of 30
+--S 92
aa:=integrate(1/(p^2-q^2*sin(a*x)^2),x)
--R
--R
@@ -658,6 +1678,154 @@
--R ]
--R Type: Union(List Expression Integer,...)
--E
+
+--S 93
+bb1:=1/(a*p*sqrt(p^2-q^2))*atan((sqrt(p^2-q^2)*tan(a*x))/p)
+--R
+--R +---------+
+--R | 2 2
+--R tan(a x)\|- q + p
+--R atan(--------------------)
+--R p
+--R (2) --------------------------
+--R +---------+
+--R | 2 2
+--R a p\|- q + p
+--R Type: Expression Integer
+--E
+
+--S 94
+bb2:=1/(2*a*p*sqrt(q^2-p^2))*log((sqrt(q^2-p^2)*tan(a*x)+p)/(sqrt(q^2-p^2)*tan(a*x)-p))
+--R
+--R +-------+
+--R | 2 2
+--R tan(a x)\|q - p + p
+--R log(----------------------)
+--R +-------+
+--R | 2 2
+--R tan(a x)\|q - p - p
+--R (3) ---------------------------
+--R +-------+
+--R | 2 2
+--R 2a p\|q - p
+--R Type: Expression Integer
+--E
+
+--S 95
+cc1:=aa.1-bb1
+--R
+--R (4)
+--R +---------+
+--R | 2 2
+--R \|- q + p
+--R *
+--R log
+--R +-------+
+--R 2 2 2 2 2 | 2 2
+--R ((- q + 2p )cos(a x) + q - p )\|q - p
+--R +
+--R 2 3
+--R (2p q - 2p )cos(a x)sin(a x)
+--R /
+--R 2 2 2 2
+--R q cos(a x) - q + p
+--R +
+--R +---------+
+--R +-------+ | 2 2
+--R | 2 2 tan(a x)\|- q + p
+--R - 2\|q - p atan(--------------------)
+--R p
+--R /
+--R +---------+ +-------+
+--R | 2 2 | 2 2
+--R 2a p\|- q + p \|q - p
+--R Type: Expression Integer
+--E
+
+--S 96
+cc2:=aa.2-bb1
+--R
+--R (5)
+--R +---------+ +---------+
+--R | 2 2 | 2 2
+--R tan(a x)\|- q + p p sin(a x)\|- q + p
+--R - atan(--------------------) - atan(-------------------------------)
+--R p 2 2 2 2
+--R (2q - 2p )cos(a x) + 2q - 2p
+--R +
+--R 2 2 2 2
+--R ((2q - p )cos(a x) + 2q - 2p )sin(a x)
+--R - atan(-------------------------------------------)
+--R +---------+
+--R 2 | 2 2
+--R (p cos(a x) + 2p cos(a x) + p)\|- q + p
+--R /
+--R +---------+
+--R | 2 2
+--R a p\|- q + p
+--R Type: Expression Integer
+--E
+
+--S 97
+cc3:=aa.1-bb2
+--R
+--R (6)
+--R +-------+
+--R | 2 2
+--R tan(a x)\|q - p + p
+--R - log(----------------------)
+--R +-------+
+--R | 2 2
+--R tan(a x)\|q - p - p
+--R +
+--R log
+--R +-------+
+--R 2 2 2 2 2 | 2 2
+--R ((- q + 2p )cos(a x) + q - p )\|q - p
+--R +
+--R 2 3
+--R (2p q - 2p )cos(a x)sin(a x)
+--R /
+--R 2 2 2 2
+--R q cos(a x) - q + p
+--R /
+--R +-------+
+--R | 2 2
+--R 2a p\|q - p
+--R Type: Expression Integer
+--E
+
+--S 98 14:363 Axiom cannot simplify this expression
+cc4:=aa.2-bb2
+--R
+--R (7)
+--R +-------+
+--R +---------+ | 2 2
+--R | 2 2 tan(a x)\|q - p + p
+--R - \|- q + p log(----------------------)
+--R +-------+
+--R | 2 2
+--R tan(a x)\|q - p - p
+--R +
+--R +---------+
+--R +-------+ | 2 2
+--R | 2 2 p sin(a x)\|- q + p
+--R - 2\|q - p atan(-------------------------------)
+--R 2 2 2 2
+--R (2q - 2p )cos(a x) + 2q - 2p
+--R +
+--R +-------+ 2 2 2 2
+--R | 2 2 ((2q - p )cos(a x) + 2q - 2p )sin(a x)
+--R - 2\|q - p atan(-------------------------------------------)
+--R +---------+
+--R 2 | 2 2
+--R (p cos(a x) + 2p cos(a x) + p)\|- q + p
+--R /
+--R +---------+ +-------+
+--R | 2 2 | 2 2
+--R 2a p\|- q + p \|q - p
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.364~~~~~$\displaystyle
@@ -669,7 +1837,7 @@
<<*>>=
)clear all
---S 26 of 30
+--S 99 14:364 Axiom cannot compute this integral
aa:=integrate(x^m*sin(a*x),x)
--R
--R
@@ -689,7 +1857,7 @@
<<*>>=
)clear all
---S 26 of 30
+--S 100 14:365 Axiom cannot compute this integral
aa:=integrate(sin(a*x)/x^n,x)
--R
--R
@@ -710,7 +1878,7 @@
<<*>>=
)clear all
---S 28 of 30
+--S 101 14:366 Axiom cannot compute this integral
aa:=integrate(sin(a*x)^n,x)
--R
--R
@@ -731,7 +1899,7 @@
<<*>>=
)clear all
---S 29 of 30
+--S 102 14:367 Axiom cannot compute this integral
aa:=integrate(1/(sin(a*x))^n,x)
--R
--R
@@ -739,7 +1907,7 @@
--R ++ 1
--I (1) | ---------- d%I
--R ++ n
---i sin(%I a)
+--I sin(%I a)
--R Type: Union(Expression Integer,...)
--E
@
@@ -754,7 +1922,7 @@
<<*>>=
)clear all
---S 30 of 30
+--S 103 14:368 Axiom cannot compute this integral
aa:=integrate(x/sin(a*x)^n,x)
--R
--R
Modified: trunk/axiom/src/input/schaum18.input.pamphlet
===================================================================
--- trunk/axiom/src/input/schaum18.input.pamphlet 2008-04-22 05:50:59 UTC (rev 861)
+++ trunk/axiom/src/input/schaum18.input.pamphlet 2008-04-24 05:55:47 UTC (rev 862)
@@ -18,7 +18,7 @@
)set message auto off
)clear all
---S 1 of 30
+--S 1
aa:=integrate(cos(a*x),x)
--R
--R
@@ -27,6 +27,22 @@
--R a
--R Type: Union(Expression Integer,...)
--E
+
+--S 2
+bb:=sin(a*x)/a
+--R
+--R sin(a x)
+--R (2) --------
+--R a
+--R Type: Expression Integer
+--E
+
+--S 3 14:369 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.370~~~~~$\displaystyle
@@ -37,7 +53,7 @@
<<*>>=
)clear all
---S 2 of 30
+--S 4
aa:=integrate(x*cos(a*x),x)
--R
--R
@@ -47,6 +63,23 @@
--R a
--R Type: Union(Expression Integer,...)
--E
+
+--S 5
+bb:=cos(a*x)/a^2+(x*sin(a*x))/a
+--R
+--R a x sin(a x) + cos(a x)
+--R (2) -----------------------
+--R 2
+--R a
+--R Type: Expression Integer
+--E
+
+--S 6 14:370 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.371~~~~~$\displaystyle
@@ -57,7 +90,7 @@
<<*>>=
)clear all
---S 3 of 30
+--S 7
aa:=integrate(x^2*cos(a*x),x)
--R
--R
@@ -68,6 +101,24 @@
--R a
--R Type: Union(Expression Integer,...)
--E
+
+--S 8
+bb:=(2*x)/a^2*cos(a*x)+(x^2/a-2/a^3)*sin(a*x)
+--R
+--R 2 2
+--R (a x - 2)sin(a x) + 2a x cos(a x)
+--R (2) ----------------------------------
+--R 3
+--R a
+--R Type: Expression Integer
+--E
+
+--S 9 14:371 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.372~~~~~$\displaystyle
@@ -79,7 +130,7 @@
<<*>>=
)clear all
---S 4 of 30
+--S 10
aa:=integrate(x^3*cos(a*x),x)
--R
--R
@@ -90,6 +141,24 @@
--R a
--R Type: Union(Expression Integer,...)
--E
+
+--S 11
+bb:=((3*x^2)/a^2-6/a^4)*cos(a*x)+(x^3/a-(6*x)/a^3)*sin(a*x)
+--R
+--R 3 3 2 2
+--R (a x - 6a x)sin(a x) + (3a x - 6)cos(a x)
+--R (2) -------------------------------------------
+--R 4
+--R a
+--R Type: Expression Integer
+--E
+
+--S 12 14:372 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.373~~~~~$\displaystyle
@@ -101,7 +170,7 @@
<<*>>=
)clear all
---S 5 of 30
+--S 13 14:373 Schaums and Axiom agree by definition
aa:=integrate(cos(x)/x,x)
--R
--R
@@ -118,7 +187,7 @@
<<*>>=
)clear all
---S 6 of 30
+--S 14 14:374 Axiom cannot compute this integral
aa:=integrate(cos(a*x)/x^2,x)
--R
--R
@@ -140,7 +209,7 @@
<<*>>=
)clear all
---S 7 of 30
+--S 15
aa:=integrate(1/cos(a*x),x)
--R
--R
@@ -151,6 +220,59 @@
--R a
--R Type: Union(Expression Integer,...)
--E
+
+--S 16
+bb1:=1/a*log(sec(a*x)+tan(a*x))
+--R
+--R log(tan(a x) + sec(a x))
+--R (2) ------------------------
+--R a
+--R Type: Expression Integer
+--E
+
+--S 17
+bb2:=1/a*log(tan(%pi/4+(a*x)/2))
+--R
+--R 2a x + %pi
+--R log(tan(----------))
+--R 4
+--R (3) --------------------
+--R a
+--R Type: Expression Integer
+--E
+
+--S 18
+cc1:=aa-bb1
+--R
+--R (4)
+--R sin(a x) + cos(a x) + 1
+--R - log(tan(a x) + sec(a x)) + log(-----------------------)
+--R cos(a x) + 1
+--R +
+--R sin(a x) - cos(a x) - 1
+--R - log(-----------------------)
+--R cos(a x) + 1
+--R /
+--R a
+--R Type: Expression Integer
+--E
+
+--S 19 14:375 Axiom cannot simplify these expressions
+cc2:=aa-bb2
+--R
+--R (5)
+--R 2a x + %pi sin(a x) + cos(a x) + 1
+--R - log(tan(----------)) + log(-----------------------)
+--R 4 cos(a x) + 1
+--R +
+--R sin(a x) - cos(a x) - 1
+--R - log(-----------------------)
+--R cos(a x) + 1
+--R /
+--R a
+--R Type: Expression Integer
+--E
+
@
\section{\cite{1}:14.376~~~~~$\displaystyle
@@ -164,7 +286,7 @@
<<*>>=
)clear all
---S 8 of 30
+--S 20 14:376 Axiom cannot compute this integral
aa:=integrate(x/cos(a*x),x)
--R
--R
@@ -184,7 +306,7 @@
<<*>>=
)clear all
---S 9 of 30
+--S 21
aa:=integrate(cos(a*x)^2,x)
--R
--R
@@ -193,6 +315,40 @@
--R 2a
--R Type: Union(Expression Integer,...)
--E
+
+--S 22
+bb:=x/2+sin(2*a*x)/(4*a)
+--R
+--R sin(2a x) + 2a x
+--R (2) ----------------
+--R 4a
+--R Type: Expression Integer
+--E
+
+--S 23
+cc:=aa-bb
+--R
+--R - sin(2a x) + 2cos(a x)sin(a x)
+--R (3) -------------------------------
+--R 4a
+--R Type: Expression Integer
+--E
+
+--S 24
+cossinrule:=rule(cos(b)*sin(a) == 1/2*(sin(a-b)+sin(a+b)))
+--R
+--I %S sin(b + a) - %S sin(b - a)
+--I (4) %S cos(b)sin(a) == -----------------------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 25 14:377 Schaums and Axiom agree
+dd:=cossinrule cc
+--R
+--R (5) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.378~~~~~$\displaystyle
@@ -203,7 +359,7 @@
<<*>>=
)clear all
---S 10 of 30
+--S 26
aa:=integrate(x*cos(a*x)^2,x)
--R
--R
@@ -214,6 +370,87 @@
--R 4a
--R Type: Union(Expression Integer,...)
--E
+
+--S 27
+bb:=x^2/4+(x*sin(2*a*x))/(4*a)+cos(2*a*x)/(8*a^2)
+--R
+--R 2 2
+--R 2a x sin(2a x) + cos(2a x) + 2a x
+--R (2) ----------------------------------
+--R 2
+--R 8a
+--R Type: Expression Integer
+--E
+
+--S 28
+cc:=aa-bb
+--R
+--R 2
+--R - 2a x sin(2a x) + 4a x cos(a x)sin(a x) - cos(2a x) + 2cos(a x)
+--R (3) -----------------------------------------------------------------
+--R 2
+--R 8a
+--R Type: Expression Integer
+--E
+
+--S 29
+cossinrule:=rule(cos(b)*sin(a) == 1/2*(sin(a-b)+sin(a+b)))
+--R
+--I %T sin(b + a) - %T sin(b - a)
+--I (4) %T cos(b)sin(a) == -----------------------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 30
+dd:=cossinrule cc
+--R
+--R 2
+--R - cos(2a x) + 2cos(a x)
+--R (5) ------------------------
+--R 2
+--R 8a
+--R Type: Expression Integer
+--E
+
+--S 31
+coscosrule:=rule(cos(a)*cos(b) == 1/2*(cos(a-b)+cos(a+b)))
+--R
+--I %U cos(b + a) + %U cos(b - a)
+--I (6) %U cos(a)cos(b) == -----------------------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 32
+ee:=coscosrule dd
+--R
+--R 2
+--R - cos(2a x) + 2cos(a x)
+--R (7) ------------------------
+--R 2
+--R 8a
+--R Type: Expression Integer
+--E
+
+--S 33
+cossqrrule1:=rule(cos(a)^2 == 1/2+1/2*cos(2*a))
+--R
+--R 2 cos(2a) + 1
+--R (8) cos(a) == -----------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 34 14:378 Schaums and Axiom differ by a constant
+ff:=cossqrrule1 ee
+--R
+--R 1
+--R (9) ---
+--R 2
+--R 8a
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.379~~~~~$\displaystyle
@@ -224,7 +461,7 @@
<<*>>=
)clear all
---S 11 of 30
+--S 35
aa:=integrate(cos(a*x)^3,x)
--R
--R
@@ -233,7 +470,42 @@
--R (1) -----------------------
--R 3a
--R Type: Union(Expression Integer,...)
+--E
+
+--S 36
+bb:=sin(a*x)/a-sin(a*x)^3/(3*a)
+--R
+--R 3
+--R - sin(a x) + 3sin(a x)
+--R (2) -----------------------
+--R 3a
+--R Type: Expression Integer
--E
+
+--S 37
+cc:=aa-bb
+--R
+--R 3 2
+--R sin(a x) + (cos(a x) - 1)sin(a x)
+--R (3) -----------------------------------
+--R 3a
+--R Type: Expression Integer
+--E
+
+--S 38
+cossqrrule:=rule(cos(a)^2 == 1-sin(a)^2)
+--R
+--R 2 ...
[truncated message content] |
|
From: <gi...@ax...> - 2008-04-24 05:54:09
|
changelog | 6 +
src/input/schaum17.input.pamphlet | 1230 ++++++++++++++++++++++++-
src/input/schaum18.input.pamphlet | 1047 +++++++++++++++++++++-
src/input/schaum19.input.pamphlet | 1809 ++++++++++++++++++++++++++++++++++++-
src/input/schaum20.input.pamphlet | 461 +++++++++-
src/input/schaum21.input.pamphlet | 425 +++++++++-
src/input/schaum22.input.pamphlet | 555 +++++++++++-
7 files changed, 5403 insertions(+), 130 deletions(-)
New commits:
commit d0a2b40bcd2a153122660e5b1f4ef96d17b7ce23
Author: Tim Daly <root@localhost.localdomain>
Date: Wed Apr 23 15:57:26 2008 -0400
20080423 tpd src/input/schaum22.input show Schaums-Axiom equivalence
20080423 tpd src/input/schaum21.input show Schaums-Axiom equivalence
20080423 tpd src/input/schaum20.input show Schaums-Axiom equivalence
20080423 tpd src/input/schaum19.input show Schaums-Axiom equivalence
20080423 tpd src/input/schaum18.input show Schaums-Axiom equivalence
20080423 tpd src/input/schaum17.input show Schaums-Axiom equivalence
|
|
From: <da...@us...> - 2008-04-22 05:50:54
|
Revision: 861
http://axiom.svn.sourceforge.net/axiom/?rev=861&view=rev
Author: daly
Date: 2008-04-21 22:50:59 -0700 (Mon, 21 Apr 2008)
Log Message:
-----------
20080421 tpd src/input/schaum16.input show Schaums-Axiom equivalence
20080421 tpd src/input/schaum15.input show Schaums-Axiom equivalence
20080421 tpd src/input/schaum14.input show Schaums-Axiom equivalence
Modified Paths:
--------------
trunk/axiom/changelog
trunk/axiom/src/input/schaum14.input.pamphlet
trunk/axiom/src/input/schaum15.input.pamphlet
trunk/axiom/src/input/schaum16.input.pamphlet
Modified: trunk/axiom/changelog
===================================================================
--- trunk/axiom/changelog 2008-04-21 07:17:21 UTC (rev 860)
+++ trunk/axiom/changelog 2008-04-22 05:50:59 UTC (rev 861)
@@ -1,3 +1,6 @@
+20080421 tpd src/input/schaum16.input show Schaums-Axiom equivalence
+20080421 tpd src/input/schaum15.input show Schaums-Axiom equivalence
+20080421 tpd src/input/schaum14.input show Schaums-Axiom equivalence
20080420 tpd src/input/schaum13.input show Schaums-Axiom equivalence
20080419 tpd src/input/schaum12.input show Schaums-Axiom equivalence
20080418 tpd src/input/schaum11.input show Schaums-Axiom equivalence
Modified: trunk/axiom/src/input/schaum14.input.pamphlet
===================================================================
--- trunk/axiom/src/input/schaum14.input.pamphlet 2008-04-21 07:17:21 UTC (rev 860)
+++ trunk/axiom/src/input/schaum14.input.pamphlet 2008-04-22 05:50:59 UTC (rev 861)
@@ -19,7 +19,7 @@
)set message auto off
)clear all
---S 1 of 12
+--S 1
aa:=integrate(1/(x^3+a^3),x)
--R
--R
@@ -32,6 +32,41 @@
--R 6a \|3
--R Type: Union(Expression Integer,...)
--E
+
+--S 2
+bb:=1/(6*a^2)*log((x+a)^2/(x^2-a*x+a^2))+1/(a^2*sqrt(3))*atan((2*x-a)/(a*sqrt(3)))
+--R
+--R 2 2 +-+
+--R x + 2a x + a +-+ (2x - a)\|3
+--R log(--------------) + 2\|3 atan(------------)
+--R 2 2 3a
+--R x - a x + a
+--R (2) ---------------------------------------------
+--R 2
+--R 6a
+--R Type: Expression Integer
+--E
+
+--S 3
+cc:=aa-bb
+--R
+--R 2 2
+--R 2 2 x + 2a x + a
+--R - log(x - a x + a ) + 2log(x + a) - log(--------------)
+--R 2 2
+--R x - a x + a
+--R (3) --------------------------------------------------------
+--R 2
+--R 6a
+--R Type: Expression Integer
+--E
+
+--S 4 14:299 Schaums and Axiom agree
+dd:=expandLog cc
+--R
+--R (4) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.300~~~~~$\displaystyle
@@ -43,7 +78,7 @@
<<*>>=
)clear all
---S 2 of 12
+--S 5
aa:=integrate(x/(x^3+a^3),x)
--R
--R
@@ -56,6 +91,39 @@
--R 6a\|3
--R Type: Union(Expression Integer,...)
--E
+
+--S 6
+bb:=1/(6*a)*log((x^2-a*x+a^2)/(x+a)^2)+1/(a*sqrt(3))*atan((2*x-a)/(a*sqrt(3)))
+--R
+--R 2 2 +-+
+--R x - a x + a +-+ (2x - a)\|3
+--R log(--------------) + 2\|3 atan(------------)
+--R 2 2 3a
+--R x + 2a x + a
+--R (2) ---------------------------------------------
+--R 6a
+--R Type: Expression Integer
+--E
+
+--S 7
+cc:=aa-bb
+--R
+--R 2 2
+--R 2 2 x - a x + a
+--R log(x - a x + a ) - 2log(x + a) - log(--------------)
+--R 2 2
+--R x + 2a x + a
+--R (3) ------------------------------------------------------
+--R 6a
+--R Type: Expression Integer
+--E
+
+--S 8 14:300 Schaums and Axiom agree
+dd:=expandLog cc
+--R
+--R (4) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.301~~~~~$\displaystyle
@@ -66,7 +134,7 @@
<<*>>=
)clear all
---S 3 of 12
+--S 9
aa:=integrate(x^2/(x^3+a^3),x)
--R
--R
@@ -76,6 +144,23 @@
--R 3
--R Type: Union(Expression Integer,...)
--E
+
+--S 10
+bb:=1/3*log(x^3+a^3)
+--R
+--R 3 3
+--R log(x + a )
+--R (2) ------------
+--R 3
+--R Type: Expression Integer
+--E
+
+--S 11 14:301 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.302~~~~~$\displaystyle
@@ -86,7 +171,7 @@
<<*>>=
)clear all
---S 4 of 12
+--S 12
aa:=integrate(1/(x*(x^3+a^3)),x)
--R
--R
@@ -97,6 +182,41 @@
--R 3a
--R Type: Union(Expression Integer,...)
--E
+
+--S 13
+bb:=1/(3*a^3)*log(x^3/(x^3+a^3))
+--R
+--R 3
+--R x
+--R log(-------)
+--R 3 3
+--R x + a
+--R (2) ------------
+--R 3
+--R 3a
+--R Type: Expression Integer
+--E
+
+--S 14
+cc:=aa-bb
+--R
+--R 3
+--R 3 3 x
+--R - log(x + a ) + 3log(x) - log(-------)
+--R 3 3
+--R x + a
+--R (3) ---------------------------------------
+--R 3
+--R 3a
+--R Type: Expression Integer
+--E
+
+--S 15 14:302 Schaums and Axiom agree
+dd:=expandLog cc
+--R
+--R (4) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.303~~~~~$\displaystyle
@@ -108,7 +228,7 @@
<<*>>=
)clear all
---S 5 of 12
+--S 15
aa:=integrate(1/(x^2*(x^3+a^3)),x)
--R
--R
@@ -121,7 +241,42 @@
--R 4 +-+
--R 6a x\|3
--R Type: Union(Expression Integer,...)
+--E
+
+--S 16
+bb:=-1/(a^3*x)-1/(6*a^4)*log((x^2-a*x+a^2)/(x+a)^2)-1/(a^4*sqrt(3))*atan((2*x-a)/(a*sqrt(3)))
+--R
+--R 2 2 +-+
+--R x - a x + a +-+ (2x - a)\|3
+--R - x log(--------------) - 2x\|3 atan(------------) - 6a
+--R 2 2 3a
+--R x + 2a x + a
+--R (2) -------------------------------------------------------
+--R 4
+--R 6a x
+--R Type: Expression Integer
--E
+
+--S 17
+cc:=aa-bb
+--R
+--R 2 2
+--R 2 2 x - a x + a
+--R - log(x - a x + a ) + 2log(x + a) + log(--------------)
+--R 2 2
+--R x + 2a x + a
+--R (3) --------------------------------------------------------
+--R 4
+--R 6a
+--R Type: Expression Integer
+--E
+
+--S 18 14:303 Schaums and Axiom agree
+dd:=expandLog cc
+--R
+--R (4) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.304~~~~~$\displaystyle
@@ -134,7 +289,7 @@
<<*>>=
)clear all
---S 6 of 12
+--S 19
aa:=integrate(1/(x^3+a^3)^2,x)
--R
--R
@@ -151,6 +306,42 @@
--R (9a x + 9a )\|3
--R Type: Union(Expression Integer,...)
--E
+
+--S 20
+bb:=x/(3*a^3*(x^3+a^3))+1/(9*a^5)*log((x+a)^2/(x^2-a*x+a^2))+2/(3*a^5*sqrt(3))*atan((2*x-a)/(a*sqrt(3)))
+--R
+--R (2)
+--R 2 2 +-+
+--R 3 3 x + 2a x + a 3 3 +-+ (2x - a)\|3 2
+--R (x + a )log(--------------) + (2x + 2a )\|3 atan(------------) + 3a x
+--R 2 2 3a
+--R x - a x + a
+--R -----------------------------------------------------------------------
+--R 5 3 8
+--R 9a x + 9a
+--R Type: Expression Integer
+--E
+
+--S 21
+cc:=aa-bb
+--R
+--R 2 2
+--R 2 2 x + 2a x + a
+--R - log(x - a x + a ) + 2log(x + a) - log(--------------)
+--R 2 2
+--R x - a x + a
+--R (3) --------------------------------------------------------
+--R 5
+--R 9a
+--R Type: Expression Integer
+--E
+
+--S 22 14:304 Schaums and Axiom agree
+dd:=expandLog cc
+--R
+--R (4) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.305~~~~~$\displaystyle
@@ -163,7 +354,7 @@
<<*>>=
)clear all
---S 7 of 12
+--S 23
aa:=integrate(x/(x^3+a^3)^2,x)
--R
--R
@@ -180,6 +371,42 @@
--R (18a x + 18a )\|3
--R Type: Union(Expression Integer,...)
--E
+
+--S 24
+bb:=x^2/(3*a^3*(x^3+a^3))+1/(18*a^4)*log((x^2-a*x+a^2)/(x+a)^2)+1/(3*a^4*sqrt(3))*atan((2*x-a)/(a*sqrt(3)))
+--R
+--R (2)
+--R 2 2 +-+
+--R 3 3 x - a x + a 3 3 +-+ (2x - a)\|3 2
+--R (x + a )log(--------------) + (2x + 2a )\|3 atan(------------) + 6a x
+--R 2 2 3a
+--R x + 2a x + a
+--R ------------------------------------------------------------------------
+--R 4 3 7
+--R 18a x + 18a
+--R Type: Expression Integer
+--E
+
+--S 25
+cc:=aa-bb
+--R
+--R 2 2
+--R 2 2 x - a x + a
+--R log(x - a x + a ) - 2log(x + a) - log(--------------)
+--R 2 2
+--R x + 2a x + a
+--R (3) ------------------------------------------------------
+--R 4
+--R 18a
+--R Type: Expression Integer
+--E
+
+--S 26 14:305 Schaums and Axiom agree
+dd:=expandLog cc
+--R
+--R (4) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.306~~~~~$\displaystyle
@@ -190,7 +417,7 @@
<<*>>=
)clear all
---S 8 of 12
+--S 27
aa:=integrate(x^2/(x^3+a^3)^2,x)
--R
--R
@@ -199,7 +426,24 @@
--R 3 3
--R 3x + 3a
--R Type: Union(Expression Integer,...)
+--E
+
+--S 28
+bb:=-1/(3*(x^3+a^3))
+--R
+--R 1
+--R (2) - ---------
+--R 3 3
+--R 3x + 3a
+--R Type: Fraction Polynomial Integer
--E
+
+--S 29 14:306 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.307~~~~~$\displaystyle
@@ -210,7 +454,7 @@
<<*>>=
)clear all
---S 9 of 12
+--S 30
aa:=integrate(1/(x*(x^3+a^3)^2),x)
--R
--R
@@ -221,6 +465,41 @@
--R 3a x + 3a
--R Type: Union(Expression Integer,...)
--E
+
+--S 31
+bb:=1/(3*a^3*(x^3+a^3))+1/(3*a^6)*log(x^3/(x^3+a^3))
+--R
+--R 3
+--R 3 3 x 3
+--R (x + a )log(-------) + a
+--R 3 3
+--R x + a
+--R (2) --------------------------
+--R 6 3 9
+--R 3a x + 3a
+--R Type: Expression Integer
+--E
+
+--S 32
+cc:=aa-bb
+--R
+--R 3
+--R 3 3 x
+--R - log(x + a ) + 3log(x) - log(-------)
+--R 3 3
+--R x + a
+--R (3) ---------------------------------------
+--R 6
+--R 3a
+--R Type: Expression Integer
+--E
+
+--S 33 14:307 Schaums and Axiom agree
+dd:=expandLog cc
+--R
+--R (4) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.308~~~~~$\displaystyle
@@ -232,7 +511,7 @@
<<*>>=
)clear all
---S 10 of 12
+--S 34
aa:=integrate(1/(x^2*(x^3+a^3)^2),x)
--R
--R
@@ -248,7 +527,44 @@
--R 7 4 10 +-+
--R (9a x + 9a x)\|3
--R Type: Union(Expression Integer,...)
+--E
+
+--S 35
+t1:=integrate(x/(x^3+a^3),x)
+--R
+--R +-+
+--R +-+ 2 2 +-+ (2x - a)\|3
+--R \|3 log(x - a x + a ) - 2\|3 log(x + a) + 6atan(------------)
+--R 3a
+--R (2) --------------------------------------------------------------
+--R +-+
+--R 6a\|3
+--R Type: Union(Expression Integer,...)
+--E
+
+--S 36
+bb:=-1/(a^6*x)-x^2/(3*a^6*(x^3+a^3))-4/(3*a^6)*t1
+--R
+--R (3)
+--R 4 3 +-+ 2 2 4 3 +-+
+--R (- 2x - 2a x)\|3 log(x - a x + a ) + (4x + 4a x)\|3 log(x + a)
+--R +
+--R +-+
+--R 4 3 (2x - a)\|3 3 4 +-+
+--R (- 12x - 12a x)atan(------------) + (- 12a x - 9a )\|3
+--R 3a
+--R /
+--R 7 4 10 +-+
+--R (9a x + 9a x)\|3
+--R Type: Expression Integer
--E
+
+--S 37 14:308 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (4) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.309~~~~~$\displaystyle
@@ -259,7 +575,7 @@
<<*>>=
)clear all
---S 11 of 12
+--S 38 14:309 Axiom cannot compute this integral
aa:=integrate(x^m/(x^3+a^3),x)
--R
--R
@@ -269,7 +585,7 @@
--R ++ 3 3
--I a + %L
--R Type: Union(Expression Integer,...)
---E
+--E
@
\section{\cite{1}:14.310~~~~~$\displaystyle
@@ -280,7 +596,7 @@
<<*>>=
)clear all
---S 12 of 12
+--S 39 14:310 Axiom cannot compute this integral
aa:=integrate(1/(x^n*(x^3+a^3)),x)
--R
--R
Modified: trunk/axiom/src/input/schaum15.input.pamphlet
===================================================================
--- trunk/axiom/src/input/schaum15.input.pamphlet 2008-04-21 07:17:21 UTC (rev 860)
+++ trunk/axiom/src/input/schaum15.input.pamphlet 2008-04-22 05:50:59 UTC (rev 861)
@@ -20,7 +20,7 @@
)set message auto off
)clear all
---S 1 of 14
+--S 1
aa:=integrate(1/(x^4+a^4),x)
--R
--R
@@ -53,6 +53,202 @@
--R \|2
--R Type: Union(Expression Integer,...)
--E
+
+--S 2
+bb:=1/(4*a^3*sqrt(2))*log((x^2+a*x*sqrt(2)+a^2)/(x^2-a*x*sqrt(2)+a^2))-1/(2*a^3*sqrt(2))*atan((a*x*sqrt(2))/(x^2-a^2))
+--R
+--R +-+ 2 2 +-+
+--R +-+ - a x\|2 - x - a +-+ a x\|2
+--R \|2 log(-------------------) - 2\|2 atan(-------)
+--R +-+ 2 2 2 2
+--R a x\|2 - x - a x - a
+--R (2) -------------------------------------------------
+--R 3
+--R 8a
+--R Type: Expression Integer
+--E
+
+--S 3
+cc:=aa-bb
+--R
+--R (3)
+--R +------+ +------+2 +------+
+--R 3 | 1 8 | 1 4 +-+ | 1 2
+--R 4a |------ log(16a |------ + 4a x\|2 |------ + x )
+--R 4| 12 4| 12 4| 12
+--R \|256a \|256a \|256a
+--R +
+--R +------+ +------+2 +------+
+--R 3 | 1 8 | 1 4 +-+ | 1 2
+--R - 4a |------ log(16a |------ - 4a x\|2 |------ + x )
+--R 4| 12 4| 12 4| 12
+--R \|256a \|256a \|256a
+--R +
+--R +------+
+--R 4 | 1
+--R 4a |------
+--R +------+ 4| 12
+--R 3 | 1 \|256a
+--R 8a |------ atan(--------------------)
+--R 4| 12 +------+
+--R \|256a 4 | 1 +-+
+--R 4a |------ - x\|2
+--R 4| 12
+--R \|256a
+--R +
+--R +------+
+--R 4 | 1
+--R 4a |------
+--R +------+ 4| 12 +-+ 2 2
+--R 3 | 1 \|256a - a x\|2 - x - a
+--R - 8a |------ atan(-------------------- - log(-------------------)
+--R 4| 12 +------+ +-+ 2 2
+--R \|256a 4 | 1 +-+ a x\|2 - x - a
+--R 4a |------ + x\|2
+--R 4| 12
+--R \|256a
+--R +
+--R +-+
+--R a x\|2
+--R 2atan(-------)
+--R 2 2
+--R x - a
+--R /
+--R 3 +-+
+--R 4a \|2
+--R Type: Expression Integer
+--E
+
+--S 4
+atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x)))
+--R
+--R - x + %i
+--R %i log(--------)
+--R x + %i
+--R (4) atan(x) == - ----------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer)
+--E
+
+--S 5
+dd:=atanrule cc
+--R
+--R (5)
+--R +------+ +------+2 +------+
+--R 3 | 1 8 | 1 4 +-+ | 1 2
+--R 4a |------ log(16a |------ + 4a x\|2 |------ + x )
+--R 4| 12 4| 12 4| 12
+--R \|256a \|256a \|256a
+--R +
+--R +------+ +------+2 +------+
+--R 3 | 1 8 | 1 4 +-+ | 1 2
+--R - 4a |------ log(16a |------ - 4a x\|2 |------ + x )
+--R 4| 12 4| 12 4| 12
+--R \|256a \|256a \|256a
+--R +
+--R +------+
+--R 4 | 1 +-+
+--R (- 4 + 4%i)a |------ + %i x\|2
+--R +------+ 4| 12
+--R 3 | 1 \|256a
+--R 4%i a |------ log(---------------------------------)
+--R 4| 12 +------+
+--R \|256a 4 | 1 +-+
+--R (4 + 4%i)a |------ + %i x\|2
+--R 4| 12
+--R \|256a
+--R +
+--R +------+
+--R 4 | 1 +-+
+--R (- 4 + 4%i)a |------ - %i x\|2
+--R +------+ 4| 12
+--R 3 | 1 \|256a
+--R - 4%i a |------ log(---------------------------------)
+--R 4| 12 +------+
+--R \|256a 4 | 1 +-+
+--R (4 + 4%i)a |------ - %i x\|2
+--R 4| 12
+--R \|256a
+--R +
+--R +-+ 2 2 +-+ 2 2
+--R - a x\|2 + %i x - %i a - a x\|2 - x - a
+--R - %i log(-------------------------) - log(-------------------)
+--R +-+ 2 2 +-+ 2 2
+--R a x\|2 + %i x - %i a a x\|2 - x - a
+--R /
+--R 3 +-+
+--R 4a \|2
+--R Type: Expression Complex Integer
+--E
+
+--S 6
+ee:=rootSimp dd
+--R
+--R (6)
+--R +-+
+--R +-+ 2 2 x\|2 + (1 + %i)a
+--R log(a x\|2 + x + a ) + %i log(-----------------)
+--R +-+
+--R x\|2 + (1 - %i)a
+--R +
+--R +-+ +-+ 2 2
+--R x\|2 + (- 1 - %i)a - a x\|2 + %i x - %i a
+--R - %i log(-------------------) - %i log(-------------------------)
+--R +-+ +-+ 2 2
+--R x\|2 + (- 1 + %i)a a x\|2 + %i x - %i a
+--R +
+--R +-+ 2 2
+--R - a x\|2 - x - a +-+ 2 2
+--R - log(-------------------) - log(- a x\|2 + x + a )
+--R +-+ 2 2
+--R a x\|2 - x - a
+--R /
+--R 3 +-+
+--R 4a \|2
+--R Type: Expression Complex Integer
+--E
+
+--S 7
+ff:=expandLog ee
+--R
+--R (7)
+--R +-+ 2 2 +-+ 2 2
+--R %i log(a x\|2 + %i x - %i a ) - %i log(a x\|2 - %i x + %i a )
+--R +
+--R +-+ +-+
+--R %i log(x\|2 + (1 + %i)a) - %i log(x\|2 + (1 - %i)a)
+--R +
+--R +-+ +-+
+--R %i log(x\|2 + (- 1 + %i)a) - %i log(x\|2 + (- 1 - %i)a)
+--R +
+--R (- 2 - %i)log(- 1)
+--R /
+--R 3 +-+
+--R 4a \|2
+--R Type: Expression Complex Integer
+--E
+
+--S 8
+gg:=complexNormalize ff
+--R
+--R %i %i
+--R %i log(--) - %i log(- --) + (- 2 - %i)log(- 1)
+--R 2 2
+--R (8) ----------------------------------------------
+--R 3 +-+
+--R 4a \|2
+--R Type: Expression Complex Integer
+--E
+
+--S 9 14:311 Schaums and Axiom differ by a constant
+hh:=expandLog gg
+--R
+--R %i log(%i) - %i log(- %i) + (- 2 - %i)log(- 1)
+--R (9) ----------------------------------------------
+--R 3 +-+
+--R 4a \|2
+--R Type: Expression Complex Integer
+--E
@
\section{\cite{1}:14.312~~~~~$\displaystyle
@@ -63,7 +259,7 @@
<<*>>=
)clear all
---S 2 of 14
+--S 10
aa:=integrate(x/(x^4+a^4),x)
--R
--R
@@ -77,6 +273,27 @@
--R 2a
--R Type: Union(Expression Integer,...)
--E
+
+--S 11
+bb:=1/(2*a^2)*atan(x^2/a^2)
+--R
+--R 2
+--R x
+--R atan(--)
+--R 2
+--R a
+--R (2) --------
+--R 2
+--R 2a
+--R Type: Expression Integer
+--E
+
+--S 12 14:312 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.313~~~~~$\displaystyle
@@ -89,7 +306,7 @@
<<*>>=
)clear all
---S 3 of 14
+--S 13
aa:=integrate(x^2/(x^4+a^4),x)
--R
--R
@@ -122,6 +339,213 @@
--R \|2
--R Type: Union(Expression Integer,...)
--E
+
+--S 14
+bb:=1/(4*a*sqrt(2))*log((x^2-a*x*sqrt(2)+a^2)/(x^2+a*x*sqrt(2)+a^2))-1/(2*a*sqrt(2))*atan((a*x*sqrt(2))/(x^2-a^2))
+--R
+--R +-+ 2 2 +-+
+--R +-+ - a x\|2 + x + a +-+ a x\|2
+--R \|2 log(-------------------) - 2\|2 atan(-------)
+--R +-+ 2 2 2 2
+--R a x\|2 + x + a x - a
+--R (2) -------------------------------------------------
+--R 8a
+--R Type: Expression Integer
+--E
+
+--S 15
+cc:=aa-bb
+--R
+--R (3)
+--R +-----+ +-----+3 +-----+2
+--R | 1 4 +-+ | 1 4 | 1 2
+--R - 4a |----- log(64a x\|2 |----- + 16a |----- + x )
+--R 4| 4 4| 4 4| 4
+--R \|256a \|256a \|256a
+--R +
+--R +-----+ +-----+3 +-----+2
+--R | 1 4 +-+ | 1 4 | 1 2
+--R 4a |----- log(- 64a x\|2 |----- + 16a |----- + x )
+--R 4| 4 4| 4 4| 4
+--R \|256a \|256a \|256a
+--R +
+--R +-----+3
+--R 4 | 1
+--R 64a |-----
+--R +-----+ 4| 4
+--R | 1 \|256a
+--R 8a |----- atan(---------------------)
+--R 4| 4 +-----+3
+--R \|256a 4 | 1 +-+
+--R 64a |----- - x\|2
+--R 4| 4
+--R \|256a
+--R +
+--R +-----+3
+--R 4 | 1
+--R 64a |-----
+--R +-----+ 4| 4 +-+ 2 2
+--R | 1 \|256a - a x\|2 + x + a
+--R - 8a |----- atan(--------------------- - log(-------------------)
+--R 4| 4 +-----+3 +-+ 2 2
+--R \|256a 4 | 1 +-+ a x\|2 + x + a
+--R 64a |----- + x\|2
+--R 4| 4
+--R \|256a
+--R +
+--R +-+
+--R a x\|2
+--R 2atan(-------)
+--R 2 2
+--R x - a
+--R /
+--R +-+
+--R 4a\|2
+--R Type: Expression Integer
+--E
+
+--S 16
+atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x)))
+--R
+--R - x + %i
+--R %i log(--------)
+--R x + %i
+--R (4) atan(x) == - ----------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer)
+--E
+
+--S 17
+dd:=atanrule cc
+--R
+--R (5)
+--R +-----+ +-----+3 +-----+2
+--R | 1 4 +-+ | 1 4 | 1 2
+--R - 4a |----- log(64a x\|2 |----- + 16a |----- + x )
+--R 4| 4 4| 4 4| 4
+--R \|256a \|256a \|256a
+--R +
+--R +-----+3
+--R 4 | 1 +-+
+--R (- 64 + 64%i)a |----- + %i x\|2
+--R +-----+ 4| 4
+--R | 1 \|256a
+--R 4%i a |----- log(-----------------------------------)
+--R 4| 4 +-----+3
+--R \|256a 4 | 1 +-+
+--R (64 + 64%i)a |----- + %i x\|2
+--R 4| 4
+--R \|256a
+--R +
+--R +-----+3
+--R 4 | 1 +-+
+--R (- 64 + 64%i)a |----- - %i x\|2
+--R +-----+ 4| 4
+--R | 1 \|256a
+--R - 4%i a |----- log(-----------------------------------)
+--R 4| 4 +-----+3
+--R \|256a 4 | 1 +-+
+--R (64 + 64%i)a |----- - %i x\|2
+--R 4| 4
+--R \|256a
+--R +
+--R +-----+ +-----+3 +-----+2
+--R | 1 4 +-+ | 1 4 | 1 2
+--R 4a |----- log(- 64a x\|2 |----- + 16a |----- + x )
+--R 4| 4 4| 4 4| 4
+--R \|256a \|256a \|256a
+--R +
+--R +-+ 2 2 +-+ 2 2
+--R - a x\|2 + x + a - a x\|2 + %i x - %i a
+--R - log(-------------------) - %i log(-------------------------)
+--R +-+ 2 2 +-+ 2 2
+--R a x\|2 + x + a a x\|2 + %i x - %i a
+--R /
+--R +-+
+--R 4a\|2
+--R Type: Expression Complex Integer
+--E
+
+--S 18
+ee:=expandLog dd
+--R
+--R (6)
+--R +-----+ +-----+3 +-----+2
+--R | 1 4 +-+ | 1 4 | 1 2
+--R - 4a |----- log(64a x\|2 |----- + 16a |----- + x )
+--R 4| 4 4| 4 4| 4
+--R \|256a \|256a \|256a
+--R +
+--R +-----+ +-----+3 +-----+2
+--R | 1 4 +-+ | 1 4 | 1 2
+--R 4a |----- log(64a x\|2 |----- - 16a |----- - x )
+--R 4| 4 4| 4 4| 4
+--R \|256a \|256a \|256a
+--R +
+--R +-----+ +-----+3
+--R | 1 4 | 1 +-+
+--R 4%i a |----- log((64 + 64%i)a |----- + x\|2 )
+--R 4| 4 4| 4
+--R \|256a \|256a
+--R +
+--R +-----+ +-----+3
+--R | 1 4 | 1 +-+
+--R - 4%i a |----- log((64 + 64%i)a |----- + %i x\|2 )
+--R 4| 4 4| 4
+--R \|256a \|256a
+--R +
+--R +-----+ +-----+3
+--R | 1 4 | 1 +-+
+--R 4%i a |----- log((64 + 64%i)a |----- - %i x\|2 )
+--R 4| 4 4| 4
+--R \|256a \|256a
+--R +
+--R +-----+ +-----+3 +-----+
+--R | 1 4 | 1 +-+ | 1
+--R - 4%i a |----- log((64 + 64%i)a |----- - x\|2 + 4a log(- 1) |-----
+--R 4| 4 4| 4 4| 4
+--R \|256a \|256a \|256a
+--R +
+--R +-+ 2 2 +-+ 2 2
+--R log(a x\|2 + x + a ) + %i log(a x\|2 + %i x - %i a )
+--R +
+--R +-+ 2 2 +-+ 2 2
+--R - %i log(a x\|2 - %i x + %i a ) - log(a x\|2 - x - a )
+--R +
+--R (- 1 - %i)log(- 1)
+--R /
+--R +-+
+--R 4a\|2
+--R Type: Expression Complex Integer
+--E
+
+--S 19
+ff:=rootSimp ee
+--R
+--R (7)
+--R +-+ 2 2 +-+ 2 2
+--R %i log(a x\|2 + %i x - %i a ) - %i log(a x\|2 - %i x + %i a )
+--R +
+--R +-+ +-+
+--R %i log(x\|2 + (1 + %i)a) - %i log(%i x\|2 + (1 + %i)a)
+--R +
+--R +-+ +-+
+--R %i log(- %i x\|2 + (1 + %i)a) - %i log(- x\|2 + (1 + %i)a) - %i log(- 1)
+--R /
+--R +-+
+--R 4a\|2
+--R Type: Expression Complex Integer
+--E
+
+--S 20 14:313 Schaums and Axiom differ by a constant
+gg:=complexNormalize ff
+--R
+--R %i log(2) - %i log(- 1) - %i log(- 2)
+--R (8) -------------------------------------
+--R +-+
+--R 4a\|2
+--R Type: Expression Complex Integer
+--E
@
\section{\cite{1}:14.314~~~~~$\displaystyle
@@ -132,7 +556,7 @@
<<*>>=
)clear all
---S 4 of 14
+--S 21
aa:=integrate(x^3/(x^4+a^4),x)
--R
--R
@@ -141,7 +565,24 @@
--R (1) ------------
--R 4
--R Type: Union(Expression Integer,...)
+--E
+
+--S 22
+bb:=1/4*log(x^4+a^4)
+--R
+--R 4 4
+--R log(x + a )
+--R (2) ------------
+--R 4
+--R Type: Expression Integer
--E
+
+--S 23 14:314 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.315~~~~~$\displaystyle
@@ -152,7 +593,7 @@
<<*>>=
)clear all
---S 5 of 14
+--S 24
aa:=integrate(1/(x*(x^4+a^4)),x)
--R
--R
@@ -163,6 +604,41 @@
--R 4a
--R Type: Union(Expression Integer,...)
--E
+
+--S 25
+bb:=1/(4*a^4)*log(x^4/(x^4+a^4))
+--R
+--R 4
+--R x
+--R log(-------)
+--R 4 4
+--R x + a
+--R (2) ------------
+--R 4
+--R 4a
+--R Type: Expression Integer
+--E
+
+--S 26
+cc:=aa-bb
+--R
+--R 4
+--R 4 4 x
+--R - log(x + a ) + 4log(x) - log(-------)
+--R 4 4
+--R x + a
+--R (3) ---------------------------------------
+--R 4
+--R 4a
+--R Type: Expression Integer
+--E
+
+--S 27 14:315 Schaums and Axiom agree
+dd:=expandLog cc
+--R
+--R (4) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.316~~~~~$\displaystyle
@@ -175,7 +651,7 @@
<<*>>=
)clear all
---S 6 of 14
+--S 28
aa:=integrate(1/(x^2*(x^4+a^4)),x)
--R
--R
@@ -220,6 +696,220 @@
--R a x\|2
--R Type: Union(Expression Integer,...)
--E
+
+--S 29
+bb:=-1/(a^4*x)-1/(4*a^5*sqrt(2))*log((x^2-a*x*sqrt(2)+a^2)/(x^2+a*x*sqrt(2)+a^2))+1/(2*a^5*sqrt(2))*atan((a*x*sqrt(2))/(x^2-a^2))
+--R
+--R +-+ 2 2 +-+
+--R +-+ - a x\|2 + x + a +-+ a x\|2
+--R - x\|2 log(-------------------) + 2x\|2 atan(-------) - 8a
+--R +-+ 2 2 2 2
+--R a x\|2 + x + a x - a
+--R (2) ----------------------------------------------------------
+--R 5
+--R 8a x
+--R Type: Expression Integer
+--E
+
+--S 30
+cc:=aa-bb
+--R
+--R (3)
+--R +------+ +------+3 +------+2
+--R 5 | 1 16 +-+ | 1 12 | 1 2
+--R 4a |------ log(64a x\|2 |------ + 16a |------ + x )
+--R 4| 20 4| 20 4| 20
+--R \|256a \|256a \|256a
+--R +
+--R +------+ +------+3 +------+2
+--R 5 | 1 16 +-+ | 1 12 | 1 2
+--R - 4a |------ log(- 64a x\|2 |------ + 16a |------ + x )
+--R 4| 20 4| 20 4| 20
+--R \|256a \|256a \|256a
+--R +
+--R +------+3
+--R 16 | 1
+--R 64a |------
+--R +------+ 4| 20
+--R 5 | 1 \|256a
+--R - 8a |------ atan(-----------------------)
+--R 4| 20 +------+3
+--R \|256a 16 | 1 +-+
+--R 64a |------ - x\|2
+--R 4| 20
+--R \|256a
+--R +
+--R +------+3
+--R 16 | 1
+--R 64a |------
+--R +------+ 4| 20 +-+ 2 2
+--R 5 | 1 \|256a - a x\|2 + x + a
+--R 8a |------ atan(----------------------- + log(-------------------)
+--R 4| 20 +------+3 +-+ 2 2
+--R \|256a 16 | 1 +-+ a x\|2 + x + a
+--R 64a |------ + x\|2
+--R 4| 20
+--R \|256a
+--R +
+--R +-+
+--R a x\|2
+--R - 2atan(-------)
+--R 2 2
+--R x - a
+--R /
+--R 5 +-+
+--R 4a \|2
+--R Type: Expression Integer
+--E
+
+--S 31
+atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x)))
+--R
+--R - x + %i
+--R %i log(--------)
+--R x + %i
+--R (4) atan(x) == - ----------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer)
+--E
+
+--S 32
+dd:=atanrule cc
+--R
+--R (5)
+--R +------+ +------+3 +------+2
+--R 5 | 1 16 +-+ | 1 12 | 1 2
+--R 4a |------ log(64a x\|2 |------ + 16a |------ + x )
+--R 4| 20 4| 20 4| 20
+--R \|256a \|256a \|256a
+--R +
+--R +------+3
+--R 16 | 1 +-+
+--R (- 64 + 64%i)a |------ + %i x\|2
+--R +------+ 4| 20
+--R 5 | 1 \|256a
+--R - 4%i a |------ log(-------------------------------------)
+--R 4| 20 +------+3
+--R \|256a 16 | 1 +-+
+--R (64 + 64%i)a |------ + %i x\|2
+--R 4| 20
+--R \|256a
+--R +
+--R +------+3
+--R 16 | 1 +-+
+--R (- 64 + 64%i)a |------ - %i x\|2
+--R +------+ 4| 20
+--R 5 | 1 \|256a
+--R 4%i a |------ log(-------------------------------------)
+--R 4| 20 +------+3
+--R \|256a 16 | 1 +-+
+--R (64 + 64%i)a |------ - %i x\|2
+--R 4| 20
+--R \|256a
+--R +
+--R +------+ +------+3 +------+2
+--R 5 | 1 16 +-+ | 1 12 | 1 2
+--R - 4a |------ log(- 64a x\|2 |------ + 16a |------ + x )
+--R 4| 20 4| 20 4| 20
+--R \|256a \|256a \|256a
+--R +
+--R +-+ 2 2 +-+ 2 2
+--R - a x\|2 + x + a - a x\|2 + %i x - %i a
+--R log(-------------------) + %i log(-------------------------)
+--R +-+ 2 2 +-+ 2 2
+--R a x\|2 + x + a a x\|2 + %i x - %i a
+--R /
+--R 5 +-+
+--R 4a \|2
+--R Type: Expression Complex Integer
+--E
+
+--S 33
+ee:=expandLog dd
+--R
+--R (6)
+--R +------+ +------+3 +------+2
+--R 5 | 1 16 +-+ | 1 12 | 1 2
+--R 4a |------ log(64a x\|2 |------ + 16a |------ + x )
+--R 4| 20 4| 20 4| 20
+--R \|256a \|256a \|256a
+--R +
+--R +------+ +------+3 +------+2
+--R 5 | 1 16 +-+ | 1 12 | 1 2
+--R - 4a |------ log(64a x\|2 |------ - 16a |------ - x )
+--R 4| 20 4| 20 4| 20
+--R \|256a \|256a \|256a
+--R +
+--R +------+ +------+3
+--R 5 | 1 16 | 1 +-+
+--R - 4%i a |------ log((64 + 64%i)a |------ + x\|2 )
+--R 4| 20 4| 20
+--R \|256a \|256a
+--R +
+--R +------+ +------+3
+--R 5 | 1 16 | 1 +-+
+--R 4%i a |------ log((64 + 64%i)a |------ + %i x\|2 )
+--R 4| 20 4| 20
+--R \|256a \|256a
+--R +
+--R +------+ +------+3
+--R 5 | 1 16 | 1 +-+
+--R - 4%i a |------ log((64 + 64%i)a |------ - %i x\|2 )
+--R 4| 20 4| 20
+--R \|256a \|256a
+--R +
+--R +------+ +------+3
+--R 5 | 1 16 | 1 +-+
+--R 4%i a |------ log((64 + 64%i)a |------ - x\|2 )
+--R 4| 20 4| 20
+--R \|256a \|256a
+--R +
+--R +------+
+--R 5 | 1 +-+ 2 2
+--R - 4a log(- 1) |------ - log(a x\|2 + x + a )
+--R 4| 20
+--R \|256a
+--R +
+--R +-+ 2 2 +-+ 2 2
+--R - %i log(a x\|2 + %i x - %i a ) + %i log(a x\|2 - %i x + %i a )
+--R +
+--R +-+ 2 2
+--R log(a x\|2 - x - a ) + (1 + %i)log(- 1)
+--R /
+--R 5 +-+
+--R 4a \|2
+--R Type: Expression Complex Integer
+--E
+
+--S 34
+ff:=rootSimp ee
+--R
+--R (7)
+--R +-+ 2 2 +-+ 2 2
+--R - %i log(a x\|2 + %i x - %i a ) + %i log(a x\|2 - %i x + %i a )
+--R +
+--R +-+ +-+
+--R - %i log(x\|2 + (1 + %i)a) + %i log(%i x\|2 + (1 + %i)a)
+--R +
+--R +-+ +-+
+--R - %i log(- %i x\|2 + (1 + %i)a) + %i log(- x\|2 + (1 + %i)a)
+--R +
+--R %i log(- 1)
+--R /
+--R 5 +-+
+--R 4a \|2
+--R Type: Expression Complex Integer
+--E
+
+--S 35 14:316 Schaums and Axiom differ by a constant
+gg:=complexNormalize ff
+--R
+--R - %i log(2) + %i log(- 1) + %i log(- 2)
+--R (8) ---------------------------------------
+--R 5 +-+
+--R 4a \|2
+--R Type: Expression Complex Integer
+--E
@
\section{\cite{1}:14.317~~~~~$\displaystyle
@@ -230,7 +920,7 @@
<<*>>=
)clear all
---S 7 of 14
+--S 36
aa:=integrate(1/(x^3*(x^4+a^4)),x)
--R
--R
@@ -244,6 +934,27 @@
--R 2a x
--R Type: Union(Expression Integer,...)
--E
+
+--S 37
+bb:=-1/(2*a^4*x^2)-1/(2*a^6)*atan(x^2/a^2)
+--R
+--R 2
+--R 2 x 2
+--R - x atan(--) - a
+--R 2
+--R a
+--R (2) -----------------
+--R 6 2
+--R 2a x
+--R Type: Expression Integer
+--E
+
+--S 38 14:317 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.318~~~~~$\displaystyle
@@ -255,7 +966,7 @@
<<*>>=
)clear all
---S 8 of 14
+--S 39
aa:=integrate(1/(x^4-a^4),x)
--R
--R
@@ -267,6 +978,37 @@
--R 4a
--R Type: Union(Expression Integer,...)
--E
+
+--S 40
+bb:=1/(4*a^3)*log((x-a)/(x+a))-1/(2*a^3)*atan(x/a)
+--R
+--R x - a x
+--R log(-----) - 2atan(-)
+--R x + a a
+--R (2) ---------------------
+--R 3
+--R 4a
+--R Type: Expression Integer
+--E
+
+--S 41
+cc:=aa-bb
+--R
+--R x - a
+--R - log(x + a) + log(x - a) - log(-----)
+--R x + a
+--R (3) --------------------------------------
+--R 3
+--R 4a
+--R Type: Expression Integer
+--E
+
+--S 42 14:318 Schaums and Axiom agree
+dd:=expandLog cc
+--R
+--R (4) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.319~~~~~$\displaystyle
@@ -277,7 +1019,7 @@
<<*>>=
)clear all
---S 9 of 14
+--S 43
aa:=integrate(x/(x^4-a^4),x)
--R
--R
@@ -288,6 +1030,41 @@
--R 4a
--R Type: Union(Expression Integer,...)
--E
+
+--S 44
+bb:=1/(4*a^2)*log((x^2-a^2)/(x^2+a^2))
+--R
+--R 2 2
+--R x - a
+--R log(-------)
+--R 2 2
+--R x + a
+--R (2) ------------
+--R 2
+--R 4a
+--R Type: Expression Integer
+--E
+
+--S 45
+cc:=aa-bb
+--R
+--R 2 2
+--R 2 2 2 2 x - a
+--R - log(x + a ) + log(x - a ) - log(-------)
+--R 2 2
+--R x + a
+--R (3) --------------------------------------------
+--R 2
+--R 4a
+--R Type: Expression Integer
+--E
+
+--S 46 14:319 Schaums and Axiom agree
+dd:=expandLog cc
+--R
+--R (4) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.320~~~~~$\displaystyle
@@ -299,7 +1076,7 @@
<<*>>=
)clear all
---S 10 of 14
+--S 47
aa:=integrate(x^2/(x^4-a^4),x)
--R
--R
@@ -309,7 +1086,36 @@
--R (1) ------------------------------------
--R 4a
--R Type: Union(Expression Integer,...)
+--E
+
+--S 48
+bb:=1/(4*a)*log((x-a)/(x+a))+1/(2*a)*atan(x/a)
+--R
+--R x - a x
+--R log(-----) + 2atan(-)
+--R x + a a
+--R (2) ---------------------
+--R 4a
+--R Type: Expression Integer
--E
+
+--S 49
+cc:=aa-bb
+--R
+--R x - a
+--R - log(x + a) + log(x - a) - log(-----)
+--R x + a
+--R (3) --------------------------------------
+--R 4a
+--R Type: Expression Integer
+--E
+
+--S 50 14:320 Schaums and Axiom agree
+dd:=expandLog cc
+--R
+--R (4) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.321~~~~~$\displaystyle
@@ -320,7 +1126,7 @@
<<*>>=
)clear all
---S 11 of 14
+--S 51
aa:=integrate(x^3/(x^4-a^4),x)
--R
--R
@@ -330,6 +1136,23 @@
--R 4
--R Type: Union(Expression Integer,...)
--E
+
+--S 52
+bb:=1/4*log(x^4-a^4)
+--R
+--R 4 4
+--R log(x - a )
+--R (2) ------------
+--R 4
+--R Type: Expression Integer
+--E
+
+--S 53 14:321 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.322~~~~~$\displaystyle
@@ -340,7 +1163,7 @@
<<*>>=
)clear all
---S 12 of 14
+--S 54
aa:=integrate(1/(x*(x^4-a^4)),x)
--R
--R
@@ -350,7 +1173,42 @@
--R 4
--R 4a
--R Type: Union(Expression Integer,...)
+--E
+
+--S 55
+bb:=1/(4*a^4)*log((x^4-a^4)/x^4)
+--R
+--R 4 4
+--R x - a
+--R log(-------)
+--R 4
+--R x
+--R (2) ------------
+--R 4
+--R 4a
+--R Type: Expression Integer
--E
+
+--S 56
+cc:=aa-bb
+--R
+--R 4 4
+--R 4 4 x - a
+--R log(x - a ) - 4log(x) - log(-------)
+--R 4
+--R x
+--R (3) -------------------------------------
+--R 4
+--R 4a
+--R Type: Expression Integer
+--E
+
+--S 57 14:322 Schaums and Axiom agree
+dd:=expandLog cc
+--R
+--R (4) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.323~~~~~$\displaystyle
@@ -362,7 +1220,7 @@
<<*>>=
)clear all
---S 13 of 14
+--S 58
aa:=integrate(1/(x^2*(x^4-a^4)),x)
--R
--R
@@ -374,6 +1232,37 @@
--R 4a x
--R Type: Union(Expression Integer,...)
--E
+
+--S 59
+bb:=1/(a^4*x)+1/(4*a^5)*log((x-a)/(x+a))+1/(2*a^5)*atan(x/a)
+--R
+--R x - a x
+--R x log(-----) + 2x atan(-) + 4a
+--R x + a a
+--R (2) ------------------------------
+--R 5
+--R 4a x
+--R Type: Expression Integer
+--E
+
+--S 60
+cc:=aa-bb
+--R
+--R x - a
+--R - log(x + a) + log(x - a) - log(-----)
+--R x + a
+--R (3) --------------------------------------
+--R 5
+--R 4a
+--R Type: Expression Integer
+--E
+
+--S 61 14:323 Schaums and Axiom agree
+dd:=expandLog cc
+--R
+--R (4) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.324~~~~~$\displaystyle
@@ -384,7 +1273,7 @@
<<*>>=
)clear all
---S 14 of 14
+--S 62
aa:=integrate(1/(x^3*(x^4-a^4)),x)
--R
--R
@@ -396,6 +1285,40 @@
--R Type: Union(Expression Integer,...)
--E
+--S 63
+bb:=1/(2*a^4*x^2)+1/(4*a^6)*log((x^2-a^2)/(x^2+a^2))
+--R
+--R 2 2
+--R 2 x - a 2
+--R x log(-------) + 2a
+--R 2 2
+--R x + a
+--R (2) --------------------
+--R 6 2
+--R 4a x
+--R Type: Expression Integer
+--E
+
+--S 64
+cc:=aa-bb
+--R
+--R 2 2
+--R 2 2 2 2 x - a
+--R - log(x + a ) + log(x - a ) - log(-------)
+--R 2 2
+--R x + a
+--R (3) --------------------------------------------
+--R 6
+--R 4a
+--R Type: Expression Integer
+--E
+
+--S 65 14:324 Schaums and Axiom agree
+dd:=expandLog cc
+--R
+--R (4) 0
+--R Type: Expression Integer
+--E
)spool
)lisp (bye)
@
Modified: trunk/axiom/src/input/schaum16.input.pamphlet
===================================================================
--- trunk/axiom/src/input/schaum16.input.pamphlet 2008-04-21 07:17:21 UTC (rev 860)
+++ trunk/axiom/src/input/schaum16.input.pamphlet 2008-04-22 05:50:59 UTC (rev 861)
@@ -18,7 +18,7 @@
)set message auto off
)clear all
---S 1 of 14
+--S 1
aa:=integrate(1/x*(x^n+a^n),x)
--R
--R
@@ -28,6 +28,79 @@
--R n
--R Type: Union(Expression Integer,...)
--E
+
+--S 2
+bb:=1/(n*a^n)*log(x^n/(x^n+a^n))
+--R
+--R n
+--R x
+--R log(-------)
+--R n n
+--R x + a
+--R (2) ------------
+--R n
+--R n a
+--R Type: Expression Integer
+--E
+
+--S 3
+cc:=aa-bb
+--R
+--R n
+--R x n n log(x) n 2
+--R - log(-------) + a %e + n log(x)(a )
+--R n n
+--R x + a
+--R (3) ---------------------------------------------
+--R n
+--R n a
+--R Type: Expression Integer
+--E
+
+--S 4
+dd:=expandLog cc
+--R
+--R n n n n n log(x) 2n
+--R log(x + a ) - log(x ) + a %e + n log(x)a
+--R (4) ---------------------------------------------------
+--R n
+--R n a
+--R Type: Expression Integer
+--E
+
+--S 5
+ee:=complexNormalize dd
+--R
+--R (5)
+--R n log(x) n log(a) n log(a) n log(x)
+--R log(%e + %e ) + %e %e
+--R +
+--R n log(a) 2
+--R n log(x)(%e ) - n log(x)
+--R /
+--R n log(a)
+--R n %e
+--R Type: Expression Integer
+--E
+
+--S 6
+explog:=rule(%e^(n*log(x)) == x^n)
+--R
+--R n log(x) n
+--R (6) %e == x
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 7 14:325 Axiom cannot simplify this expression
+ff:=explog ee
+--R
+--R n n n n 2n
+--R log(x + a ) + a x + n log(x)a - n log(x)
+--R (7) --------------------------------------------
+--R n
+--R n a
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.326~~~~~$\displaystyle
@@ -38,7 +111,7 @@
<<*>>=
)clear all
---S 2 of 14
+--S 8
aa:=integrate(x^(n-1)/(x^n+a^n),x)
--R
--R
@@ -48,6 +121,41 @@
--R n
--R Type: Union(Expression Integer,...)
--E
+
+--S 9
+bb:=1/n*log(x^n+a^n)
+--R
+--R n n
+--R log(x + a )
+--R (2) ------------
+--R n
+--R Type: Expression Integer
+--E
+
+--S 10
+cc:=aa-bb
+--R
+--R n log(x) n n n
+--R log(%e + a ) - log(x + a )
+--R (3) -----------------------------------
+--R n
+--R Type: Expression Integer
+--E
+
+--S 11
+explog:=rule(%e^(n*log(x)) == x^n)
+--R
+--R n log(x) n
+--R (4) %e == x
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 12 14:326 Schaums and Axiom agree
+dd:=explog cc
+--R
+--R (5) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.327~~~~~$\displaystyle
@@ -59,7 +167,7 @@
<<*>>=
)clear all
---S 3 of 14
+--S 13 14:327 Axiom cannot compute this integral
aa:=integrate(x^m/(x^n+a^n)^r,x)
--R
--R
@@ -81,7 +189,7 @@
<<*>>=
)clear all
---S 4 of 14
+--S 14 14:328 Axiom cannot compute this integral
aa:=integrate(1/(x^m*(x^n+a^n)^r),x)
--R
--R
@@ -103,7 +211,7 @@
<<*>>=
)clear all
---S 5 of 14
+--S 15
aa:=integrate(1/(x*sqrt(x^n+a^n)),x)
--R
--R
@@ -130,6 +238,114 @@
--R n\|- a
--R Type: Union(List Expression Integer,...)
--E
+
+--S 16
+bb:=1/(n*sqrt(a^n))*log((sqrt(x^n+a^n)-sqrt(a^n))/(sqrt(x^n+a^n)+sqrt(a^n)))
+--R
+--R +-------+ +--+
+--R | n n | n
+--R \|x + a - \|a
+--R log(-----------------...
[truncated message content] |
|
From: <gi...@ax...> - 2008-04-22 05:50:09
|
changelog | 3 +
src/input/schaum14.input.pamphlet | 342 +++++++++++++-
src/input/schaum15.input.pamphlet | 951 ++++++++++++++++++++++++++++++++++++-
src/input/schaum16.input.pamphlet | 409 +++++++++++++++-
4 files changed, 1664 insertions(+), 41 deletions(-)
New commits:
commit 7cd02a4b67c2fb90491b8a2bab541868d792a808
Author: Tim Daly <root@localhost.localdomain>
Date: Thu Apr 3 17:55:17 2008 -0400
20080421 tpd src/input/schaum16.input show Schaums-Axiom equivalence
20080421 tpd src/input/schaum15.input show Schaums-Axiom equivalence
20080421 tpd src/input/schaum14.input show Schaums-Axiom equivalence
|
|
From: <da...@us...> - 2008-04-21 07:17:14
|
Revision: 860
http://axiom.svn.sourceforge.net/axiom/?rev=860&view=rev
Author: daly
Date: 2008-04-21 00:17:21 -0700 (Mon, 21 Apr 2008)
Log Message:
-----------
20080420 tpd src/input/schaum13.input show Schaums-Axiom equivalence
Modified Paths:
--------------
trunk/axiom/changelog
trunk/axiom/src/input/schaum13.input.pamphlet
Modified: trunk/axiom/changelog
===================================================================
--- trunk/axiom/changelog 2008-04-20 06:56:05 UTC (rev 859)
+++ trunk/axiom/changelog 2008-04-21 07:17:21 UTC (rev 860)
@@ -1,3 +1,4 @@
+20080420 tpd src/input/schaum13.input show Schaums-Axiom equivalence
20080419 tpd src/input/schaum12.input show Schaums-Axiom equivalence
20080418 tpd src/input/schaum11.input show Schaums-Axiom equivalence
20080418 tpd src/input/schaum10.input fix Schaums-Axiom equivalence
Modified: trunk/axiom/src/input/schaum13.input.pamphlet
===================================================================
--- trunk/axiom/src/input/schaum13.input.pamphlet 2008-04-20 06:56:05 UTC (rev 859)
+++ trunk/axiom/src/input/schaum13.input.pamphlet 2008-04-21 07:17:21 UTC (rev 860)
@@ -29,7 +29,7 @@
)set message auto off
)clear all
---S 1 of 19
+--S 1
aa:=integrate(1/sqrt(a*x^2+b*x+c),x)
--R
--R
@@ -60,6 +60,183 @@
--R \|- a
--R Type: Union(List Expression Integer,...)
--E
+
+--S 2
+bb1:=1/sqrt(a)*log(2*sqrt(a)*sqrt(a*x^2+b*x+c)*2*a*x+b)
+--R
+--R +--------------+
+--R +-+ | 2
+--R log(4a x\|a \|a x + b x + c + b)
+--R (2) ----------------------------------
+--R +-+
+--R \|a
+--R Type: Expression Integer
+--E
+
+--S 3
+bb2:=-1/sqrt(-a)*asin((2*a*x+b)/sqrt(b^2-4*a*c))
+--R
+--R 2a x + b
+--R asin(--------------)
+--R +-----------+
+--R | 2
+--R \|- 4a c + b
+--R (3) - --------------------
+--R +---+
+--R \|- a
+--R Type: Expression Integer
+--E
+
+--S 4
+bb3:=1/sqrt(a)*asinh((2*a*x+b)/sqrt(4*a*c-b^2))
+--R
+--R 2a x + b
+--R asinh(------------)
+--R +---------+
+--R | 2
+--R \|4a c - b
+--R (4) -------------------
+--R +-+
+--R \|a
+--R Type: Expression Integer
+--E
+
+--S 5
+cc1:=bb1-aa.1
+--R
+--R (5)
+--R +--------------+
+--R +-+ | 2
+--R log(4a x\|a \|a x + b x + c + b)
+--R +
+--R -
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R /
+--R +-+
+--R \|a
+--R Type: Expression Integer
+--E
+
+--S 6
+cc2:=bb1-aa.2
+--R
+--R (6)
+--R +--------------+
+--R +---+ +-+ | 2
+--R \|- a log(4a x\|a \|a x + b x + c + b)
+--R +
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R +-+ \|- a \|a x + b x + c - \|- a \|c
+--R - 2\|a atan(------------------------------------)
+--R a x
+--R /
+--R +---+ +-+
+--R \|- a \|a
+--R Type: Expression Integer
+--E
+
+--S 7
+cc3:=bb2-aa.1
+--R
+--R (7)
+--R -
+--R +---+
+--R \|- a
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R +-+ 2a x + b
+--R - \|a asin(--------------)
+--R +-----------+
+--R | 2
+--R \|- 4a c + b
+--R /
+--R +---+ +-+
+--R \|- a \|a
+--R Type: Expression Integer
+--E
+
+--S 8
+cc4:=bb2-aa.2
+--R
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R \|- a \|a x + b x + c - \|- a \|c 2a x + b
+--R - 2atan(------------------------------------) - asin(--------------)
+--R a x +-----------+
+--R | 2
+--R \|- 4a c + b
+--R (8) --------------------------------------------------------------------
+--R +---+
+--R \|- a
+--R Type: Expression Integer
+--E
+
+--S 9
+cc5:=bb3-aa.1
+--R
+--R (9)
+--R -
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R 2a x + b
+--R asinh(------------)
+--R +---------+
+--R | 2
+--R \|4a c - b
+--R /
+--R +-+
+--R \|a
+--R Type: Expression Integer
+--E
+
+--S 10 14:280 Axiom cannot simplify these expressions
+cc6:=bb3-aa.2
+--R
+--R (10)
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R +-+ \|- a \|a x + b x + c - \|- a \|c +---+ 2a x + b
+--R - 2\|a atan(------------------------------------) + \|- a asinh(------------)
+--R a x +---------+
+--R | 2
+--R \|4a c - b
+--R -----------------------------------------------------------------------------
+--R +---+ +-+
+--R \|- a \|a
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.281~~~~~$\displaystyle
@@ -70,7 +247,7 @@
<<*>>=
)clear all
---S 2 of 19
+--S 11
aa:=integrate(x/sqrt(a*x^2+b*x+c),x)
--R
--R
@@ -121,6 +298,220 @@
--R ]
--R Type: Union(List Expression Integer,...)
--E
+
+--S 12
+t1:=integrate(1/sqrt(a*x^2+b*x+c),x)
+--R
+--R (2)
+--R [
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R /
+--R +-+
+--R \|a
+--R ,
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R \|- a \|a x + b x + c - \|- a \|c
+--R 2atan(------------------------------------)
+--R a x
+--R -------------------------------------------]
+--R +---+
+--R \|- a
+--R Type: Union(List Expression Integer,...)
+--E
+
+--S 13
+bb1:=sqrt(a*x^2+b*x+c)/a-b/(2*a)*t1.1
+--R
+--R (3)
+--R -
+--R b
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R +--------------+
+--R +-+ | 2
+--R 2\|a \|a x + b x + c
+--R /
+--R +-+
+--R 2a\|a
+--R Type: Expression Integer
+--E
+
+--S 14
+bb2:=sqrt(a*x^2+b*x+c)/a-b/(2*a)*t1.2
+--R
+--R (4)
+--R +--------------+
+--R +---+ | 2 +---+ +-+ +--------------+
+--R \|- a \|a x + b x + c - \|- a \|c +---+ | 2
+--R - b atan(------------------------------------) + \|- a \|a x + b x + c
+--R a x
+--R ------------------------------------------------------------------------
+--R +---+
+--R a\|- a
+--R Type: Expression Integer
+--E
+
+--S 15
+cc1:=bb1-aa.1
+--R
+--R (5)
+--R +--------------+
+--R +-+ | 2 2
+--R (- 2b\|c \|a x + b x + c + b x + 2b c)
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c + 2a x)\|a x + b x + c - 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R +--------------+
+--R +-+ | 2 2
+--R (- 2b\|c \|a x + b x + c + b x + 2b c)
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R +--------------+
+--R +-+ | 2 +-+ +-+
+--R - 4c\|a \|a x + b x + c + (2b x + 4c)\|a \|c
+--R /
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R 4a\|a \|c \|a x + b x + c + (- 2a b x - 4a c)\|a
+--R Type: Expression Integer
+--E
+
+--S 16
+cc2:=bb1-aa.2
+--R
+--R (6)
+--R +--------------+
+--R +---+ +-+ | 2 2 +---+
+--R (- 2b\|- a \|c \|a x + b x + c + (b x + 2b c)\|- a )
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R +--------------+
+--R +-+ +-+ | 2 2 +-+
+--R (4b\|a \|c \|a x + b x + c + (- 2b x - 4b c)\|a )
+--R *
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R \|- a \|a x + b x + c - \|- a \|c
+--R atan(------------------------------------)
+--R a x
+--R +
+--R +--------------+
+--R +---+ +-+ | 2 +---+ +-+ +-+
+--R - 4c\|- a \|a \|a x + b x + c + (2b x + 4c)\|- a \|a \|c
+--R /
+--R +--------------+
+--R +---+ +-+ +-+ | 2 +---+ +-+
+--R 4a\|- a \|a \|c \|a x + b x + c + (- 2a b x - 4a c)\|- a \|a
+--R Type: Expression Integer
+--E
+
+--S 17
+cc3:=bb2-aa.1
+--R
+--R (7)
+--R +--------------+
+--R +---+ +-+ | 2 2 +---+
+--R (- 2b\|- a \|c \|a x + b x + c + (b x + 2b c)\|- a )
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c + 2a x)\|a x + b x + c - 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R +--------------+
+--R +-+ +-+ | 2 2 +-+
+--R (- 4b\|a \|c \|a x + b x + c + (2b x + 4b c)\|a )
+--R *
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R \|- a \|a x + b x + c - \|- a \|c
+--R atan(------------------------------------)
+--R a x
+--R +
+--R +--------------+
+--R +---+ +-+ | 2 +---+ +-+ +-+
+--R - 4c\|- a \|a \|a x + b x + c + (2b x + 4c)\|- a \|a \|c
+--R /
+--R +--------------+
+--R +---+ +-+ +-+ | 2 +---+ +-+
+--R 4a\|- a \|a \|c \|a x + b x + c + (- 2a b x - 4a c)\|- a \|a
+--R Type: Expression Integer
+--E
+
+--S 18 14:281 Axiom cannot simplify these expressions
+cc4:=bb2-aa.2
+--R
+--R +--------------+
+--R | 2 +-+
+--R - 2c\|a x + b x + c + (b x + 2c)\|c
+--R (8) --------------------------------------
+--R +--------------+
+--R +-+ | 2
+--R 2a\|c \|a x + b x + c - a b x - 2a c
+--R Type: Expression Integer
+--E
+
@
\section{\cite{1}:14.282~~~~~$\displaystyle
@@ -132,7 +523,7 @@
<<*>>=
)clear all
---S 3 of 19
+--S 19
aa:=integrate(x^2/(a*x^2+b*x+c),x)
--R
--R
@@ -177,7 +568,247 @@
--R 2a \|4a c - b
--R ]
--R Type: Union(List Expression Integer,...)
---E
+--E
+
+--S 20
+t1:=integrate(1/sqrt(a*x^2+b*x+c),x)
+--R
+--R (2)
+--R [
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R /
+--R +-+
+--R \|a
+--R ,
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R \|- a \|a x + b x + c - \|- a \|c
+--R 2atan(------------------------------------)
+--R a x
+--R -------------------------------------------]
+--R +---+
+--R \|- a
+--R Type: Union(List Expression Integer,...)
+--E
+
+--S 21
+bb1:=(2*a*x-3*b)/(4*a^2)*sqrt(a*x^2+b*x+c)+(3*b^2-4*a*c)/(8*a^2)*t1.1
+--R
+--R (3)
+--R 2
+--R (- 4a c + 3b )
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R +--------------+
+--R +-+ | 2
+--R (4a x - 6b)\|a \|a x + b x + c
+--R /
+--R 2 +-+
+--R 8a \|a
+--R Type: Expression Integer
+--E
+
+--S 22
+bb2:=(2*a*x-3*b)/(4*a^2)*sqrt(a*x^2+b*x+c)+(3*b^2-4*a*c)/(8*a^2)*t1.2
+--R
+--R (4)
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R 2 \|- a \|a x + b x + c - \|- a \|c
+--R (- 4a c + 3b )atan(------------------------------------)
+--R a x
+--R +
+--R +--------------+
+--R +---+ | 2
+--R (2a x - 3b)\|- a \|a x + b x + c
+--R /
+--R 2 +---+
+--R 4a \|- a
+--R Type: Expression Integer
+--E
+
+--S 23
+cc1:=aa.1-bb1
+--R
+--R (5)
+--R +-----------+
+--R 2 | 2
+--R (4a c - 3b )\|- 4a c + b
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R 2 +-+
+--R (8a c - 4b )\|a
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x
+--R +
+--R 3
+--R - 4a b c + b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R +-----------+ +--------------+
+--R | 2 +-+ | 2
+--R (- 4a x + 6b)\|- 4a c + b \|a \|a x + b x + c
+--R +
+--R +-----------+
+--R 2 | 2 +-+
+--R (- 4b log(a x + b x + c) + 8a x)\|- 4a c + b \|a
+--R /
+--R +-----------+
+--R 2 | 2 +-+
+--R 8a \|- 4a c + b \|a
+--R Type: Expression Integer
+--E
+
+--S 24
+cc2:=aa.2-bb1
+--R
+--R (6)
+--R +---------+
+--R 2 | 2
+--R (4a c - 3b )\|4a c - b
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R +---------+
+--R | 2
+--R 2 +-+ (2a x + b)\|4a c - b
+--R (- 16a c + 8b )\|a atan(----------------------)
+--R 2
+--R 4a c - b
+--R +
+--R +---------+ +--------------+
+--R +-+ | 2 | 2
+--R (- 4a x + 6b)\|a \|4a c - b \|a x + b x + c
+--R +
+--R +---------+
+--R 2 +-+ | 2
+--R (- 4b log(a x + b x + c) + 8a x)\|a \|4a c - b
+--R /
+--R +---------+
+--R 2 +-+ | 2
+--R 8a \|a \|4a c - b
+--R Type: Expression Integer
+--E
+
+--S 25
+cc3:=aa.2-bb1
+--R
+--R (7)
+--R +---------+
+--R 2 | 2
+--R (4a c - 3b )\|4a c - b
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R +---------+
+--R | 2
+--R 2 +-+ (2a x + b)\|4a c - b
+--R (- 16a c + 8b )\|a atan(----------------------)
+--R 2
+--R 4a c - b
+--R +
+--R +---------+ +--------------+
+--R +-+ | 2 | 2
+--R (- 4a x + 6b)\|a \|4a c - b \|a x + b x + c
+--R +
+--R +---------+
+--R 2 +-+ | 2
+--R (- 4b log(a x + b x + c) + 8a x)\|a \|4a c - b
+--R /
+--R +---------+
+--R 2 +-+ | 2
+--R 8a \|a \|4a c - b
+--R Type: Expression Integer
+--E
+
+--S 26 14:282 Axiom cannot simplify these expressions
+cc4:=aa.2-bb2
+--R
+--R (8)
+--R +--------------+
+--R +---------+ +---+ | 2 +---+ +-+
+--R 2 | 2 \|- a \|a x + b x + c - \|- a \|c
+--R (4a c - 3b )\|4a c - b atan(------------------------------------)
+--R a x
+--R +
+--R +---------+
+--R | 2
+--R 2 +---+ (2a x + b)\|4a c - b
+--R (- 8a c + 4b )\|- a atan(----------------------)
+--R 2
+--R 4a c - b
+--R +
+--R +---------+ +--------------+
+--R +---+ | 2 | 2
+--R (- 2a x + 3b)\|- a \|4a c - b \|a x + b x + c
+--R +
+--R +---------+
+--R 2 +---+ | 2
+--R (- 2b log(a x + b x + c) + 4a x)\|- a \|4a c - b
+--R /
+--R +---------+
+--R 2 +---+ | 2
+--R 4a \|- a \|4a c - b
+--R Type: Expression Integer
+--E
+
@
\section{\cite{1}:14.283~~~~~$\displaystyle
@@ -200,7 +831,7 @@
<<*>>=
)clear all
---S 4 of 19
+--S 27
aa:=integrate(1/x*(a*x^2+b*x+c),x)
--R
--R
@@ -210,6 +841,91 @@
--R 2
--R Type: Union(Expression Integer,...)
--E
+
+--S 28
+bb1:=-1/sqrt(c)*log((2*sqrt(c)*sqrt(a*x^2+b*x+c)+b*x+2*c)/x)
+--R
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c + b x + 2c
+--R log(---------------------------------)
+--R x
+--R (2) - --------------------------------------
+--R +-+
+--R \|c
+--R Type: Expression Integer
+--E
+
+--S 29
+bb2:=1/sqrt(-c)*asin((b*x+2*c)/(x*sqrt(b^2-4*a*c)))
+--R
+--R b x + 2c
+--R asin(---------------)
+--R +-----------+
+--R | 2
+--R x\|- 4a c + b
+--R (3) ---------------------
+--R +---+
+--R \|- c
+--R Type: Expression Integer
+--E
+
+--S 30
+bb3:=-1/sqrt(c)*asinh((b*x+2*c)/(x*sqrt(b^2-4*a*c)))
+--R
+--R b x + 2c
+--R asinh(---------------)
+--R +-----------+
+--R | 2
+--R x\|- 4a c + b
+--R (4) - ----------------------
+--R +-+
+--R \|c
+--R Type: Expression Integer
+--E
+
+--S 31
+cc1:=aa-bb1
+--R
+--R (5)
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c + b x + 2c 2 +-+
+--R 2log(---------------------------------) + (2c log(x) + a x + 2b x)\|c
+--R x
+--R -----------------------------------------------------------------------
+--R +-+
+--R 2\|c
+--R Type: Expression Integer
+--E
+
+--S 32
+cc2:=aa-bb2
+--R
+--R b x + 2c 2 +---+
+--R - 2asin(---------------) + (2c log(x) + a x + 2b x)\|- c
+--R +-----------+
+--R | 2
+--R x\|- 4a c + b
+--R (6) ----------------------------------------------------------
+--R +---+
+--R 2\|- c
+--R Type: Expression Integer
+--E
+
+--S 33 14:283 Axiom cannot simplify these expressions
+cc3:=aa-bb3
+--R
+--R b x + 2c 2 +-+
+--R 2asinh(---------------) + (2c log(x) + a x + 2b x)\|c
+--R +-----------+
+--R | 2
+--R x\|- 4a c + b
+--R (7) -------------------------------------------------------
+--R +-+
+--R 2\|c
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.284~~~~~$\displaystyle
@@ -220,7 +936,7 @@
<<*>>=
)clear all
---S 5 of 19
+--S 34
aa:=integrate(1/(x^2*(a*x^2+b*x+c)),x)
--R
--R
@@ -266,6 +982,93 @@
--R ]
--R Type: Union(List Expression Integer,...)
--E
+
+--S 35
+t1:=integrate(1/(x*sqrt(a*x^2+b*x+c)),x)
+--R
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R log(---------------------------------)
+--R x
+--R (2) --------------------------------------
+--R +-+
+--R \|c
+--R Type: Union(Expression Integer,...)
+--E
+
+--S 36
+bb:=-sqrt(a*x^2+b*x+c)/(c*x)-b/(2*x)*t1
+--R
+--R +--------------+
+--R +-+ | 2 +--------------+
+--R 2\|c \|a x + b x + c - b x - 2c +-+ | 2
+--R - b c log(---------------------------------) - 2\|c \|a x + b x + c
+--R x
+--R (3) ---------------------------------------------------------------------
+--R +-+
+--R 2c x\|c
+--R Type: Expression Integer
+--E
+
+--S 37
+cc1:=aa.1-bb1
+--R
+--R (4)
+--R 2
+--R (2a c - b )x
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x
+--R +
+--R 3
+--R - 4a b c + b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R +-----------+
+--R 2 2 | 2
+--R (b x log(a x + b x + c) - 2b x log(x) - 2bb1 c x - 2c)\|- 4a c + b
+--R /
+--R +-----------+
+--R 2 | 2
+--R 2c x\|- 4a c + b
+--R Type: Expression Integer
+--E
+
+--S 38 14:284 Axiom cannot simplify these expressions
+cc2:=aa.2-bb
+--R
+--R (5)
+--R +--------------+
+--R +---------+ +-+ | 2
+--R 2 | 2 2\|c \|a x + b x + c - b x - 2c
+--R b c \|4a c - b log(---------------------------------)
+--R x
+--R +
+--R +---------+
+--R | 2
+--R 2 +-+ (2a x + b)\|4a c - b
+--R (- 4a c + 2b )x\|c atan(----------------------)
+--R 2
+--R 4a c - b
+--R +
+--R +---------+ +--------------+
+--R +-+ | 2 | 2
+--R 2c\|c \|4a c - b \|a x + b x + c
+--R +
+--R +---------+
+--R 2 +-+ | 2
+--R (b x log(a x + b x + c) - 2b x log(x) - 2c)\|c \|4a c - b
+--R /
+--R +---------+
+--R 2 +-+ | 2
+--R 2c x\|c \|4a c - b
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.285~~~~~$\displaystyle
@@ -277,7 +1080,7 @@
<<*>>=
)clear all
---S 6 of 19
+--S 39
aa:=integrate(sqrt(a*x^2+b*x+c),x)
--R
--R
@@ -361,6 +1164,228 @@
--R ]
--R Type: Union(List Expression Integer,...)
--E
+
+--S 40
+t1:=integrate(1/sqrt(a*x^2+b*x+c),x)
+--R
+--R (2)
+--R [
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R /
+--R +-+
+--R \|a
+--R ,
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R \|- a \|a x + b x + c - \|- a \|c
+--R 2atan(------------------------------------)
+--R a x
+--R -------------------------------------------]
+--R +---+
+--R \|- a
+--R Type: Union(List Expression Integer,...)
+--E
+
+--S 41
+bb1:=((2*a*x+b)*sqrt(a*x^2+b*x+c))/(4*a)+(4*a*c-b^2)/(8*a)*t1.1
+--R
+--R (3)
+--R 2
+--R (4a c - b )
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R +--------------+
+--R +-+ | 2
+--R (4a x + 2b)\|a \|a x + b x + c
+--R /
+--R +-+
+--R 8a\|a
+--R Type: Expression Integer
+--E
+
+--S 42
+bb2:=((2*a*x+b)*sqrt(a*x^2+b*x+c))/(4*a)+(4*a*c-b^2)/(8*a)*t1.2
+--R
+--R (4)
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R 2 \|- a \|a x + b x + c - \|- a \|c
+--R (4a c - b )atan(------------------------------------)
+--R a x
+--R +
+--R +--------------+
+--R +---+ | 2
+--R (2a x + b)\|- a \|a x + b x + c
+--R /
+--R +---+
+--R 4a\|- a
+--R Type: Expression Integer
+--E
+
+--S 43
+cc1:=aa.1-bb1
+--R
+--R (5)
+--R +--------------+
+--R 2 2 | 2
+--R (4b c x + 8b c )\|a x + b x + c
+--R +
+--R 3 2 2 2 +-+
+--R ((- 4a b c - b )x - 8b c x - 8b c )\|c
+--R /
+--R +--------------+
+--R +-+ | 2 2 2 2
+--R (16a b x + 32a c)\|c \|a x + b x + c + (- 16a c - 4a b )x - 32a b c x
+--R +
+--R 2
+--R - 32a c
+--R Type: Expression Integer
+--E
+
+--S 44
+cc2:=aa.2-bb1
+--R
+--R (6)
+--R +--------------+
+--R 3 2 2 +---+ +-+ | 2
+--R ((- 16a b c + 4b )x - 32a c + 8b c)\|- a \|c \|a x + b x + c
+--R +
+--R 2 2 4 2 2 3 3 2 2 +---+
+--R ((16a c - b )x + (32a b c - 8b c)x + 32a c - 8b c )\|- a
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R +--------------+
+--R 3 2 2 +-+ +-+ | 2
+--R ((32a b c - 8b )x + 64a c - 16b c)\|a \|c \|a x + b x + c
+--R +
+--R 2 2 4 2 2 3 3 2 2 +-+
+--R ((- 32a c + 2b )x + (- 64a b c + 16b c)x - 64a c + 16b c )\|a
+--R *
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R \|- a \|a x + b x + c - \|- a \|c
+--R atan(------------------------------------)
+--R a x
+--R +
+--R +--------------+
+--R 2 2 +---+ +-+ | 2
+--R (8b c x + 16b c )\|- a \|a \|a x + b x + c
+--R +
+--R 3 2 2 2 +---+ +-+ +-+
+--R ((- 8a b c - 2b )x - 16b c x - 16b c )\|- a \|a \|c
+--R /
+--R +--------------+
+--R +---+ +-+ +-+ | 2
+--R (32a b x + 64a c)\|- a \|a \|c \|a x + b x + c
+--R +
+--R 2 2 2 2 +---+ +-+
+--R ((- 32a c - 8a b )x - 64a b c x - 64a c )\|- a \|a
+--R Type: Expression Integer
+--E
+
+--S 45
+cc3:=aa.1-bb2
+--R
+--R (7)
+--R +--------------+
+--R 3 2 2 +---+ +-+ | 2
+--R ((16a b c - 4b )x + 32a c - 8b c)\|- a \|c \|a x + b x + c
+--R +
+--R 2 2 4 2 2 3 3 2 2 +---+
+--R ((- 16a c + b )x + (- 32a b c + 8b c)x - 32a c + 8b c )\|- a
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R +--------------+
+--R 3 2 2 +-+ +-+ | 2
+--R ((- 32a b c + 8b )x - 64a c + 16b c)\|a \|c \|a x + b x + c
+--R +
+--R 2 2 4 2 2 3 3 2 2 +-+
+--R ((32a c - 2b )x + (64a b c - 16b c)x + 64a c - 16b c )\|a
+--R *
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R \|- a \|a x + b x + c - \|- a \|c
+--R atan(------------------------------------)
+--R a x
+--R +
+--R +--------------+
+--R 2 2 +---+ +-+ | 2
+--R (8b c x + 16b c )\|- a \|a \|a x + b x + c
+--R +
+--R 3 2 2 2 +---+ +-+ +-+
+--R ((- 8a b c - 2b )x - 16b c x - 16b c )\|- a \|a \|c
+--R /
+--R +--------------+
+--R +---+ +-+ +-+ | 2
+--R (32a b x + 64a c)\|- a \|a \|c \|a x + b x + c
+--R +
+--R 2 2 2 2 +---+ +-+
+--R ((- 32a c - 8a b )x - 64a b c x - 64a c )\|- a \|a
+--R Type: Expression Integer
+--E
+
+--S 46 14:285 Axiom cannot simplify these expressions
+cc4:=aa.2-bb2
+--R
+--R (8)
+--R +--------------+
+--R 2 2 | 2
+--R (4b c x + 8b c )\|a x + b x + c
+--R +
+--R 3 2 2 2 +-+
+--R ((- 4a b c - b )x - 8b c x - 8b c )\|c
+--R /
+--R +--------------+
+--R +-+ | 2 2 2 2
+--R (16a b x + 32a c)\|c \|a x + b x + c + (- 16a c - 4a b )x - 32a b c x
+--R +
+--R 2
+--R - 32a c
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.286~~~~~$\displaystyle
@@ -378,7 +1403,7 @@
<<*>>=
)clear all
---S 7 of 19
+--S 47
aa:=integrate(x*sqrt(a*x^2+b*x+c),x)
--R
--R
@@ -522,6 +1547,416 @@
--R ]
--R Type: Union(List Expression Integer,...)
--E
+
+--S 48
+t1:=integrate(1/sqrt(a*x^2+b*x+c),x)
+--R
+--R (2)
+--R [
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R /
+--R +-+
+--R \|a
+--R ,
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R \|- a \|a x + b x + c - \|- a \|c
+--R 2atan(------------------------------------)
+--R a x
+--R -------------------------------------------]
+--R +---+
+--R \|- a
+--R Type: Union(List Expression Integer,...)
+--E
+
+--S 49
+bb1:=(a*x^2+b*x+c)^(3/2)/(3*a)-(b*(2*a*x+b))/(8*a^2)*sqrt(a*x^2+b*x+c)-(b*(4*a*c-b^2))/(16*a^2)*t1.1
+--R
+--R (3)
+--R 3
+--R (- 12a b c + 3b )
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R +--------------+
+--R 2 2 2 +-+ | 2
+--R (16a x + 4a b x + 16a c - 6b )\|a \|a x + b x + c
+--R /
+--R 2 +-+
+--R 48a \|a
+--R Type: Expression Integer
+--E
+
+--S 50
+bb2:=(a*x^2+b*x+c)^(3/2)/(3*a)-(b*(2*a*x+b))/(8*a^2)*sqrt(a*x^2+b*x+c)-(b*(4*a*c-b^2))/(16*a^2)*t1.2
+--R
+--R (4)
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R 3 \|- a \|a x + b x + c - \|- a \|c
+--R (- 12a b c + 3b )atan(------------------------------------)
+--R a x
+--R +
+--R +--------------+
+--R 2 2 2 +---+ | 2
+--R (8a x + 2a b x + 8a c - 3b )\|- a \|a x + b x + c
+--R /
+--R 2 +---+
+--R 24a \|- a
+--R Type: Expression Integer
+--E
+
+--S 51
+cc1:=aa.1-bb1
+--R
+--R (5)
+--R 2 2 3 5 2 2 2 4 3
+--R (96a b c + 48a b c - 18b )x + (384a b c - 96b c)x + 384a b c
+--R +
+--R 3 2
+--R - 96b c
+--R *
+--R +--------------+
+--R +-+ | 2
+--R \|c \|a x + b x + c
+--R +
+--R 2 2 2 4 6 3 2 3 3 2 5 2
+--R (- 144a b c + 24a b c + 3b )x + (- 288a b c - 144a b c + 54b c)x
+--R +
+--R 2 3 4 2 4 3 3
+--R (- 576a b c + 144b c )x - 384a b c + 96b c
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c + 2a x)\|a x + b x + c - 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R 2 2 3 5 2 2 2 4 3
+--R (96a b c + 48a b c - 18b )x + (384a b c - 96b c)x + 384a b c
+--R +
+--R 3 2
+--R - 96b c
+--R *
+--R +--------------+
+--R +-+ | 2
+--R \|c \|a x + b x + c
+--R +
+--R 2 2 2 4 6 3 2 3 3 2 5 2
+--R (- 144a b c + 24a b c + 3b )x + (- 288a b c - 144a b c + 54b c)x
+--R +
+--R 2 3 4 2 4 3 3
+--R (- 576a b c + 144b c )x - 384a b c + 96b c
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R 2 3 2 2 4 2 3 3 2 4
+--R (128a c + 48a b c - 36b c)x + (512a b c - 192b c )x + 512a c
+--R +
+--R 2 3
+--R - 192b c
+--R *
+--R +--------------+
+--R +-+ | 2
+--R \|a \|a x + b x + c
+--R +
+--R 2 2 3 5 3 2 3 2 2 4 2
+--R (- 192a b c + 56a b c + 6b )x + (- 384a c - 144a b c + 108b c)x
+--R +
+--R 3 3 2 4 2 3
+--R (- 768a b c + 288b c )x - 512a c + 192b c
+--R *
+--R +-+ +-+
+--R \|a \|c
+--R /
+--R +--------------+
+--R 3 2 2 2 2 2 2 +-+ +-+ | 2
+--R ((384a c + 288a b )x + 1536a b c x + 1536a c )\|a \|c \|a x + b x + c
+--R +
+--R 3 2 3 3 3 2 2 2 2 2 2
+--R (- 576a b c - 48a b )x + (- 1152a c - 864a b c)x - 2304a b c x
+--R +
+--R 2 3
+--R - 1536a c
+--R *
+--R +-+
+--R \|a
+--R Type: Expression Integer
+--E
+
+--S 52
+cc2:=aa.2-bb1
+--R
+--R (6)
+--R 2 2 3 5 2 2 2 4 3
+--R (96a b c + 48a b c - 18b )x + (384a b c - 96b c)x + 384a b c
+--R +
+--R 3 2
+--R - 96b c
+--R *
+--R +--------------+
+--R +---+ +-+ | 2
+--R \|- a \|c \|a x + b x + c
+--R +
+--R 2 2 2 4 6 3
+--R (- 144a b c + 24a b c + 3b )x
+--R +
+--R 2 3 3 2 5 2 2 3 4 2
+--R (- 288a b c - 144a b c + 54b c)x + (- 576a b c + 144b c )x
+--R +
+--R 4 3 3
+--R - 384a b c + 96b c
+--R *
+--R +---+
+--R \|- a
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R 2 2 3 5 2 2 2 4
+--R (- 192a b c - 96a b c + 36b )x + (- 768a b c + 192b c)x
+--R +
+--R 3 3 2
+--R - 768a b c + 192b c
+--R *
+--R +--------------+
+--R +-+ +-+ | 2
+--R \|a \|c \|a x + b x + c
+--R +
+--R 2 2 2 4 6 3
+--R (288a b c - 48a b c - 6b )x
+--R +
+--R 2 3 3 2 5 2 2 3 4 2
+--R (576a b c + 288a b c - 108b c)x + (1152a b c - 288b c )x
+--R +
+--R 4 3 3
+--R 768a b c - 192b c
+--R *
+--R +-+
+--R \|a
+--R *
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R \|- a \|a x + b x + c - \|- a \|c
+--R atan(------------------------------------)
+--R a x
+--R +
+--R 2 3 2 2 4 2 3 3 2 4
+--R (128a c + 48a b c - 36b c)x + (512a b c - 192b c )x + 512a c
+--R +
+--R 2 3
+--R - 192b c
+--R *
+--R +--------------+
+--R +---+ +-+ | 2
+--R \|- a \|a \|a x + b x + c
+--R +
+--R 2 2 3 5 3 2 3 2 2 4 2
+--R (- 192a b c + 56a b c + 6b )x + (- 384a c - 144a b c + 108b c)x
+--R +
+--R 3 3 2 4 2 3
+--R (- 768a b c + 288b c )x - 512a c + 192b c
+--R *
+--R +---+ +-+ +-+
+--R \|- a \|a \|c
+--R /
+--R 3 2 2 2 2 2 2 +---+ +-+ +-+
+--R ((384a c + 288a b )x + 1536a b c x + 1536a c )\|- a \|a \|c
+--R *
+--R +--------------+
+--R | 2
+--R \|a x + b x + c
+--R +
+--R 3 2 3 3 3 2 2 2 2 2 2
+--R (- 576a b c - 48a b )x + (- 1152a c - 864a b c)x - 2304a b c x
+--R +
+--R 2 3
+--R - 1536a c
+--R *
+--R +---+ +-+
+--R \|- a \|a
+--R Type: Expression Integer
+--E
+
+--S 53
+cc3:=aa.1-bb2
+--R
+--R (7)
+--R 2 2 3 5 2 2 2 4 3
+--R (96a b c + 48a b c - 18b )x + (384a b c - 96b c)x + 384a b c
+--R +
+--R 3 2
+--R - 96b c
+--R *
+--R +--------------+
+--R +---+ +-+ | 2
+--R \|- a \|c \|a x + b x + c
+--R +
+--R 2 2 2 4 6 3
+--R (- 144a b c + 24a b c + 3b )x
+--R +
+--R 2 3 3 2 5 2 2 3 4 2
+--R (- 288a b c - 144a b c + 54b c)x + (- 576a b c + 144b c )x
+--R +
+--R 4 3 3
+--R - 384a b c + 96b c
+--R *
+--R +---+
+--R \|- a
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c + 2a x)\|a x + b x + c - 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R 2 2 3 5 2 2 2 4
+--R (192a b c + 96a b c - 36b )x + (768a b c - 192b c)x
+--R +
+--R 3 3 2
+--R 768a b c - 192b c
+--R *
+--R +--------------+
+--R +-+ +-+ | 2
+--R \|a \|c \|a x + b x + c
+--R +
+--R 2 2 2 4 6 3
+--R (- 288a b c + 48a b c + 6b )x
+--R +
+--R 2 3 3 2 5 2 2 3 4 2
+--R (- 576a b c - 288a b c + 108b c)x + (- 1152a b c + 288b c )x
+--R +
+--R 4 3 3
+--R - 768a b c + 192b c
+--R *
+--R +-+
+--R \|a
+--R *
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R \|- a \|a x + b x + c - \|- a \|c
+--R atan(------------------------------------)
+--R a x
+--R +
+--R 2 3 2 2 4 2 3 3 2 4
+--R (128a c + 48a b c - 36b c)x + (512a b c - 192b c )x + 512a c
+--R +
+--R 2 3
+--R - 192b c
+--R *
+--R +--------------+
+--R +---+ +-+ | 2
+--R \|- a \|a \|a x + b x + c
+--R +
+--R 2 2 3 5 3 2 3 2 2 4 2
+--R (- 192a b c + 56a b c + 6b )x + (- 384a c - 144a b c + 108b c)x
+--R +
+--R 3 3 2 4 2 3
+--R (- 768a b c + 288b c )x - 512a c + 192b c
+--R *
+--R +---+ +-+ +-+
+--R \|- a \|a \|c
+--R /
+--R 3 2 2 2 2 2 2 +---+ +-+ +-+
+--R ((384a c + 288a b )x + 1536a b c x + 1536a c )\|- a \|a \|c
+--R *
+--R +--------------+
+--R | 2
+--R \|a x + b x + c
+--R +
+--R 3 2 3 3 3 2 2 2 2 2 2
+--R (- 576a b c - 48a b )x + (- 1152a c - 864a b c)x - 2304a b c x
+--R +
+--R 2 3
+--R - 1536a c
+--R *
+--R +---+ +-+
+--R \|- a \|a
+--R Type: Expression Integer
+--E
+
+--S 54 14:286 Axiom cannot simplify these expressions
+cc4:=aa.2-bb2
+--R
+--R (8)
+--R 2 3 2 2 4 2 3 3 2 4
+--R (64a c + 24a b c - 18b c)x + (256a b c - 96b c )x + 256a c
+--R +
+--R 2 3
+--R - 96b c
+--R *
+--R +--------------+
+--R | 2
+--R \|a x + b x + c
+--R +
+--R 2 2 3 5 3 2 3 2 2 4 2
+--R (- 96a b c + 28a b c + 3b )x + (- 192a c - 72a b c + 54b c)x
+--R +
+--R 3 3 2 4 2 3
+--R (- 384a b c + 144b c )x - 256a c + 96b c
+--R *
+--R +-+
+--R \|c
+--R /
+--R +--------------+
+--R 3 2 2 2 2 2 2 +-+ | 2
+--R ((192a c + 144a b )x + 768a b c x + 768a c )\|c \|a x + b x + c
+--R +
+--R 3 2 3 3 3 2 2 2 2 2 2 2 3
+--R (- 288a b c - 24a b )x + (- 576a c - 432a b c)x - 1152a b c x - 768a c
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.287~~~~~$\displaystyle
@@ -533,7 +1968,7 @@
<<*>>=
)clear all
---S 8 of 19
+--S 55
aa:=integrate(x^2*sqrt(a*x^2+b*x+c),x)
--R
--R
@@ -763,7 +2198,891 @@
--R \|- a
--R ]
--R Type: Union(List Expression Integer,...)
---E
+--E
+
+--S 56
+t1:=integrate(sqrt(a*x^2+b*x+c),x)
+--R
+--R (2)
+--R [
+--R +--------------+
+--R 3 2 2 +-+ | 2
+--R ((16a b c - 4b )x + 32a c - 8b c)\|c \|a x + b x + c
+--R +
+--R 2 2 4 2 2 3 3 2 2
+--R (- 16a c + b )x + (- 32a b c + 8b c)x - 32a c + 8b c
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R 2 2 3 3 2 2 2 +-+
+--R ((- 16a c - 4a b )x + (- 40a b c - 2b )x + (- 32a c - 8b c)x)\|a
+--R *
+--R +--------------+
+--R | 2
+--R \|a x + b x + c
+--R +
+--R 2 4 2 2 3 3 2
+--R 16a b x + (32a c + 24a b )x + (56a b c + 6b )x
+--R +
+--R 2 2
+--R (32a c + 8b c)x
+--R *
+--R +-+ +-+
+--R \|a \|c
+--R /
+--R +--------------+
+--R +-+ +-+ | 2
+--R (32a b x + 64a c)\|a \|c \|a x + b x + c
+--R +
+--R 2 2 2 2 +-+
+--R ((- 32a c - 8a b )x - 64a b c x - 64a c )\|a
+--R ,
+--R
+--R +--------------+
+--R 3 2 2 +-+ | 2
+--R ((16a b c - 4b )x + 32a c - 8b c)\|c \|a x + b x + c
+--R +
+--R 2 2 4 2 2 3 3 2 2
+--R (- 16a c + b )x + (- 32a b c + 8b c)x - 32a c + 8b c
+--R *
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R \|- a \|a x + b x + c - \|- a \|c
+--R atan(------------------------------------)
+--R a x
+--R +
+--R 2 2 3 3 2 2 2 +---+
+--R ((- 8a c - 2a b )x + (- 20a b c - b )x + (- 16a c - 4b c)x)\|- a
+--R *
+--R +--------------+
+--R | 2
+--R \|a x + b x + c
+--R +
+--R 2 4 2 2 3 3 2 2 2
+--R (8a b x + (16a c + 12a b )x + (28a b c + 3b )x + (16a c + 4b c)x)
+--R *
+--R +---+ +-+
+--R \|- a \|c
+--R /
+--R +--------------+
+--R +---+ +-+ | 2
+--R (16a b x + 32a c)\|- a \|c \|a x + b x + c
+--R +
+--R 2 2 2 2 +---+
+--R ((- 16a c - 4a b )x - 32a b c x - 32a c )\|- a
+--R ]
+--R Type: Union(List Expression Integer,...)
+--E
+
+--S 57
+bb1:=(6*a*x-5*b)/(24*a^2)*(a*x^2+b*x+c)^(3/2)+(5*b^2-4*a*c)/(16*a^2)*t1.1
+--R
+--R (3)
+--R 2 2 3 5 2 3 2 2 4
+--R ((- 192a b c + 288a b c - 60b )x - 384a c + 576a b c - 120b c)
+--R *
+--R +--------------+
+--R +-+ | 2
+--R \|c \|a x + b x + c
+--R +
+--R 3 3 2 2 2 4 6 2
+--R (192a c - 240a b c - 12a b c + 15b )x
+--R +
+--R 2 3 3 2 5 2 4 2 3 4 2
+--R (384a b c - 576a b c + 120b c)x + 384a c - 576a b c + 120b c
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R 4 3 2 5 3 2 3 4
+--R (- 384a c - 96a b )x + (- 832a b c - 16a b )x
+--R +
+--R 3 2 2 2 4 3 2 2 3 5 2
+--R (- 960a c - 96a b c + 20a b )x + (- 96a b c + 144a...
[truncated message content] |
|
From: <gi...@ax...> - 2008-04-21 07:15:49
|
changelog | 1 +
src/input/schaum13.input.pamphlet | 3649 ++++++++++++++++++++++++++++++++++++-
2 files changed, 3628 insertions(+), 22 deletions(-)
New commits:
commit 0cc7eb6a986ad58b183d3c805f93bed31d6faea7
Author: Tim Daly <root@localhost.localdomain>
Date: Thu Apr 3 04:45:20 2008 -0400
20080420 tpd src/input/schaum13.input show Schaums-Axiom equivalence
|
|
From: <da...@us...> - 2008-04-20 06:56:02
|
Revision: 859
http://axiom.svn.sourceforge.net/axiom/?rev=859&view=rev
Author: daly
Date: 2008-04-19 23:56:05 -0700 (Sat, 19 Apr 2008)
Log Message:
-----------
20080419 tpd src/input/schaum12.input show Schaums-Axiom equivalence
Modified Paths:
--------------
trunk/axiom/changelog
trunk/axiom/src/input/schaum12.input.pamphlet
Modified: trunk/axiom/changelog
===================================================================
--- trunk/axiom/changelog 2008-04-19 06:06:09 UTC (rev 858)
+++ trunk/axiom/changelog 2008-04-20 06:56:05 UTC (rev 859)
@@ -1,3 +1,4 @@
+20080419 tpd src/input/schaum12.input show Schaums-Axiom equivalence
20080418 tpd src/input/schaum11.input show Schaums-Axiom equivalence
20080418 tpd src/input/schaum10.input fix Schaums-Axiom equivalence
20080418 tpd src/input/schaum9.input fix Schaums-Axiom equivalence
Modified: trunk/axiom/src/input/schaum12.input.pamphlet
===================================================================
--- trunk/axiom/src/input/schaum12.input.pamphlet 2008-04-19 06:06:09 UTC (rev 858)
+++ trunk/axiom/src/input/schaum12.input.pamphlet 2008-04-20 06:56:05 UTC (rev 859)
@@ -24,7 +24,7 @@
)set message auto off
)clear all
---S 1 of 15
+--S 1
aa:=integrate(1/sqrt(a*x^2+b*x+c),x)
--R
--R
@@ -55,6 +55,152 @@
--R \|- a
--R Type: Union(List Expression Integer,...)
--E
+
+--S 2
+bb1:=2/sqrt(4*a*c-b^2)*atan((2*a*x+b)/sqrt(4*a*c-b^2))
+--R
+--R
+--R 2a x + b
+--R 2atan(------------)
+--R +---------+
+--R | 2
+--R \|4a c - b
+--R (2) -------------------
+--R +---------+
+--R | 2
+--R \|4a c - b
+--R Type: Expression Integer
+--E
+
+--S 3
+bb2:=1/sqrt(b^2-4*a*c)*log((2*a*x+b-sqrt(b^2-4*a*c))/(2*a*x+b+sqrt(b^2-4*a*c)))
+--R
+--R
+--R +-----------+
+--R | 2
+--R - \|- 4a c + b + 2a x + b
+--R log(---------------------------)
+--R +-----------+
+--R | 2
+--R \|- 4a c + b + 2a x + b
+--R (3) --------------------------------
+--R +-----------+
+--R | 2
+--R \|- 4a c + b
+--R Type: Expression Integer
+--E
+
+--S 4
+cc1:=aa.1-bb1
+--R
+--R (4)
+--R +---------+
+--R | 2
+--R \|4a c - b
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R +-+ 2a x + b
+--R - 2\|a atan(------------)
+--R +---------+
+--R | 2
+--R \|4a c - b
+--R /
+--R +---------+
+--R +-+ | 2
+--R \|a \|4a c - b
+--R Type: Expression Integer
+--E
+
+--S 5
+cc2:=aa.1-bb2
+--R
+--R (5)
+--R +-----------+
+--R | 2
+--R \|- 4a c + b
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R +-----------+
+--R | 2
+--R +-+ - \|- 4a c + b + 2a x + b
+--R - \|a log(---------------------------)
+--R +-----------+
+--R | 2
+--R \|- 4a c + b + 2a x + b
+--R /
+--R +-----------+
+--R | 2 +-+
+--R \|- 4a c + b \|a
+--R Type: Expression Integer
+--E
+
+--S 6
+cc3:=aa.2-bb1
+--R
+--R (6)
+--R +--------------+
+--R +---------+ +---+ | 2 +---+ +-+
+--R | 2 \|- a \|a x + b x + c - \|- a \|c
+--R 2\|4a c - b atan(------------------------------------)
+--R a x
+--R +
+--R +---+ 2a x + b
+--R - 2\|- a atan(------------)
+--R +---------+
+--R | 2
+--R \|4a c - b
+--R /
+--R +---------+
+--R +---+ | 2
+--R \|- a \|4a c - b
+--R Type: Expression Integer
+--E
+
+--S 7 14:265 Axiom cannot simplify these expressions
+cc4:=aa.2-bb2
+--R
+--R (7)
+--R +-----------+
+--R | 2
+--R +---+ - \|- 4a c + b + 2a x + b
+--R - \|- a log(---------------------------)
+--R +-----------+
+--R | 2
+--R \|- 4a c + b + 2a x + b
+--R +
+--R +--------------+
+--R +-----------+ +---+ | 2 +---+ +-+
+--R | 2 \|- a \|a x + b x + c - \|- a \|c
+--R 2\|- 4a c + b atan(------------------------------------)
+--R a x
+--R /
+--R +-----------+
+--R | 2 +---+
+--R \|- 4a c + b \|- a
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.266~~~~~$\displaystyle
@@ -65,7 +211,7 @@
<<*>>=
)clear all
---S 2 of 15
+--S 8
aa:=integrate(x/sqrt(a*x^2+b*x+c),x)
--R
--R
@@ -116,6 +262,316 @@
--R ]
--R Type: Union(List Expression Integer,...)
--E
+
+--S 9
+t1:=integrate(1/(a*x^2+b*x+c),x)
+--R
+--R
+--R (2)
+--R [
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R /
+--R +-----------+
+--R | 2
+--R \|- 4a c + b
+--R ,
+--R +---------+
+--R | 2
+--R (2a x + b)\|4a c - b
+--R 2atan(----------------------)
+--R 2
+--R 4a c - b
+--R -----------------------------]
+--R +---------+
+--R | 2
+--R \|4a c - b
+--R Type: Union(List Expression Integer,...)
+--E
+
+--S 10
+bb1:=1/(2*a)*log(a*x^2+b*x+c)-b/(2*a)*t1.1
+--R
+--R
+--R (3)
+--R -
+--R b
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R +-----------+
+--R 2 | 2
+--R log(a x + b x + c)\|- 4a c + b
+--R /
+--R +-----------+
+--R | 2
+--R 2a\|- 4a c + b
+--R Type: Expression Integer
+--E
+
+--S 11
+bb2:=1/(2*a)*log(a*x^2+b*x+c)-b/(2*a)*t1.2
+--R
+--R
+--R +---------+
+--R | 2 +---------+
+--R (2a x + b)\|4a c - b 2 | 2
+--R - 2b atan(----------------------) + log(a x + b x + c)\|4a c - b
+--R 2
+--R 4a c - b
+--R (4) -------------------------------------------------------------------
+--R +---------+
+--R | 2
+--R 2a\|4a c - b
+--R Type: Expression Integer
+--E
+
+--S 12
+cc1:=aa.1-bb1
+--R
+--R (5)
+--R +-----------+ +--------------+ +-----------+
+--R | 2 +-+ | 2 2 | 2
+--R (2b\|- 4a c + b \|c \|a x + b x + c + (- b x - 2b c)\|- 4a c + b )
+--R *
+--R log
+--R +--------------+
+--R +-+ +-+ | 2 +-+
+--R (2\|a \|c + 2a x)\|a x + b x + c - 2a x\|c
+--R +
+--R 2 +-+
+--R (- 2a x - b x - 2c)\|a
+--R /
+--R +--------------+
+--R +-+ | 2
+--R 2\|c \|a x + b x + c - b x - 2c
+--R +
+--R +--------------+
+--R +-+ +-+ | 2 2 +-+
+--R (2b\|a \|c \|a x + b x + c + (- b x - 2b c)\|a )
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R +-----------+ +-----------+
+--R 2 | 2 +-+ +-+ | 2 +-+
+--R (- 2log(a x + b x + c)\|- 4a c + b \|a \|c - 2b x\|- 4a c + b \|a )
+--R *
+--R +--------------+
+--R | 2
+--R \|a x + b x + c
+--R +
+--R +-----------+
+--R 2 | 2 +-+ +-+
+--R (4a x + 2b x)\|- 4a c + b \|a \|c
+--R +
+--R +-----------+
+--R 2 | 2 +-+
+--R (b x + 2c)log(a x + b x + c)\|- 4a c + b \|a
+--R /
+--R +-----------+ +--------------+
+--R | 2 +-+ +-+ | 2
+--R 4a\|- 4a c + b \|a \|c \|a x + b x + c
+--R +
+--R +-----------+
+--R | 2 +-+
+--R (- 2a b x - 4a c)\|- 4a c + b \|a
+--R Type: Expression Integer
+--E
+
+--S 13
+cc2:=aa.2-bb1
+--R
+--R (6)
+--R +--------------+
+--R +---+ +-+ | 2 2 +---+
+--R (2b\|- a \|c \|a x + b x + c + (- b x - 2b c)\|- a )
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R +-----------+ +--------------+ +-----------+
+--R | 2 +-+ | 2 2 | 2
+--R (- 4b\|- 4a c + b \|c \|a x + b x + c + (2b x + 4b c)\|- 4a c + b )
+--R *
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R \|- a \|a x + b x + c - \|- a \|c
+--R atan(------------------------------------)
+--R a x
+--R +
+--R +-----------+
+--R 2 | 2 +---+ +-+
+--R - 2log(a x + b x + c)\|- 4a c + b \|- a \|c
+--R +
+--R +-----------+
+--R | 2 +---+
+--R - 2b x\|- 4a c + b \|- a
+--R *
+--R +--------------+
+--R | 2
+--R \|a x + b x + c
+--R +
+--R +-----------+
+--R 2 | 2 +---+ +-+
+--R (4a x + 2b x)\|- 4a c + b \|- a \|c
+--R +
+--R +-----------+
+--R 2 | 2 +---+
+--R (b x + 2c)log(a x + b x + c)\|- 4a c + b \|- a
+--R /
+--R +-----------+ +--------------+
+--R | 2 +---+ +-+ | 2
+--R 4a\|- 4a c + b \|- a \|c \|a x + b x + c
+--R +
+--R +-----------+
+--R | 2 +---+
+--R (- 2a b x - 4a c)\|- 4a c + b \|- a
+--R Type: Expression Integer
+--E
+
+--S 14
+cc3:=aa.2-bb1
+--R
+--R (7)
+--R +--------------+
+--R +---+ +-+ | 2 2 +---+
+--R (2b\|- a \|c \|a x + b x + c + (- b x - 2b c)\|- a )
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R +-----------+ +--------------+ +-----------+
+--R | 2 +-+ | 2 2 | 2
+--R (- 4b\|- 4a c + b \|c \|a x + b x + c + (2b x + 4b c)\|- 4a c + b )
+--R *
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R \|- a \|a x + b x + c - \|- a \|c
+--R atan(------------------------------------)
+--R a x
+--R +
+--R +-----------+
+--R 2 | 2 +---+ +-+
+--R - 2log(a x + b x + c)\|- 4a c + b \|- a \|c
+--R +
+--R +-----------+
+--R | 2 +---+
+--R - 2b x\|- 4a c + b \|- a
+--R *
+--R +--------------+
+--R | 2
+--R \|a x + b x + c
+--R +
+--R +-----------+
+--R 2 | 2 +---+ +-+
+--R (4a x + 2b x)\|- 4a c + b \|- a \|c
+--R +
+--R +-----------+
+--R 2 | 2 +---+
+--R (b x + 2c)log(a x + b x + c)\|- 4a c + b \|- a
+--R /
+--R +-----------+ +--------------+
+--R | 2 +---+ +-+ | 2
+--R 4a\|- 4a c + b \|- a \|c \|a x + b x + c
+--R +
+--R +-----------+
+--R | 2 +---+
+--R (- 2a b x - 4a c)\|- 4a c + b \|- a
+--R Type: Expression Integer
+--E
+
+--S 15 14:266 Axiom cannot simplify these expressions
+cc4:=aa.2-bb2
+--R
+--R (8)
+--R +---------+ +--------------+ +---------+
+--R +-+ | 2 | 2 2 | 2
+--R (- 4b\|c \|4a c - b \|a x + b x + c + (2b x + 4b c)\|4a c - b )
+--R *
+--R +--------------+
+--R +---+ | 2 +---+ +-+
+--R \|- a \|a x + b x + c - \|- a \|c
+--R atan(------------------------------------)
+--R a x
+--R +
+--R +--------------+
+--R +---+ +-+ | 2 2 +---+
+--R (4b\|- a \|c \|a x + b x + c + (- 2b x - 4b c)\|- a )
+--R *
+--R +---------+
+--R | 2
+--R (2a x + b)\|4a c - b
+--R atan(----------------------)
+--R 2
+--R 4a c - b
+--R +
+--R +---------+
+--R 2 +---+ +-+ +---+ | 2
+--R (- 2log(a x + b x + c)\|- a \|c - 2b x\|- a )\|4a c - b
+--R *
+--R +--------------+
+--R | 2
+--R \|a x + b x + c
+--R +
+--R 2 +---+ +-+ 2 +---+
+--R ((4a x + 2b x)\|- a \|c + (b x + 2c)log(a x + b x + c)\|- a )
+--R *
+--R +---------+
+--R | 2
+--R \|4a c - b
+--R /
+--R +---------+ +--------------+
+--R +---+ +-+ | 2 | 2
+--R 4a\|- a \|c \|4a c - b \|a x + b x + c
+--R +
+--R +---------+
+--R +---+ | 2
+--R (- 2a b x - 4a c)\|- a \|4a c - b
+--R Type: Expression Integer
+--E
+
@
\section{\cite{1}:14.267~~~~~$\displaystyle
@@ -127,7 +583,7 @@
<<*>>=
)clear all
---S 3 of 15
+--S 16
aa:=integrate(x^2/(a*x^2+b*x+c),x)
--R
--R
@@ -173,6 +629,141 @@
--R ]
--R Type: Union(List Expression Integer,...)
--E
+
+--S 17
+t1:=integrate(1/(a*x^2+b*x+c),x)
+--R
+--R
+--R (2)
+--R [
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R /
+--R +-----------+
+--R | 2
+--R \|- 4a c + b
+--R ,
+--R +---------+
+--R | 2
+--R (2a x + b)\|4a c - b
+--R 2atan(----------------------)
+--R 2
+--R 4a c - b
+--R -----------------------------]
+--R +---------+
+--R | 2
+--R \|4a c - b
+--R Type: Union(List Expression Integer,...)
+--E
+
+--S 18
+bb1:=x/a-b/(2*a^2)*log(a*x^2+b*x+c)+(b^2-2*a*c)/(2*a^2)*t1.1
+--R
+--R
+--R (3)
+--R 2
+--R (- 2a c + b )
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R +-----------+
+--R 2 | 2
+--R (- b log(a x + b x + c) + 2a x)\|- 4a c + b
+--R /
+--R +-----------+
+--R 2 | 2
+--R 2a \|- 4a c + b
+--R Type: Expression Integer
+--E
+
+--S 19
+bb2:=x/a-b/(2*a^2)*log(a*x^2+b*x+c)+(b^2-2*a*c)/(2*a^2)*t1.2
+--R
+--R
+--R (4)
+--R +---------+
+--R | 2
+--R 2 (2a x + b)\|4a c - b
+--R (- 4a c + 2b )atan(----------------------)
+--R 2
+--R 4a c - b
+--R +
+--R +---------+
+--R 2 | 2
+--R (- b log(a x + b x + c) + 2a x)\|4a c - b
+--R /
+--R +---------+
+--R 2 | 2
+--R 2a \|4a c - b
+--R Type: Expression Integer
+--E
+
+--S 20
+cc1:=bb1-aa.1
+--R
+--R (5)
+--R 2
+--R (- 2a c + b )
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R 2
+--R (- 2a c + b )
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x
+--R +
+--R 3
+--R - 4a b c + b
+--R /
+--R 2
+--R a x + b x + c
+--R /
+--R +-----------+
+--R 2 | 2
+--R 2a \|- 4a c + b
+--R Type: Expression Integer
+--E
+
+--S 21 14:267 Schaums and Axiom differ by a constant
+dd1:=complexNormalize cc1
+--R
+--R 2 3 2 2
+--R (- 2a c + b )log(- 16a c + 4a b )
+--R (6) ---------------------------------
+--R +-----------+
+--R 2 | 2
+--R 2a \|- 4a c + b
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.268~~~~~$\displaystyle
@@ -184,7 +775,7 @@
<<*>>=
)clear all
---S 4 of 15
+--S 22 14:268 Axiom cannot compute this integral
aa:=integrate(x^m/(a*x^2+b*x+c),x)
--R
--R
@@ -195,6 +786,7 @@
--I c + %N b + %N a
--R Type: Union(Expression Integer,...)
--E
+
@
\section{\cite{1}:14.269~~~~~$\displaystyle
@@ -206,7 +798,7 @@
<<*>>=
)clear all
---S 5 of 15
+--S 23
aa:=integrate(1/(x*(a*x^2+b*x+c)),x)
--R
--R
@@ -251,6 +843,179 @@
--R ]
--R Type: Union(List Expression Integer,...)
--E
+
+--S 24
+t1:=integrate(1/(a*x^2+b*x+c),x)
+--R
+--R
+--R (2)
+--R [
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R /
+--R +-----------+
+--R | 2
+--R \|- 4a c + b
+--R ,
+--R +---------+
+--R | 2
+--R (2a x + b)\|4a c - b
+--R 2atan(----------------------)
+--R 2
+--R 4a c - b
+--R -----------------------------]
+--R +---------+
+--R | 2
+--R \|4a c - b
+--R Type: Union(List Expression Integer,...)
+--E
+
+--S 25
+bb1:=1/(2*c)*log(x^2/(a*x^2+b*x+c))-b/(2*c)*t1.1
+--R
+--R
+--R (3)
+--R -
+--R b
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R 2 +-----------+
+--R x | 2
+--R log(--------------)\|- 4a c + b
+--R 2
+--R a x + b x + c
+--R /
+--R +-----------+
+--R | 2
+--R 2c\|- 4a c + b
+--R Type: Expression Integer
+--E
+
+--S 26
+bb2:=1/(2*c)*log(x^2/(a*x^2+b*x+c))-b/(2*c)*t1.2
+--R
+--R
+--R +---------+
+--R | 2 2 +---------+
+--R (2a x + b)\|4a c - b x | 2
+--R - 2b atan(----------------------) + log(--------------)\|4a c - b
+--R 2 2
+--R 4a c - b a x + b x + c
+--R (4) -------------------------------------------------------------------
+--R +---------+
+--R | 2
+--R 2c\|4a c - b
+--R Type: Expression Integer
+--E
+
+--S 27
+cc1:=bb1-aa.1
+--R
+--R (5)
+--R -
+--R b
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R -
+--R b
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b
+--R +
+--R 2 2 3
+--R (- 8a c + 2a b )x - 4a b c + b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R 2 +-----------+
+--R 2 x | 2
+--R (log(a x + b x + c) - 2log(x) + log(--------------))\|- 4a c + b
+--R 2
+--R a x + b x + c
+--R /
+--R +-----------+
+--R | 2
+--R 2c\|- 4a c + b
+--R Type: Expression Integer
+--E
+
+--S 28
+dd1:=expandLog cc1
+--R
+--R (6)
+--R -
+--R b
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R +
+--R -
+--R b
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x
+--R +
+--R 3
+--R - 4a b c + b
+--R +
+--R 2
+--R 2b log(a x + b x + c)
+--R /
+--R +-----------+
+--R | 2
+--R 2c\|- 4a c + b
+--R Type: Expression Integer
+--E
+
+--S 29 14:269 Schaums and Axiom differ by a constant
+ee1:=complexNormalize dd1
+--R
+--R 3 2 2
+--R b log(- 16a c + 4a b )
+--R (7) - ----------------------
+--R +-----------+
+--R | 2
+--R 2c\|- 4a c + b
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.270~~~~~$\displaystyle
@@ -262,7 +1027,7 @@
<<*>>=
)clear all
---S 6 of 15
+--S 30
aa:=integrate(1/(x^2*(a*x^2+b*x+c)),x)
--R
--R
@@ -308,6 +1073,186 @@
--R ]
--R Type: Union(List Expression Integer,...)
--E
+
+--S 31
+t1:=integrate(1/(a*x^2+b*x+c),x)
+--R
+--R
+--R (2)
+--R [
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R /
+--R +-----------+
+--R | 2
+--R \|- 4a c + b
+--R ,
+--R +---------+
+--R | 2
+--R (2a x + b)\|4a c - b
+--R 2atan(----------------------)
+--R 2
+--R 4a c - b
+--R -----------------------------]
+--R +---------+
+--R | 2
+--R \|4a c - b
+--R Type: Union(List Expression Integer,...)
+--E
+
+--S 32
+bb1:=b/(2*c^2)*log((a*x^2+b*x+c)/x^2)-1/(c*x)+(b^2-2*a*c)/(2*c^2)*t1.1
+--R
+--R
+--R (3)
+--R 2
+--R (- 2a c + b )x
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R 2 +-----------+
+--R a x + b x + c | 2
+--R (b x log(--------------) - 2c)\|- 4a c + b
+--R 2
+--R x
+--R /
+--R +-----------+
+--R 2 | 2
+--R 2c x\|- 4a c + b
+--R Type: Expression Integer
+--E
+
+--S 33
+bb2:=b/(2*c^2)*log((a*x^2+b*x+c)/x^2)-1/(c*x)+(b^2-2*a*c)/(2*c^2)*t1.2
+--R
+--R
+--R (4)
+--R +---------+
+--R | 2
+--R 2 (2a x + b)\|4a c - b
+--R (- 4a c + 2b )x atan(----------------------)
+--R 2
+--R 4a c - b
+--R +
+--R 2 +---------+
+--R a x + b x + c | 2
+--R (b x log(--------------) - 2c)\|4a c - b
+--R 2
+--R x
+--R /
+--R +---------+
+--R 2 | 2
+--R 2c x\|4a c - b
+--R Type: Expression Integer
+--E
+
+--S 34
+cc1:=bb1-aa.1
+--R
+--R (5)
+--R 2
+--R (- 2a c + b )
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R 2
+--R (- 2a c + b )
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x
+--R +
+--R 3
+--R - 4a b c + b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R 2 +-----------+
+--R 2 a x + b x + c | 2
+--R (- b log(a x + b x + c) + 2b log(x) + b log(--------------))\|- 4a c + b
+--R 2
+--R x
+--R /
+--R +-----------+
+--R 2 | 2
+--R 2c \|- 4a c + b
+--R Type: Expression Integer
+--E
+
+--S 35
+dd1:=expandLog cc1
+--R
+--R (6)
+--R 2
+--R (- 2a c + b )
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R +
+--R 2
+--R (- 2a c + b )
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x
+--R +
+--R 3
+--R - 4a b c + b
+--R +
+--R 2 2
+--R (4a c - 2b )log(a x + b x + c)
+--R /
+--R +-----------+
+--R 2 | 2
+--R 2c \|- 4a c + b
+--R Type: Expression Integer
+--E
+
+--S 36 14:270 Schaums and Axiom differ by a constant
+ee1:=complexNormalize dd1
+--R
+--R 2 3 2 2
+--R (- 2a c + b )log(- 16a c + 4a b )
+--R (7) ---------------------------------
+--R +-----------+
+--R 2 | 2
+--R 2c \|- 4a c + b
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.271~~~~~$\displaystyle
@@ -320,7 +1265,7 @@
<<*>>=
)clear all
---S 7 of 15
+--S 37 14:271 Axiom cannot compute this integral
aa:=integrate(1/(x^n*(a*x^2+b*x+c)),x)
--R
--R
@@ -331,6 +1276,7 @@
--I (c + %N b + %N a)%N
--R Type: Union(Expression Integer,...)
--E
+
@
\section{\cite{1}:14.272~~~~~$\displaystyle
@@ -342,7 +1288,7 @@
<<*>>=
)clear all
---S 8 of 15
+--S 38
aa:=integrate(1/(a*x^2+b*x+c)^2,x)
--R
--R
@@ -383,6 +1329,213 @@
--R ]
--R Type: Union(List Expression Integer,...)
--E
+
+--S 39
+t1:=integrate(1/(a*x^2+b*x+c),x)
+--R
+--R
+--R (2)
+--R [
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R /
+--R +-----------+
+--R | 2
+--R \|- 4a c + b
+--R ,
+--R +---------+
+--R | 2
+--R (2a x + b)\|4a c - b
+--R 2atan(----------------------)
+--R 2
+--R 4a c - b
+--R -----------------------------]
+--R +---------+
+--R | 2
+--R \|4a c - b
+--R Type: Union(List Expression Integer,...)
+--E
+
+--S 40
+bb1:=(b*x+2*c)/((4*a*c-b^2)*(a*x^2+b*x+c))-b/(4*a*c-b^2)*t1.1
+--R
+--R
+--R (3)
+--R 2 2
+--R (- a b x - b x - b c)
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R +-----------+
+--R | 2
+--R (b x + 2c)\|- 4a c + b
+--R /
+--R +-----------+
+--R 2 2 2 3 2 2 | 2
+--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|- 4a c + b
+--R Type: Expression Integer
+--E
+
+--S 41
+bb2:=(b*x+2*c)/((4*a*c-b^2)*(a*x^2+b*x+c))-b/(4*a*c-b^2)*t1.2
+--R
+--R
+--R (4)
+--R +---------+
+--R | 2
+--R 2 2 (2a x + b)\|4a c - b
+--R (- 2a b x - 2b x - 2b c)atan(----------------------)
+--R 2
+--R 4a c - b
+--R +
+--R +---------+
+--R | 2
+--R (b x + 2c)\|4a c - b
+--R /
+--R +---------+
+--R 2 2 2 3 2 2 | 2
+--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|4a c - b
+--R Type: Expression Integer
+--E
+
+--S 42
+cc1:=aa.1-bb1
+--R
+--R (5)
+--R 2 2 2
+--R ((a b + 2a )x + (b + 2a b)x + (b + 2a)c)
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R +-----------+
+--R | 2
+--R ((- b + 2a)x - 2c + b)\|- 4a c + b
+--R /
+--R +-----------+
+--R 2 2 2 3 2 2 | 2
+--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|- 4a c + b
+--R Type: Expression Integer
+--E
+
+--S 43
+cc2:=aa.2-bb1
+--R
+--R (6)
+--R +---------+
+--R 2 2 | 2
+--R (a b x + b x + b c)\|4a c - b
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R +---------+
+--R +-----------+ | 2
+--R 2 2 | 2 (2a x + b)\|4a c - b
+--R (4a x + 4a b x + 4a c)\|- 4a c + b atan(----------------------)
+--R 2
+--R 4a c - b
+--R +
+--R +-----------+ +---------+
+--R | 2 | 2
+--R ((- b + 2a)x - 2c + b)\|- 4a c + b \|4a c - b
+--R /
+--R +-----------+ +---------+
+--R 2 2 2 3 2 2 | 2 | 2
+--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|- 4a c + b \|4a c - b
+--R Type: Expression Integer
+--E
+
+--S 44
+cc3:=aa.1-bb2
+--R
+--R (7)
+--R +---------+
+--R 2 2 | 2
+--R (2a x + 2a b x + 2a c)\|4a c - b
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R +---------+
+--R +-----------+ | 2
+--R 2 2 | 2 (2a x + b)\|4a c - b
+--R (2a b x + 2b x + 2b c)\|- 4a c + b atan(----------------------)
+--R 2
+--R 4a c - b
+--R +
+--R +-----------+ +---------+
+--R | 2 | 2
+--R ((- b + 2a)x - 2c + b)\|- 4a c + b \|4a c - b
+--R /
+--R +-----------+ +---------+
+--R 2 2 2 3 2 2 | 2 | 2
+--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|- 4a c + b \|4a c - b
+--R Type: Expression Integer
+--E
+
+--S 45 14:272 Axiom cannot simplify these expressions
+cc4:=aa.2-bb2
+--R
+--R (8)
+--R +---------+
+--R | 2
+--R 2 2 2 (2a x + b)\|4a c - b
+--R ((2a b + 4a )x + (2b + 4a b)x + (2b + 4a)c)atan(----------------------)
+--R 2
+--R 4a c - b
+--R +
+--R +---------+
+--R | 2
+--R ((- b + 2a)x - 2c + b)\|4a c - b
+--R /
+--R +---------+
+--R 2 2 2 3 2 2 | 2
+--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|4a c - b
+--R Type: Expression Integer
+--E
+
@
\section{\cite{1}:14.273~~~~~$\displaystyle
@@ -394,7 +1547,7 @@
<<*>>=
)clear all
---S 9 of 15
+--S 46
aa:=integrate(x/(a*x^2+b*x+c)^2,x)
--R
--R
@@ -440,6 +1593,177 @@
--R ]
--R Type: Union(List Expression Integer,...)
--E
+
+--S 47
+t1:=integrate(1/(a*x^2+b*x+c),x)
+--R
+--R
+--R (2)
+--R [
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R /
+--R +-----------+
+--R | 2
+--R \|- 4a c + b
+--R ,
+--R +---------+
+--R | 2
+--R (2a x + b)\|4a c - b
+--R 2atan(----------------------)
+--R 2
+--R 4a c - b
+--R -----------------------------]
+--R +---------+
+--R | 2
+--R \|4a c - b
+--R Type: Union(List Expression Integer,...)
+--E
+
+--S 48
+bb1:=-(b*x+2*c)/((4*a*c-b^2)*(a*x^2+b*x+c))-b/(4*a*c-b^2)*t1.1
+--R
+--R
+--R (3)
+--R 2 2
+--R (- a b x - b x - b c)
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R +-----------+
+--R | 2
+--R (- b x - 2c)\|- 4a c + b
+--R /
+--R +-----------+
+--R 2 2 2 3 2 2 | 2
+--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|- 4a c + b
+--R Type: Expression Integer
+--E
+
+--S 49
+bb2:=-(b*x+2*c)/((4*a*c-b^2)*(a*x^2+b*x+c))-b/(4*a*c-b^2)*t1.2
+--R
+--R
+--R (4)
+--R +---------+
+--R | 2
+--R 2 2 (2a x + b)\|4a c - b
+--R (- 2a b x - 2b x - 2b c)atan(----------------------)
+--R 2
+--R 4a c - b
+--R +
+--R +---------+
+--R | 2
+--R (- b x - 2c)\|4a c - b
+--R /
+--R +---------+
+--R 2 2 2 3 2 2 | 2
+--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|4a c - b
+--R Type: Expression Integer
+--E
+
+--S 50
+cc1:=bb1-aa.1
+--R
+--R (5)
+--R -
+--R b
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R -
+--R b
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b
+--R +
+--R 2 2 3
+--R (- 8a c + 2a b )x - 4a b c + b
+--R /
+--R 2
+--R a x + b x + c
+--R /
+--R +-----------+
+--R 2 | 2
+--R (4a c - b )\|- 4a c + b
+--R Type: Expression Integer
+--E
+
+--S 51
+dd1:=expandLog cc1
+--R
+--R (6)
+--R -
+--R b
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R +
+--R -
+--R b
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x
+--R +
+--R 3
+--R - 4a b c + b
+--R +
+--R 2
+--R 2b log(a x + b x + c)
+--R /
+--R +-----------+
+--R 2 | 2
+--R (4a c - b )\|- 4a c + b
+--R Type: Expression Integer
+--E
+
+--S 52 14:273 Schaums and Axiom differ by a constant
+ee1:=complexNormalize dd1
+--R
+--R 3 2 2
+--R b log(- 16a c + 4a b )
+--R (7) - -------------------------
+--R +-----------+
+--R 2 | 2
+--R (4a c - b )\|- 4a c + b
+--R Type: Expression Integer
+--E
+
@
\section{\cite{1}:14.274~~~~~$\displaystyle
@@ -451,7 +1775,7 @@
<<*>>=
)clear all
---S 10 of 15
+--S 53
aa:=integrate(x^2/(a*x^2+b*x+c)^2,x)
--R
--R
@@ -497,6 +1821,96 @@
--R ]
--R Type: Union(List Expression Integer,...)
--E
+
+--S 54
+t1:=integrate(1/(a*x^2+b*x+c),x)
+--R
+--R
+--R (2)
+--R [
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R /
+--R +-----------+
+--R | 2
+--R \|- 4a c + b
+--R ,
+--R +---------+
+--R | 2
+--R (2a x + b)\|4a c - b
+--R 2atan(----------------------)
+--R 2
+--R 4a c - b
+--R -----------------------------]
+--R +---------+
+--R | 2
+--R \|4a c - b
+--R Type: Union(List Expression Integer,...)
+--E
+
+--S 55
+bb1:=((b^2-2*a*c)*x+b*c)/(a*(4*a*c-b^2)*(a*x^2+b*x+c))+(2*c)/(4*a*c-b^2)*t1.1
+--R
+--R (3)
+--R 2 2 2
+--R (2a c x + 2a b c x + 2a c )
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R +-----------+
+--R 2 | 2
+--R ((- 2a c + b )x + b c)\|- 4a c + b
+--R /
+--R +-----------+
+--R 3 2 2 2 2 3 2 2 2 | 2
+--R ((4a c - a b )x + (4a b c - a b )x + 4a c - a b c)\|- 4a c + b
+--R Type: Expression Integer
+--E
+
+--S 56
+bb2:=((b^2-2*a*c)*x+b*c)/(a*(4*a*c-b^2)*(a*x^2+b*x+c))+(2*c)/(4*a*c-b^2)*t1.2
+--R
+--R (4)
+--R +---------+
+--R | 2
+--R 2 2 2 (2a x + b)\|4a c - b
+--R (4a c x + 4a b c x + 4a c )atan(----------------------)
+--R 2
+--R 4a c - b
+--R +
+--R +---------+
+--R 2 | 2
+--R ((- 2a c + b )x + b c)\|4a c - b
+--R /
+--R +---------+
+--R 3 2 2 2 2 3 2 2 2 | 2
+--R ((4a c - a b )x + (4a b c - a b )x + 4a c - a b c)\|4a c - b
+--R Type: Expression Integer
+--E
+
+--S 57 14:274 Schaums and Axiom agree
+cc1:=aa.1-bb1
+--R
+--R (5) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.275~~~~~$\displaystyle
@@ -514,7 +1928,7 @@
<<*>>=
)clear all
---S 11 of 15
+--S 58 14:275 Axiom cannot compute this integral
aa:=integrate(x^m/(a*x^2+b*x+c)^n,x)
--R
--R
@@ -525,6 +1939,7 @@
--I (c + %N b + %N a)
--R Type: Union(Expression Integer,...)
--E
+
@
\section{\cite{1}:14.276~~~~~$\displaystyle
@@ -541,7 +1956,7 @@
<<*>>=
)clear all
---S 12 of 15
+--S 59 14:276 Axiom cannot compute this integral
aa:=integrate(x^(2*n-1)/(a*x^2+b*x+c)^n,x)
--R
--R
@@ -552,6 +1967,7 @@
--I (c + %N b + %N a)
--R Type: Union(Expression Integer,...)
--E
+
@
\section{\cite{1}:14.277~~~~~$\displaystyle
@@ -568,7 +1984,7 @@
<<*>>=
)clear all
---S 13 of 15
+--S 60
aa:=integrate(1/(x*(a*x^2+b*x+c)^2),x)
--R
--R
@@ -641,6 +2057,380 @@
--R ]
--R Type: Union(List Expression Integer,...)
--E
+
+--S 61
+t1:=integrate(1/(a*x^2+b*x+c)^2,x)
+--R
+--R
+--R (2)
+--R [
+--R 2 2
+--R (2a x + 2a b x + 2a c)
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R +-----------+
+--R | 2
+--R (2a x + b)\|- 4a c + b
+--R /
+--R +-----------+
+--R 2 2 2 3 2 2 | 2
+--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|- 4a c + b
+--R ,
+--R +---------+
+--R | 2 +---------+
+--R 2 2 (2a x + b)\|4a c - b | 2
+--R (4a x + 4a b x + 4a c)atan(----------------------) + (2a x + b)\|4a c - b
+--R 2
+--R 4a c - b
+--R ----------------------------------------------------------------------------
+--R +---------+
+--R 2 2 2 3 2 2 | 2
+--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|4a c - b
+--R ]
+--R Type: Union(List Expression Integer,...)
+--E
+
+--S 62
+t2:=integrate(1/(x*(a*x^2+b*x+c)),x)
+--R
+--R
+--R (3)
+--R [
+--R b
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x
+--R +
+--R 3
+--R - 4a b c + b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R +-----------+
+--R 2 | 2
+--R (- log(a x + b x + c) + 2log(x))\|- 4a c + b
+--R /
+--R +-----------+
+--R | 2
+--R 2c\|- 4a c + b
+--R ,
+--R
+--R +---------+
+--R | 2
+--R (2a x + b)\|4a c - b
+--R - 2b atan(----------------------)
+--R 2
+--R 4a c - b
+--R +
+--R +---------+
+--R 2 | 2
+--R (- log(a x + b x + c) + 2log(x))\|4a c - b
+--R /
+--R +---------+
+--R | 2
+--R 2c\|4a c - b
+--R ]
+--R Type: Union(List Expression Integer,...)
+--E
+
+--S 63
+bb1:=1/(2*c*(a*x^2+b*x+c))-b/(2*c)*t1.1+1/c*t2.1
+--R
+--R
+--R (4)
+--R 2 2 2 2
+--R (- 2a b c x - 2a b c x - 2a b c )
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x
+--R +
+--R 3
+--R 4a b c - b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R 2 3 2 2 4 2 3
+--R ((4a b c - a b )x + (4a b c - b )x + 4a b c - b c)
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x
+--R +
+--R 3
+--R - 4a b c + b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R 2 2 2 3 2 2
+--R ((- 4a c + a b )x + (- 4a b c + b )x - 4a c + b c)
+--R *
+--R 2
+--R log(a x + b x + c)
+--R +
+--R 2 2 2 3 2 2
+--R ((8a c - 2a b )x + (8a b c - 2b )x + 8a c - 2b c)log(x) - 2a b c x
+--R +
+--R 2 2
+--R 4a c - 2b c
+--R *
+--R +-----------+
+--R | 2
+--R \|- 4a c + b
+--R /
+--R +-----------+
+--R 2 3 2 2 2 3 3 2 4 2 3 | 2
+--R ((8a c - 2a b c )x + (8a b c - 2b c )x + 8a c - 2b c )\|- 4a c + b
+--R Type: Expression Integer
+--E
+
+--S 64
+bb2:=1/(2*c*(a*x^2+b*x+c))-b/(2*c)*t1.2+1/c*t2.1
+--R
+--R
+--R (5)
+--R +---------+
+--R 2 3 2 2 4 2 3 | 2
+--R ((4a b c - a b )x + (4a b c - b )x + 4a b c - b c)\|4a c - b
+--R *
+--R log
+--R +-----------+
+--R 2 2 2 | 2 2 2
+--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x
+--R +
+--R 3
+--R - 4a b c + b
+--R /
+--R 2
+--R a x + b x + c
+--R +
+--R +-----------+
+--R 2 2 2 2 | 2
+--R (- 4a b c x - 4a b c x - 4a b c )\|- 4a c + b
+--R *
+--R +---------+
+--R | 2
+--R (2a x + b)\|4a c - b
+--R atan(----------------------)
+--R 2
+--R 4a c - b
+--R +
+--R 2 2 2 3 2 2
+--R ((- 4a c + a b )x + (- 4a b c + b )x - 4a c + b c)
+--R *
+--R 2
+--R log(a x + b x + c)
+--R +
+--R 2 2 2 3 2 2
+--R ((8a c - 2a b )x + (8a b c - 2b )x + 8a c - 2b c)log(x) - 2a b c x
+--R +
+--R 2 2
+--R 4a c - 2b c...
[truncated message content] |
|
From: <gi...@ax...> - 2008-04-20 06:53:53
|
changelog | 1 +
src/input/schaum12.input.pamphlet | 2240 ++++++++++++++++++++++++++++++++++++-
2 files changed, 2226 insertions(+), 15 deletions(-)
New commits:
commit 1ea268fc7b41807e5e002ca9364d16062b21a4cb
Author: Tim Daly <root@localhost.localdomain>
Date: Wed Apr 2 14:30:39 2008 -0400
20080419 tpd src/input/schaum12.input show Schaums-Axiom equivalence
|
|
From: <da...@us...> - 2008-04-19 06:06:04
|
Revision: 858
http://axiom.svn.sourceforge.net/axiom/?rev=858&view=rev
Author: daly
Date: 2008-04-18 23:06:09 -0700 (Fri, 18 Apr 2008)
Log Message:
-----------
20080418 tpd src/input/schaum11.input show Schaums-Axiom equivalence
20080418 tpd src/input/schaum10.input fix Schaums-Axiom equivalence
20080418 tpd src/input/schaum9.input fix Schaums-Axiom equivalence
20080418 tpd src/input/schaum8.input fix Schaums-Axiom equivalence
20080418 tpd src/input/schaum7.input fix Schaums-Axiom equivalence
20080418 tpd src/input/schaum6.input fix Schaums-Axiom equivalence
20080418 tpd src/input/schaum2.input fix Schaums-Axiom equivalence
20080418 tpd src/input/schaum1.input fix Schaums-Axiom equivalence
Modified Paths:
--------------
trunk/axiom/changelog
trunk/axiom/src/input/schaum1.input.pamphlet
trunk/axiom/src/input/schaum10.input.pamphlet
trunk/axiom/src/input/schaum11.input.pamphlet
trunk/axiom/src/input/schaum2.input.pamphlet
trunk/axiom/src/input/schaum6.input.pamphlet
trunk/axiom/src/input/schaum7.input.pamphlet
trunk/axiom/src/input/schaum8.input.pamphlet
trunk/axiom/src/input/schaum9.input.pamphlet
Modified: trunk/axiom/changelog
===================================================================
--- trunk/axiom/changelog 2008-04-18 06:42:58 UTC (rev 857)
+++ trunk/axiom/changelog 2008-04-19 06:06:09 UTC (rev 858)
@@ -1,3 +1,11 @@
+20080418 tpd src/input/schaum11.input show Schaums-Axiom equivalence
+20080418 tpd src/input/schaum10.input fix Schaums-Axiom equivalence
+20080418 tpd src/input/schaum9.input fix Schaums-Axiom equivalence
+20080418 tpd src/input/schaum8.input fix Schaums-Axiom equivalence
+20080418 tpd src/input/schaum7.input fix Schaums-Axiom equivalence
+20080418 tpd src/input/schaum6.input fix Schaums-Axiom equivalence
+20080418 tpd src/input/schaum2.input fix Schaums-Axiom equivalence
+20080418 tpd src/input/schaum1.input fix Schaums-Axiom equivalence
20080417 tpd src/input/schaum10.input show Schaums-Axiom equivalence
20080417 tpd src/input/schaum9.input show Schaums-Axiom equivalence
20080417 tpd src/input/schaum8.input show Schaums-Axiom equivalence
Modified: trunk/axiom/src/input/schaum1.input.pamphlet
===================================================================
--- trunk/axiom/src/input/schaum1.input.pamphlet 2008-04-18 06:42:58 UTC (rev 857)
+++ trunk/axiom/src/input/schaum1.input.pamphlet 2008-04-19 06:06:09 UTC (rev 858)
@@ -1201,7 +1201,7 @@
--R Type: RewriteRule(Integer,Integer,Expression Integer)
--E
---S 94 14:80 Schaums and Axiom agree
+--S 94
dd:=explog cc
--R
--R n + 1 n
@@ -1210,8 +1210,14 @@
--R a n + a
--R Type: Expression Integer
--E
+
+--S 95 14:80 Schaums and Axiom agree
+ee:=complexNormalize dd
+--R
+--R (6) 0
+--R Type: Expression Integer
+--E
@
-The numerator is clearly zero but I cannot get Axiom to simplify it.
\section{\cite{1}:14.81~~~~~$\displaystyle
\int{x(ax+b)^n~dx}$}
@@ -1221,7 +1227,7 @@
$$
<<*>>=
)clear all
---S 95
+--S 96
aa:=integrate(x*(a*x+b)^n,x)
--R
--R 2 2 2 2 n log(a x + b)
@@ -1232,7 +1238,7 @@
--R Type: Union(Expression Integer,...)
--E
---S 96
+--S 97
bb:=((a*x+b)^(n+2))/((n+2)*a^2)-(b*(a*x+b)^(n+1))/((n+1)*a^2)
--R
--R n + 2 n + 1
@@ -1243,7 +1249,7 @@
--R Type: Expression Integer
--E
---S 97
+--S 98
cc:=aa-bb
--R
--R (3)
@@ -1258,7 +1264,7 @@
--R Type: Expression Integer
--E
---S 98
+--S 99
explog:=rule(%e^(n*log(x)) == x^n)
--R
--R n log(x) n
@@ -1266,7 +1272,7 @@
--R Type: RewriteRule(Integer,Integer,Expression Integer)
--E
---S 99 14:81 Schaums and Axiom agreement cannot be determined
+--S 100
dd:=explog cc
--R
--R (5)
@@ -1280,6 +1286,13 @@
--R a n + 3a n + 2a
--R Type: Expression Integer
--E
+
+--S 101
+ee:=complexNormalize dd
+--R
+--R (6) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.82~~~~~$\displaystyle
\int{x^2(ax+b)^n~dx}$}
@@ -1292,7 +1305,7 @@
<<*>>=
)clear all
---S 100
+--S 102
aa:=integrate(x^2*(a*x+b)^n,x)
--R
--R (1)
@@ -1304,7 +1317,7 @@
--R Type: Union(Expression Integer,...)
--E
---S 101
+--S 103
bb:=(a*x+b)^(n+3)/((n+3)*a^3)-(2*b*(a*x+b)^(n+2))/((n+2)*a^3)+(b^2*(a*x+b)^(n+1))/((n+1)*a^3)
--R
--R (2)
@@ -1319,7 +1332,7 @@
--R Type: Expression Integer
--E
---S 102 14:82 Schaums and Axiom agreement cannot be determined
+--S 104
cc:=aa-bb
--R
--R (3)
@@ -1339,6 +1352,39 @@
--R a n + 6a n + 11a n + 6a
--R Type: Expression Integer
--E
+
+--S 105
+explog:=rule(%e^(n*log(x)) == x^n)
+--R
+--R n log(x) n
+--R (4) %e == x
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 106
+dd:=explog cc
+--R
+--R (5)
+--R 2 n + 3 2 n + 2
+--R (- n - 3n - 2)(a x + b) + (2b n + 8b n + 6b)(a x + b)
+--R +
+--R 2 2 2 2 n + 1
+--R (- b n - 5b n - 6b )(a x + b)
+--R +
+--R 3 2 3 3 3 2 2 2 2 2 3 n
+--R ((a n + 3a n + 2a )x + (a b n + a b n)x - 2a b n x + 2b )(a x + b)
+--R /
+--R 3 3 3 2 3 3
+--R a n + 6a n + 11a n + 6a
+--R Type: Expression Integer
+--E
+
+--S 107 14:82 Schaums and Axiom agree
+ee:=complexNormalize dd
+--R
+--R (6) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.83~~~~~$\displaystyle
\int{x^m(ax+b)^n}~dx$}
@@ -1361,7 +1407,7 @@
$$
<<*>>=
---S 103 14:83 Axiom cannot do this integration
+--S 108 14:83 Axiom cannot do this integration
aa:=integrate(x^m*(a*x+b)^n,x)
--R
--R x
Modified: trunk/axiom/src/input/schaum10.input.pamphlet
===================================================================
--- trunk/axiom/src/input/schaum10.input.pamphlet 2008-04-18 06:42:58 UTC (rev 857)
+++ trunk/axiom/src/input/schaum10.input.pamphlet 2008-04-19 06:06:09 UTC (rev 858)
@@ -133,7 +133,7 @@
--R Type: Expression Integer
--E
---S 11 14:211 Axiom cannot simplify this expression
+--S 11
cc:=aa-bb
--R
--R +-------+ +-------+
@@ -144,6 +144,16 @@
--R Type: Expression Integer
--E
+--S 12 14:211 Schaums and Axiom differ by a constant
+dd:=complexNormalize cc
+--R
+--R 2 2
+--R a log(- a )
+--R (4) - -----------
+--R 2
+--R Type: Expression Integer
+--E
+
@
\section{\cite{1}:14.212~~~~~$\displaystyle
@@ -154,7 +164,7 @@
<<*>>=
)clear all
---S 12
+--S 13
aa:=integrate(x^3/sqrt(x^2-a^2),x)
--R
--R
@@ -168,7 +178,7 @@
--R Type: Union(Expression Integer,...)
--E
---S 13
+--S 14
bb:=(x^2-a^2)^(3/2)/3+a^2*sqrt(x^2-a^2)
--R
--R +-------+
@@ -179,7 +189,7 @@
--R Type: Expression Integer
--E
---S 14 14:212 Schaums and Axiom agree
+--S 15 14:212 Schaums and Axiom agree
cc:=aa-bb
--R
--R (3) 0
@@ -194,7 +204,7 @@
<<*>>=
)clear all
---S 15
+--S 16
aa:=integrate(1/(x*sqrt(x^2-a^2)),x)
--R
--R
@@ -208,29 +218,134 @@
--R Type: Union(Expression Integer,...)
--E
---S 16
-bb:=1/a*asec(abs(x/a))
+--S 17
+bb:=1/a*asec(x/a)
--R
---R x
---R asec(abs(-))
---R a
---R (2) ------------
---R a
+--R x
+--R asec(-)
+--R a
+--R (2) -------
+--R a
--R Type: Expression Integer
--E
---S 17 14:213 Axiom cannot simplify this expression
+--S 18
cc:=aa-bb
--R
--R +-------+
--R | 2 2
---R \|x - a - x x
---R 2atan(--------------) - asec(abs(-))
---R a a
---R (3) ------------------------------------
---R a
+--R \|x - a - x x
+--R 2atan(--------------) - asec(-)
+--R a a
+--R (3) -------------------------------
+--R a
--R Type: Expression Integer
--E
+
+--S 19
+asecrule:=rule(asec(x) == 1/2*%pi+%i*log(sqrt(1-1/x^2)+%i/x))
+--R
+--R +------+
+--R | 2
+--R |x - 1
+--R x |------ + %i
+--R | 2
+--R \| x
+--R 2%i log(---------------) + %pi
+--R x
+--R (4) asec(x) == ------------------------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer)
+--E
+
+--S 20
+dd:=asecrule cc
+--R
+--R +-------+
+--R | 2 2
+--R |x - a
+--R x |------- + %i a +-------+
+--R | 2 | 2 2
+--R \| x \|x - a - x
+--R - 2%i log(------------------) + 4atan(--------------) - %pi
+--R x a
+--R (5) -----------------------------------------------------------
+--R 2a
+--R Type: Expression Complex Integer
+--E
+
+--S 21
+atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x)))
+--R
+--R - x + %i
+--R %i log(--------)
+--R x + %i
+--R (6) atan(x) == - ----------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer)
+--E
+
+--S 22
+ee:=atanrule dd
+--R
+--R +-------+
+--R | 2 2
+--R |x - a
+--R x |------- + %i a +-------+
+--R | 2 | 2 2
+--R \| x - \|x - a + x + %i a
+--R - 2%i log(------------------) - 2%i log(-----------------------) - %pi
+--R x +-------+
+--R | 2 2
+--R \|x - a - x + %i a
+--R (7) ----------------------------------------------------------------------
+--R 2a
+--R Type: Expression Complex Integer
+--E
+
+--S 23
+ff:=expandLog ee
+--R
+--R (8)
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R 2%i log(\|x - a - x + %i a) - 2%i log(\|x - a - x - %i a)
+--R +
+--R +-------+
+--R | 2 2
+--R |x - a
+--R - 2%i log(x |------- + %i a) + 2%i log(x) - 2%i log(- 1) - %pi
+--R | 2
+--R \| x
+--R /
+--R 2a
+--R Type: Expression Complex Integer
+--E
+
+--S 24
+gg:=rootSimp ff
+--R
+--R (9)
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R - 2%i log(\|x - a + %i a) + 2%i log(\|x - a - x + %i a)
+--R +
+--R +-------+
+--R | 2 2
+--R - 2%i log(\|x - a - x - %i a) + 2%i log(x) - 2%i log(- 1) - %pi
+--R /
+--R 2a
+--R Type: Expression Complex Integer
+--E
+
+--S 25 14:213 Schaums and Axiom differ by a constant
+hh:=complexNormalize gg
+--R
+--R %pi
+--R (10) - ---
+--R 2a
+--R Type: Expression Complex Integer
+--E
@
\section{\cite{1}:14.214~~~~~$\displaystyle
@@ -241,7 +356,7 @@
<<*>>=
)clear all
---S 18
+--S 26
aa:=integrate(1/(x^2*sqrt(x^2-a^2)),x)
--R
--R
@@ -253,7 +368,7 @@
--R Type: Union(Expression Integer,...)
--E
---S 19
+--S 27
bb:=sqrt(x^2-a^2)/(a^2*x)
--R
--R +-------+
@@ -265,7 +380,7 @@
--R Type: Expression Integer
--E
---S 20 14:214 Schaums and Axiom differ by a constant
+--S 28 14:214 Schaums and Axiom differ by a constant
cc:=aa-bb
--R
--R 1
@@ -284,7 +399,7 @@
<<*>>=
)clear all
---S 21
+--S 29
aa:=integrate(1/(x^3*sqrt(x^2-a^2)),x)
--R
--R
@@ -305,33 +420,143 @@
--R Type: Union(Expression Integer,...)
--E
---S 22
-bb:=sqrt(x^2-a^2)/(2*a^2*x^2)+1/(2*a^3)*asec(abs(x/a))
+--S 30
+bb:=sqrt(x^2-a^2)/(2*a^2*x^2)+1/(2*a^3)*asec(x/a)
--R
---R +-------+
---R 2 x | 2 2
---R x asec(abs(-)) + a\|x - a
---R a
---R (2) ----------------------------
---R 3 2
---R 2a x
+--R +-------+
+--R | 2 2 2 x
+--R a\|x - a + x asec(-)
+--R a
+--R (2) -----------------------
+--R 3 2
+--R 2a x
--R Type: Expression Integer
--E
---S 23 14:215 Axiom cannot simplify this expression
+--S 31
cc:=aa-bb
+--R
--R
--R +-------+
--R | 2 2
---R \|x - a - x x
---R 2atan(--------------) - asec(abs(-))
---R a a
---R (3) ------------------------------------
---R 3
---R 2a
+--R \|x - a - x x
+--R 2atan(--------------) - asec(-)
+--R a a
+--R (3) -------------------------------
+--R 3
+--R 2a
--R Type: Expression Integer
--E
+--S 32
+atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x)))
+--R
+--R - x + %i
+--R %i log(--------)
+--R x + %i
+--R (4) atan(x) == - ----------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer)
+--E
+
+--S 33
+dd:=atanrule cc
+--R
+--R +-------+
+--R | 2 2
+--R - \|x - a + x + %i a x
+--R - %i log(-----------------------) - asec(-)
+--R +-------+ a
+--R | 2 2
+--R \|x - a - x + %i a
+--R (5) -------------------------------------------
+--R 3
+--R 2a
+--R Type: Expression Complex Integer
+--E
+
+--S 34
+asecrule:=rule(asec(x) == 1/2*%pi+%i*log(sqrt(1-1/x^2)+%i/x))
+--R
+--R +------+
+--R | 2
+--R |x - 1
+--R x |------ + %i
+--R | 2
+--R \| x
+--R 2%i log(---------------) + %pi
+--R x
+--R (6) asec(x) == ------------------------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer)
+--E
+
+--S 35
+ee:=asecrule dd
+--R
+--R +-------+
+--R | 2 2
+--R |x - a
+--R x |------- + %i a +-------+
+--R | 2 | 2 2
+--R \| x - \|x - a + x + %i a
+--R - 2%i log(------------------) - 2%i log(-----------------------) - %pi
+--R x +-------+
+--R | 2 2
+--R \|x - a - x + %i a
+--R (7) ----------------------------------------------------------------------
+--R 3
+--R 4a
+--R Type: Expression Complex Integer
+--E
+
+--S 36
+ff:=expandLog ee
+--R
+--R (8)
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R 2%i log(\|x - a - x + %i a) - 2%i log(\|x - a - x - %i a)
+--R +
+--R +-------+
+--R | 2 2
+--R |x - a
+--R - 2%i log(x |------- + %i a) + 2%i log(x) - 2%i log(- 1) - %pi
+--R | 2
+--R \| x
+--R /
+--R 3
+--R 4a
+--R Type: Expression Complex Integer
+--E
+
+--S 37
+gg:=rootSimp ff
+--R
+--R (9)
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R - 2%i log(\|x - a + %i a) + 2%i log(\|x - a - x + %i a)
+--R +
+--R +-------+
+--R | 2 2
+--R - 2%i log(\|x - a - x - %i a) + 2%i log(x) - 2%i log(- 1) - %pi
+--R /
+--R 3
+--R 4a
+--R Type: Expression Complex Integer
+--E
+
+--S 38 14:215 Schaums and Axiom differ by a constant
+hh:=complexNormalize gg
+--R
+--R %pi
+--R (10) - ---
+--R 3
+--R 4a
+--R Type: Expression Complex Integer
+--E
+
@
\section{\cite{1}:14.216~~~~~$\displaystyle\int{\sqrt{x^2-a^2}}~dx$}
@@ -341,7 +566,7 @@
<<*>>=
)clear all
---S 24
+--S 39
aa:=integrate(sqrt(x^2-a^2),x)
--R
--R
@@ -360,7 +585,7 @@
--R Type: Union(Expression Integer,...)
--E
---S 25
+--S 40
bb:=(x*sqrt(x^2-a^2))/2-a^2/2*log(x+sqrt(x^2-a^2))
--R
--R +-------+ +-------+
@@ -371,7 +596,7 @@
--R Type: Expression Integer
--E
---S 26 14:216 Axiom cannot simplify this expression
+--S 41
cc:=aa-bb
--R
--R +-------+ +-------+
@@ -382,6 +607,15 @@
--R Type: Expression Integer
--E
+--S 42 14:216 Schaums and Axiom differ by a constant
+dd:=complexNormalize cc
+--R
+--R 2 2
+--R a log(- a )
+--R (4) -----------
+--R 2
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.217~~~~~$\displaystyle\int{x\sqrt{x^2-a^2}}~dx$}
@@ -391,7 +625,7 @@
<<*>>=
)clear all
---S 27
+--S 43
aa:=integrate(x*sqrt(x^2-a^2),x)
--R
--R
@@ -405,7 +639,7 @@
--R Type: Union(Expression Integer,...)
--E
---S 28
+--S 44
bb:=(x^2-a^2)^(3/2)/3
--R
--R +-------+
@@ -416,7 +650,7 @@
--R Type: Expression Integer
--E
---S 29 14:217 Schaums and Axiom agree
+--S 45 14:217 Schaums and Axiom agree
cc:=aa-bb
--R
--R (3) 0
@@ -433,7 +667,7 @@
<<*>>=
)clear all
---S 30
+--S 46
aa:=integrate(x^2*sqrt(x^2-a^2),x)
--R
--R
@@ -452,7 +686,7 @@
--R Type: Union(Expression Integer,...)
--E
---S 31
+--S 47
bb:=(x*(x^2-a^2)^(3/2))/4+(a^2*x*sqrt(x^2-a^2))/8-a^4/8*log(x+sqrt(x^2-a^2))
--R
--R +-------+ +-------+
@@ -463,7 +697,7 @@
--R Type: Expression Integer
--E
---S 32 14:218 Axiom cannot simplify this expression
+--S 48
cc:=aa-bb
--R
--R +-------+ +-------+
@@ -473,6 +707,16 @@
--R 8
--R Type: Expression Integer
--E
+
+--S 49 14:218 Schaums and Axiom differ by a constant
+dd:=complexNormalize cc
+--R
+--R 4 2
+--R a log(- a )
+--R (4) -----------
+--R 8
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.219~~~~~$\displaystyle
@@ -483,7 +727,7 @@
<<*>>=
)clear all
---S 33
+--S 50
aa:=integrate(x^3*sqrt(x^2-a^2),x)
--R
--R
@@ -501,7 +745,7 @@
--R Type: Union(Expression Integer,...)
--E
---S 34
+--S 51
bb:=(x^2-a^2)^(5/2)/5+(a^2*(x^2-a^2)^(3/2))/3
--R
--R +-------+
@@ -512,7 +756,7 @@
--R Type: Expression Integer
--E
---S 35 14:219 Schaums and Axiom agree
+--S 52 14:219 Schaums and Axiom agree
cc:=aa-bb
--R
--R (3) 0
@@ -528,7 +772,7 @@
<<*>>=
)clear all
---S 36
+--S 53
aa:=integrate(sqrt(x^2-a^2)/x,x)
--R
--R
@@ -544,26 +788,129 @@
--R Type: Union(Expression Integer,...)
--E
---S 37
-bb:=sqrt(x^2-a^2)-a*asec(abs(x/a))
+--S 54
+bb:=sqrt(x^2-a^2)-a*asec(x/a)
--R
---R +-------+
---R x | 2 2
---R (2) - a asec(abs(-)) + \|x - a
---R a
+--R +-------+
+--R | 2 2 x
+--R (2) \|x - a - a asec(-)
+--R a
--R Type: Expression Integer
--E
---S 38 14:220 Axiom cannot simplify this expression
+--S 55
cc:=aa-bb
--R
--R +-------+
--R | 2 2
---R \|x - a - x x
---R (3) - 2a atan(--------------) + a asec(abs(-))
---R a a
+--R \|x - a - x x
+--R (3) - 2a atan(--------------) + a asec(-)
+--R a a
--R Type: Expression Integer
--E
+
+--S 56
+atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x)))
+--R
+--R - x + %i
+--R %i log(--------)
+--R x + %i
+--R (4) atan(x) == - ----------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer)
+--E
+
+--S 57
+dd:=atanrule cc
+--R
+--R +-------+
+--R | 2 2
+--R - \|x - a + x + %i a x
+--R (5) %i a log(-----------------------) + a asec(-)
+--R +-------+ a
+--R | 2 2
+--R \|x - a - x + %i a
+--R Type: Expression Complex Integer
+--E
+
+--S 58
+asecrule:=rule(asec(x) == 1/2*%pi+%i*log(sqrt(1-1/x^2)+%i/x))
+--R
+--R +------+
+--R | 2
+--R |x - 1
+--R x |------ + %i
+--R | 2
+--R \| x
+--R 2%i log(---------------) + %pi
+--R x
+--R (6) asec(x) == ------------------------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer)
+--E
+
+--S 59
+ee:=asecrule dd
+--R
+--R (7)
+--R +-------+
+--R | 2 2
+--R |x - a
+--R x |------- + %i a +-------+
+--R | 2 | 2 2
+--R \| x - \|x - a + x + %i a
+--R 2%i a log(------------------) + 2%i a log(-----------------------) + a %pi
+--R x +-------+
+--R | 2 2
+--R \|x - a - x + %i a
+--R --------------------------------------------------------------------------
+--R 2
+--R Type: Expression Complex Integer
+--E
+
+--S 60
+ff:=expandLog ee
+--R
+--R (8)
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R - 2%i a log(\|x - a - x + %i a) + 2%i a log(\|x - a - x - %i a)
+--R +
+--R +-------+
+--R | 2 2
+--R |x - a
+--R 2%i a log(x |------- + %i a) - 2%i a log(x) + 2%i a log(- 1) + a %pi
+--R | 2
+--R \| x
+--R /
+--R 2
+--R Type: Expression Complex Integer
+--E
+
+--S 61
+gg:=rootSimp ff
+--R
+--R (9)
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R 2%i a log(\|x - a + %i a) - 2%i a log(\|x - a - x + %i a)
+--R +
+--R +-------+
+--R | 2 2
+--R 2%i a log(\|x - a - x - %i a) - 2%i a log(x) + 2%i a log(- 1) + a %pi
+--R /
+--R 2
+--R Type: Expression Complex Integer
+--E
+
+--S 62 14:220 Schaums and Axiom differ by a constant
+hh:=complexNormalize gg
+--R
+--R a %pi
+--R (10) -----
+--R 2
+--R Type: Expression Complex Integer
+--E
@
\section{\cite{1}:14.221~~~~~$\displaystyle
@@ -574,7 +921,7 @@
<<*>>=
)clear all
---S 39
+--S 63
aa:=integrate(sqrt(x^2-a^2)/x^2,x)
--R
--R
@@ -588,7 +935,7 @@
--R Type: Union(Expression Integer,...)
--E
---S 40
+--S 64
bb:=-sqrt(x^2-a^2)/x+log(x+sqrt(x^2-a^2))
--R
--R +-------+ +-------+
@@ -599,7 +946,7 @@
--R Type: Expression Integer
--E
---S 41 14:221 Axiom cannot simplify this expression
+--S 65
cc:=aa-bb
--R
--R +-------+ +-------+
@@ -608,6 +955,14 @@
--R Type: Expression Integer
--E
+--S 66 14:221 Schaums and Axiom differ by a constant
+dd:=complexNormalize cc
+--R
+--R 2
+--R (4) - log(- a ) - 1
+--R Type: Expression Integer
+--E
+
@
\section{\cite{1}:14.222~~~~~$\displaystyle
@@ -619,7 +974,7 @@
<<*>>=
)clear all
---S 42
+--S 67
aa:=integrate(sqrt(x^2-a^2)/x^3,x)
--R
--R
@@ -640,31 +995,136 @@
--R Type: Union(Expression Integer,...)
--E
---S 43
-bb:=-sqrt(x^2-a^2)/(2*x^2)+1/(2*a)*asec(abs(x/a))
+--S 68
+bb:=-sqrt(x^2-a^2)/(2*x^2)+1/(2*a)*asec(x/a)
--R
---R +-------+
---R 2 x | 2 2
---R x asec(abs(-)) - a\|x - a
---R a
---R (2) ----------------------------
---R 2
---R 2a x
+--R +-------+
+--R | 2 2 2 x
+--R - a\|x - a + x asec(-)
+--R a
+--R (2) -------------------------
+--R 2
+--R 2a x
--R Type: Expression Integer
--E
---S 44 14:222 Axiom cannot simplify this expression
+--S 69
cc:=aa-bb
--R
--R +-------+
--R | 2 2
---R \|x - a - x x
---R 2atan(--------------) - asec(abs(-))
---R a a
---R (3) ------------------------------------
---R 2a
+--R \|x - a - x x
+--R 2atan(--------------) - asec(-)
+--R a a
+--R (3) -------------------------------
+--R 2a
--R Type: Expression Integer
--E
+
+--S 70
+asecrule:=rule(asec(x) == 1/2*%pi+%i*log(sqrt(1-1/x^2)+%i/x))
+--R
+--R +------+
+--R | 2
+--R |x - 1
+--R x |------ + %i
+--R | 2
+--R \| x
+--R 2%i log(---------------) + %pi
+--R x
+--R (4) asec(x) == ------------------------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer)
+--E
+
+--S 71
+dd:=asecrule cc
+--R
+--R +-------+
+--R | 2 2
+--R |x - a
+--R x |------- + %i a +-------+
+--R | 2 | 2 2
+--R \| x \|x - a - x
+--R - 2%i log(------------------) + 4atan(--------------) - %pi
+--R x a
+--R (5) -----------------------------------------------------------
+--R 4a
+--R Type: Expression Complex Integer
+--E
+
+--S 72
+atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x)))
+--R
+--R - x + %i
+--R %i log(--------)
+--R x + %i
+--R (6) atan(x) == - ----------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer)
+--E
+
+--S 73
+ee:=atanrule dd
+--R
+--R +-------+
+--R | 2 2
+--R |x - a
+--R x |------- + %i a +-------+
+--R | 2 | 2 2
+--R \| x - \|x - a + x + %i a
+--R - 2%i log(------------------) - 2%i log(-----------------------) - %pi
+--R x +-------+
+--R | 2 2
+--R \|x - a - x + %i a
+--R (7) ----------------------------------------------------------------------
+--R 4a
+--R Type: Expression Complex Integer
+--E
+
+--S 74
+ff:=expandLog ee
+--R
+--R (8)
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R 2%i log(\|x - a - x + %i a) - 2%i log(\|x - a - x - %i a)
+--R +
+--R +-------+
+--R | 2 2
+--R |x - a
+--R - 2%i log(x |------- + %i a) + 2%i log(x) - 2%i log(- 1) - %pi
+--R | 2
+--R \| x
+--R /
+--R 4a
+--R Type: Expression Complex Integer
+--E
+
+--S 75
+gg:=rootSimp ff
+--R
+--R (9)
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R - 2%i log(\|x - a + %i a) + 2%i log(\|x - a - x + %i a)
+--R +
+--R +-------+
+--R | 2 2
+--R - 2%i log(\|x - a - x - %i a) + 2%i log(x) - 2%i log(- 1) - %pi
+--R /
+--R 4a
+--R Type: Expression Complex Integer
+--E
+
+--S 76 14:222 Schaums and Axiom differ by a constant
+hh:=complexNormalize gg
+--R
+--R %pi
+--R (10) - ---
+--R 4a
+--R Type: Expression Complex Integer
+--E
@
\section{\cite{1}:14.223~~~~~$\displaystyle\int{\frac{dx}{(x^2-a^2)^{3/2}}}$}
@@ -674,7 +1134,7 @@
<<*>>=
)clear all
---S 45
+--S 77
aa:=integrate(1/(x^2-a^2)^(3/2),x)
--R
--R
@@ -686,7 +1146,7 @@
--R Type: Union(Expression Integer,...)
--E
---S 46
+--S 78
bb:=-x/(a^2*sqrt(x^2-a^2))
--R
--R x
@@ -697,7 +1157,7 @@
--R Type: Expression Integer
--E
---S 47 14:223 Schaums and Axiom differ by a constant
+--S 79 14:223 Schaums and Axiom differ by a constant
cc:=aa-bb
--R
--R 1
@@ -716,7 +1176,7 @@
<<*>>=
)clear all
---S 48
+--S 80
aa:=integrate(x/(x^2-a^2)^(3/2),x)
--R
--R
@@ -730,7 +1190,7 @@
--R Type: Union(Expression Integer,...)
--E
---S 49
+--S 81
bb:=-1/sqrt(x^2-a^2)
--R
--R 1
@@ -741,7 +1201,7 @@
--R Type: Expression Integer
--E
---S 50 14:224 Schaums and Axiom agree
+--S 82 14:224 Schaums and Axiom agree
cc:=aa-bb
--R
--R (3) 0
@@ -757,7 +1217,7 @@
<<*>>=
)clear all
---S 51
+--S 83
aa:=integrate(x^2/(x^2-a^2)^(3/2),x)
--R
--R
@@ -771,7 +1231,7 @@
--R Type: Union(Expression Integer,...)
--E
---S 52
+--S 84
bb:=-x/sqrt(x^2-a^2)+log(x+sqrt(x^2-a^2))
--R
--R +-------+ +-------+
@@ -784,7 +1244,7 @@
--R Type: Expression Integer
--E
---S 53 14:225 Axiom cannot simplify this expression
+--S 85
cc:=aa-bb
--R
--R +-------+ +-------+
@@ -792,6 +1252,14 @@
--R (3) - log(\|x - a + x) - log(\|x - a - x) - 1
--R Type: Expression Integer
--E
+
+--S 86 14:225 Schaums and Axiom differ by a constant
+dd:=complexNormalize cc
+--R
+--R 2
+--R (4) - log(- a ) - 1
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.226~~~~~$\displaystyle
@@ -802,7 +1270,7 @@
<<*>>=
)clear all
---S 54
+--S 87
aa:=integrate(x^3/(x^2-a^2)^(3/2),x)
--R
--R
@@ -816,7 +1284,7 @@
--R Type: Union(Expression Integer,...)
--E
---S 55
+--S 88
bb:=sqrt(x^2-a^2)-a^2/sqrt(x^2-a^2)
--R
--R 2 2
@@ -828,7 +1296,7 @@
--R Type: Expression Integer
--E
---S 56 14:226 Schaums and Axiom agree
+--S 89 14:226 Schaums and Axiom agree
cc:=aa-bb
--R
--R (3) 0
@@ -845,7 +1313,7 @@
<<*>>=
)clear all
---S 57
+--S 90
aa:=integrate(1/(x*(x^2-a^2)^(3/2)),x)
--R
--R
@@ -861,33 +1329,143 @@
--R Type: Union(Expression Integer,...)
--E
---S 58
-bb:=-1/(a^2*sqrt(x^2-a^2))-1/a^3*asec(abs(x/a))
+--S 91
+bb:=-1/(a^2*sqrt(x^2-a^2))-1/a^3*asec(x/a)
--R
---R +-------+
---R | 2 2 x
---R - \|x - a asec(abs(-)) - a
---R a
---R (2) ----------------------------
---R +-------+
---R 3 | 2 2
---R a \|x - a
+--R +-------+
+--R x | 2 2
+--R - asec(-)\|x - a - a
+--R a
+--R (2) -----------------------
+--R +-------+
+--R 3 | 2 2
+--R a \|x - a
--R Type: Expression Integer
--E
---S 59 14:227 Axiom cannot simplify this expression
+--S 92
cc:=aa-bb
--R
--R +-------+
--R | 2 2
---R \|x - a - x x
---R - 2atan(--------------) + asec(abs(-))
---R a a
---R (3) --------------------------------------
---R 3
---R a
+--R \|x - a - x x
+--R - 2atan(--------------) + asec(-)
+--R a a
+--R (3) ---------------------------------
+--R 3
+--R a
--R Type: Expression Integer
--E
+
+--S 93
+atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x)))
+--R
+--R - x + %i
+--R %i log(--------)
+--R x + %i
+--R (4) atan(x) == - ----------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer)
+--E
+
+--S 94
+dd:=atanrule cc
+--R
+--R +-------+
+--R | 2 2
+--R - \|x - a + x + %i a x
+--R %i log(-----------------------) + asec(-)
+--R +-------+ a
+--R | 2 2
+--R \|x - a - x + %i a
+--R (5) -----------------------------------------
+--R 3
+--R a
+--R Type: Expression Complex Integer
+--E
+
+--S 95
+asecrule:=rule(asec(x) == 1/2*%pi+%i*log(sqrt(1-1/x^2)+%i/x))
+--R
+--R +------+
+--R | 2
+--R |x - 1
+--R x |------ + %i
+--R | 2
+--R \| x
+--R 2%i log(---------------) + %pi
+--R x
+--R (6) asec(x) == ------------------------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer)
+--E
+
+--S 96
+ee:=asecrule dd
+--R
+--R +-------+
+--R | 2 2
+--R |x - a
+--R x |------- + %i a +-------+
+--R | 2 | 2 2
+--R \| x - \|x - a + x + %i a
+--R 2%i log(------------------) + 2%i log(-----------------------) + %pi
+--R x +-------+
+--R | 2 2
+--R \|x - a - x + %i a
+--R (7) --------------------------------------------------------------------
+--R 3
+--R 2a
+--R Type: Expression Complex Integer
+--E
+
+--S 97
+ff:=expandLog ee
+--R
+--R (8)
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R - 2%i log(\|x - a - x + %i a) + 2%i log(\|x - a - x - %i a)
+--R +
+--R +-------+
+--R | 2 2
+--R |x - a
+--R 2%i log(x |------- + %i a) - 2%i log(x) + 2%i log(- 1) + %pi
+--R | 2
+--R \| x
+--R /
+--R 3
+--R 2a
+--R Type: Expression Complex Integer
+--E
+
+--S 98
+gg:=rootSimp ff
+--R
+--R (9)
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R 2%i log(\|x - a + %i a) - 2%i log(\|x - a - x + %i a)
+--R +
+--R +-------+
+--R | 2 2
+--R 2%i log(\|x - a - x - %i a) - 2%i log(x) + 2%i log(- 1) + %pi
+--R /
+--R 3
+--R 2a
+--R Type: Expression Complex Integer
+--E
+
+--S 99 14:227 Schaums and Axiom differ by a constant
+hh:=complexNormalize gg
+--R
+--R %pi
+--R (10) ---
+--R 3
+--R 2a
+--R Type: Expression Complex Integer
+--E
+
@
\section{\cite{1}:14.228~~~~~$\displaystyle
@@ -898,7 +1476,7 @@
<<*>>=
)clear all
---S 60
+--S 100
aa:=integrate(1/(x^2*(x^2-a^2)^(3/2)),x)
--R
--R
@@ -910,7 +1488,7 @@
--R Type: Union(Expression Integer,...)
--E
---S 61
+--S 101
bb:=-sqrt(x^2-a^2)/(a^4*x)-x/(a^4*sqrt(x^2-a^2))
--R
--R 2 2
@@ -922,7 +1500,7 @@
--R Type: Expression Integer
--E
---S 62 14:228 Schaums and Axiom differ by a constant
+--S 102 14:228 Schaums and Axiom differ by a constant
cc:=aa-bb
--R
--R 2
@@ -944,7 +1522,7 @@
<<*>>=
)clear all
---S 63
+--S 103
aa:=integrate(1/(x^3*(x^2-a^2)^(3/2)),x)
--R
--R
@@ -965,35 +1543,143 @@
--R Type: Union(Expression Integer,...)
--E
---S 64
-bb:=1/(2*a^2*x^2*sqrt(x^2-a^2))-3/(2*a^4*sqrt(x^2-a^2))-3/(2*a^5)*asec(abs(x/a))
+--S 104
+bb:=1/(2*a^2*x^2*sqrt(x^2-a^2))-3/(2*a^4*sqrt(x^2-a^2))-3/(2*a^5)*asec(x/a)
--R
---R +-------+
---R 2 | 2 2 x 2 3
---R - 3x \|x - a asec(abs(-)) - 3a x + a
---R a
---R (2) ----------------------------------------
---R +-------+
---R 5 2 | 2 2
---R 2a x \|x - a
+--R +-------+
+--R 2 x | 2 2 2 3
+--R - 3x asec(-)\|x - a - 3a x + a
+--R a
+--R (2) -----------------------------------
+--R +-------+
+--R 5 2 | 2 2
+--R 2a x \|x - a
--R Type: Expression Integer
--E
---S 65 14:229 Axiom cannot simplify this expression
+--S 105
cc:=aa-bb
--R
--R +-------+
--R | 2 2
---R \|x - a - x x
---R - 6atan(--------------) + 3asec(abs(-))
---R a a
---R (3) ---------------------------------------
---R 5
---R 2a
+--R \|x - a - x x
+--R - 6atan(--------------) + 3asec(-)
+--R a a
+--R (3) ----------------------------------
+--R 5
+--R 2a
--R Type: Expression Integer
--E
+
+--S 106
+atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x)))
+--R
+--R - x + %i
+--R %i log(--------)
+--R x + %i
+--R (4) atan(x) == - ----------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer)
+--E
+
+--S 107
+dd:=atanrule cc
+--R
+--R +-------+
+--R | 2 2
+--R - \|x - a + x + %i a x
+--R 3%i log(-----------------------) + 3asec(-)
+--R +-------+ a
+--R | 2 2
+--R \|x - a - x + %i a
+--R (5) -------------------------------------------
+--R 5
+--R 2a
+--R Type: Expression Complex Integer
+--E
+
+--S 108
+asecrule:=rule(asec(x) == 1/2*%pi+%i*log(sqrt(1-1/x^2)+%i/x))
+--R
+--R +------+
+--R | 2
+--R |x - 1
+--R x |------ + %i
+--R | 2
+--R \| x
+--R 2%i log(---------------) + %pi
+--R x
+--R (6) asec(x) == ------------------------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer)
+--E
+
+--S 109
+ee:=asecrule dd
+--R
+--R +-------+
+--R | 2 2
+--R |x - a
+--R x |------- + %i a +-------+
+--R | 2 | 2 2
+--R \| x - \|x - a + x + %i a
+--R 6%i log(------------------) + 6%i log(-----------------------) + 3%pi
+--R x +-------+
+--R | 2 2
+--R \|x - a - x + %i a
+--R (7) ---------------------------------------------------------------------
+--R 5
+--R 4a
+--R Type: Expression Complex Integer
+--E
+
+--S 110
+ff:=expandLog ee
+--R
+--R (8)
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R - 6%i log(\|x - a - x + %i a) + 6%i log(\|x - a - x - %i a)
+--R +
+--R +-------+
+--R | 2 2
+--R |x - a
+--R 6%i log(x |------- + %i a) - 6%i log(x) + 6%i log(- 1) + 3%pi
+--R | 2
+--R \| x
+--R /
+--R 5
+--R 4a
+--R Type: Expression Complex Integer
+--E
+
+--S 111
+gg:=rootSimp ff
+--R
+--R (9)
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R 6%i log(\|x - a + %i a) - 6%i log(\|x - a - x + %i a)
+--R +
+--R +-------+
+--R | 2 2
+--R 6%i log(\|x - a - x - %i a) - 6%i log(x) + 6%i log(- 1) + 3%pi
+--R /
+--R 5
+--R 4a
+--R Type: Expression Complex Integer
+--E
+
+--S 112 14:229 Schaums and Axiom differ by a constant
+hh:=complexNormalize gg
+--R
+--R 3%pi
+--R (10) ----
+--R 5
+--R 4a
+--R Type: Expression Complex Integer
+--E
@
-
\section{\cite{1}:14.230~~~~~$\displaystyle\int{(x^2-a^2)^{3/2}}~dx$}
$$\int{(x^2-a^2)^{3/2}}=
\frac{x(x^2-a^2)^{3/2}}{4}-\frac{3a^2x\sqrt{x^2-a^2}}{8}+
@@ -1002,7 +1688,7 @@
<<*>>=
)clear all
---S 66
+--S 113
aa:=integrate((x^2-a^2)^(3/2),x)
--R
--R
@@ -1024,7 +1710,7 @@
--R Type: Union(Expression Integer,...)
--E
---S 67
+--S 114
bb:=(x*(x^2-a^2)^(3/2))/4-(3*a^2*x*sqrt(x^2-a^2))/8+3/8*a^4*log(x+sqrt(x^2-a^2))
--R
--R +-------+ +-------+
@@ -1035,7 +1721,7 @@
--R Type: Expression Integer
--E
---S 68 14:230 Axiom cannot simplify this expression
+--S 115
cc:=aa-bb
--R
--R +-------+ +-------+
@@ -1046,6 +1732,16 @@
--R Type: Expression Integer
--E
+--S 116 14:230 Schaums and Axiom differ by a constant
+dd:=complexNormalize cc
+--R
+--R 4 2
+--R 3a log(- a )
+--R (4) - ------------
+--R 8
+--R Type: Expression Integer
+--E
+
@
\section{\cite{1}:14.231~~~~~$\displaystyle\int{x(x^2-a^2)^{3/2}}~dx$}
@@ -1053,7 +1749,7 @@
<<*>>=
)clear all
---S 69
+--S 117
aa:=integrate(x*(x^2-a^2)^(3/2),x)
--R
--R
@@ -1071,7 +1767,7 @@
--R Type: Union(Expression Integer,...)
--E
---S 70
+--S 118
bb:=(x^2-a^2)^(5/2)/5
--R
--R +-------+
@@ -1082,7 +1778,7 @@
--R Type: Expression Integer
--E
---S 71 14:231 Schaums and Axiom agree
+--S 119 14:231 Schaums and Axiom agree
cc:=aa-bb
--R
--R (3) 0
@@ -1099,7 +1795,7 @@
<<*>>=
)clear all
---S 72
+--S 120
aa:=integrate(x^2*(x^2-a^2)^(3/2),x)
--R
--R
@@ -1128,7 +1824,7 @@
--R Type: Union(Expression Integer,...)
--E
---S 73
+--S 121
bb:=(x*(x^2-a^2)^(5/2))/6+(a^2*x*(x^2-a^2)^(3/2))/24-(a^4*x*sqrt(x^2-a^2))/16+a^6/16*log(x+sqrt(x^2-a^2))
--R
--R +-------+ +-------+
@@ -1139,7 +1835,7 @@
--R Type: Expression Integer
--E
---S 74 14:232 Axiom cannot simplify this expression
+--S 122
cc:=aa-bb
--R
--R +-------+ +-------+
@@ -1149,6 +1845,16 @@
--R 16
--R Type: Expression Integer
--E
+
+--S 123 14:232 Schaums and Axiom differ by a constant
+dd:=complexNormalize cc
+--R
+--R 6 2
+--R a log(- a )
+--R (4) - -----------
+--R 16
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.233~~~~~$\displaystyle\int{x^3(x^2-a^2)^{3/2}}~dx$}
@@ -1158,7 +1864,7 @@
<<*>>=
)clear all
---S 75
+--S 124
aa:=integrate(x^3*(x^2-a^2)^(3/2),x)
--R
--R
@@ -1188,7 +1894,7 @@
--R Type: Union(Expression Integer,...)
--E
---S 76
+--S 125
bb:=(x^2-a^2)^(7/2)/7+(a^2*(x^2-a^2)^(5/2))/5
--R
--R +-------+
@@ -1199,7 +1905,7 @@
--R Type: Expression Integer
--E
---S 77 14:233 Schaums and Axiom agree
+--S 126 14:233 Schaums and Axiom agree
cc:=aa-bb
--R
--R (3) 0
@@ -1216,7 +1922,7 @@
<<*>>=
)clear all
---S 78
+--S 127
aa:=integrate((x^2-a^2)^(3/2)/x,x)
--R
--R
@@ -1237,28 +1943,135 @@
--R Type: Union(Expression Integer,...)
--E
---S 79
-bb:=(x^2-a^2)^(3/2)/3-a^2*sqrt(x^2-a^2)+a^3*asec(abs(x/a))
+--S 128
+bb:=(x^2-a^2)^(3/2)/3-a^2*sqrt(x^2-a^2)+a^3*asec(x/a)
--R
---R +-------+
---R 3 x 2 2 | 2 2
---R 3a asec(abs(-)) + (x - 4a )\|x - a
---R a
---R (2) --------------------------------------
---R 3
+--R +-------+
+--R 2 2 | 2 2 3 x
+--R (x - 4a )\|x - a + 3a asec(-)
+--R a
+--R (2) ---------------------------------
+--R 3
--R Type: Expression Integer
--E
---S 80 14:234 Axiom cannot simplify this expression
+--S 129
cc:=aa-bb
--R
--R +-------+
--R | 2 2
---R 3 \|x - a - x 3 x
---R (3) 2a atan(--------------) - a asec(abs(-))
---R a a
+--R 3 \|x - a - x 3 x
+--R (3) 2a atan(--------------) - a asec(-)
+--R a a
--R Type: Expression Integer
--E
+
+--S 130
+asecrule:=rule(asec(x) == 1/2*%pi+%i*log(sqrt(1-1/x^2)+%i/x))
+--R
+--R +------+
+--R | 2
+--R |x - 1
+--R x |------ + %i
+--R | 2
+--R \| x
+--R 2%i log(---------------) + %pi
+--R x
+--R (4) asec(x) == ------------------------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer)
+--E
+
+--S 131
+dd:=asecrule cc
+--R
+--R +-------+
+--R | 2 2
+--R |x - a
+--R x |------- + %i a +-------+
+--R | 2 | 2 2
+--R 3 \| x 3 \|x - a - x 3
+--R - 2%i a log(------------------) + 4a atan(--------------) - a %pi
+--R x a
+--R (5) -----------------------------------------------------------------
+--R 2
+--R Type: Expression Complex Integer
+--E
+
+--S 132
+atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x)))
+--R
+--R - x + %i
+--R %i log(--------)
+--R x + %i
+--R (6) atan(x) == - ----------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer)
+--E
+
+--S 133
+ee:=atanrule dd
+--R
+--R (7)
+--R +-------+
+--R | 2 2
+--R |x - a
+--R x |------- + %i a +-------+
+--R | 2 | 2 2
+--R 3 \| x 3 - \|x - a + x + %i a 3
+--R - 2%i a log(------------------) - 2%i a log(-----------------------) - a %pi
+--R x +-------+
+--R | 2 2
+--R \|x - a - x + %i a
+--R ----------------------------------------------------------------------------
+--R 2
+--R Type: Expression Complex Integer
+--E
+
+--S 134
+ff:=expandLog ee
+--R
+--R (8)
+--R +-------+ +-------+
+--R 3 | 2 2 3 | 2 2
+--R 2%i a log(\|x - a - x + %i a) - 2%i a log(\|x - a - x - %i a)
+--R +
+--R +-------+
+--R | 2 2
+--R 3 |x - a 3 3 3
+--R - 2%i a log(x |------- + %i a) + 2%i a log(x) - 2%i a log(- 1) - a %pi
+--R | 2
+--R \| x
+--R /
+--R 2
+--R Type: Expression Complex Integer
+--E
+
+--S 135
+gg:=rootSimp ff
+--R
+--R (9)
+--R +-------+ +-------+
+--R 3 | 2 2 3 | 2 2
+--R - 2%i a log(\|x - a + %i a) + 2%i a log(\|x - a - x + %i a)
+--R +
+--R +-------+
+--R 3 | 2 2 3 3 3
+--R - 2%i a log(\|x - a - x - %i a) + 2%i a log(x) - 2%i a log(- 1) - a %pi
+--R /
+--R 2
+--R Type: Expression Complex Integer
+--E
+
+--S 136 14:234 Schaums and Axiom differ by a constant
+hh:=complexNormalize gg
+--R
+--R 3
+--R a %pi
+--R (10) - -----
+--R 2
+--R Type: Expression Complex Integer
+--E
@
\section{\cite{1}:14.235~~~~~$\displaystyle
@@ -1270,7 +2083,7 @@
<<*>>=
)clear all
---S 81
+--S 137
aa:=integrate((x^2-a^2)^{3/2}/x^2,x)
--R
--R
@@ -1289,7 +2102,7 @@
--R Type: Union(Expression Integer,...)
--E
---S 82
+--S 138
bb:=-(x^2-a^2)^(3/2)/x+3*x*sqrt(x^2-a^2)/2-3/2*a^2*log(x+sqrt(x^2-a^2))
--R
--R +-------+ +-------+
@@ -1300,7 +2113,7 @@
--R Type: Expression Integer
--E
---S 83 14:235 Axiom cannot simplify this expression
+--S 139
cc:=aa-bb
--R
--R +-------+ +-------+
@@ -1310,6 +2123,17 @@
--R 2
--R Type: Expression Integer
--E
+
+--S 140 14:235 Schaums and Axiom differ by a constant
+dd:=complexNormalize cc
+--R
+--R 2 2 2
+--R 3a log(- a ) + 2a
+--R (4) ------------------
+--R 2
+--R Type: Expression Integer
+--E
+
@
\section{\cite{1}:14.236~~~~~$\displaystyle
@@ -1321,7 +2145,7 @@
<<*>>=
)clear all
---S 84
+--S 141
aa:=integrate((x^2-a^2)^(3/2)/x^3,x)
--R
--R
@@ -1342,31 +2166,137 @@
--R Type: Union(Expression Integer,...)
--E
---S 85
-bb:=-(x^2-a^2)^(3/2)/(2*x^2)+(3*sqrt(x^2-a^2))/2-3/2*a*asec(abs(x/a))
+--S 142
+bb:=-(x^2-a^2)^(3/2)/(2*x^2)+(3*sqrt(x^2-a^2))/2-3/2*a*asec(x/a)
--R
---R +-------+
---R 2 x 2 2 | 2 2
---R - 3a x asec(abs(-)) + (2x + a )\|x - a
---R a
---R (2) ------------------------------------------
---R 2
---R 2x
+--R +-------+
+--R 2 2 | 2 2 2 x
+--R (2x + a )\|x - a - 3a x asec(-)
+--R a
+--R (2) -----------------------------------
+--R 2
+--R 2x
--R Type: Expression Integer
--E
---S 86 14:236 Axiom cannot simplify this expression
+--S 143
cc:=aa-bb
--R
--R +-------+
--R | 2 2
---R \|x - a - x x
---R - 6a atan(--------------) + 3a asec(abs(-))
---R a a
---R (3) -------------------------------------------
---R 2
+--R \|x - a - x x
+--R - 6a atan(--------------) + 3a asec(-)
+--R a a
+--R (3) --------------------------------------
+--R 2
--R Type: Expression Integer
--E
+
+--S 144
+atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x)))
+--R
+--R - x + %i
+--R %i log(--------)
+--R x + %i
+--R (4) atan(x) == - ----------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer)
+--E
+
+--S 145
+dd:=atanrule cc
+--R
+--R +-------+
+--R | 2 2
+--R - \|x - a + x + %i a x
+--R 3%i a log(-----------------------) + 3a asec(-)
+--R +-------+ a
+--R | 2 2
+--R \|x - a - x + %i a
+--R (5) -----------------------------------------------
+--R 2
+--R Type: Expression Complex Integer
+--E
+
+--S 146
+asecrule:=rule(asec(x) == 1/2*%pi+%i*log(sqrt(1-1/x^2)+%i/x))
+--R
+--R +------+
+--R | 2
+--R |x - 1
+--R ...
[truncated message content] |
|
From: <gi...@ax...> - 2008-04-19 06:05:10
|
changelog | 8 +
src/input/schaum1.input.pamphlet | 68 ++-
src/input/schaum10.input.pamphlet | 1282 +++++++++++++++++++++++----
src/input/schaum11.input.pamphlet | 1795 ++++++++++++++++++++++++++++++++++++-
src/input/schaum2.input.pamphlet | 253 ++++--
src/input/schaum6.input.pamphlet | 15 +-
src/input/schaum7.input.pamphlet | 14 +-
src/input/schaum8.input.pamphlet | 15 +-
src/input/schaum9.input.pamphlet | 483 ++++++----
9 files changed, 3430 insertions(+), 503 deletions(-)
New commits:
commit 69c3d8ce42b5e9ce8139aaf6bfca9aac3536b37b
Author: Tim Daly <root@localhost.localdomain>
Date: Wed Apr 2 02:08:08 2008 -0400
20080418 tpd src/input/schaum11.input show Schaums-Axiom equivalence
20080418 tpd src/input/schaum10.input fix Schaums-Axiom equivalence
20080418 tpd src/input/schaum9.input fix Schaums-Axiom equivalence
20080418 tpd src/input/schaum8.input fix Schaums-Axiom equivalence
20080418 tpd src/input/schaum7.input fix Schaums-Axiom equivalence
20080418 tpd src/input/schaum6.input fix Schaums-Axiom equivalence
20080418 tpd src/input/schaum2.input fix Schaums-Axiom equivalence
20080418 tpd src/input/schaum1.input fix Schaums-Axiom equivalence
|
|
From: <da...@us...> - 2008-04-18 06:42:52
|
Revision: 857
http://axiom.svn.sourceforge.net/axiom/?rev=857&view=rev
Author: daly
Date: 2008-04-17 23:42:58 -0700 (Thu, 17 Apr 2008)
Log Message:
-----------
20080417 tpd src/input/schaum10.input show Schaums-Axiom equivalence
20080417 tpd src/input/schaum9.input show Schaums-Axiom equivalence
20080417 tpd src/input/schaum8.input show Schaums-Axiom equivalence
Modified Paths:
--------------
trunk/axiom/changelog
trunk/axiom/src/input/schaum10.input.pamphlet
trunk/axiom/src/input/schaum8.input.pamphlet
trunk/axiom/src/input/schaum9.input.pamphlet
Modified: trunk/axiom/changelog
===================================================================
--- trunk/axiom/changelog 2008-04-17 15:06:30 UTC (rev 856)
+++ trunk/axiom/changelog 2008-04-18 06:42:58 UTC (rev 857)
@@ -1,3 +1,6 @@
+20080417 tpd src/input/schaum10.input show Schaums-Axiom equivalence
+20080417 tpd src/input/schaum9.input show Schaums-Axiom equivalence
+20080417 tpd src/input/schaum8.input show Schaums-Axiom equivalence
20080417 tpd src/input/schaum7.input fix typo in --S 30
20080417 tpd src/input/schaum5.input fix typo 14:88 -> 14:123
20080416 tpd src/input/schaum7.input show Schaums-Axiom equivalence
Modified: trunk/axiom/src/input/schaum10.input.pamphlet
===================================================================
--- trunk/axiom/src/input/schaum10.input.pamphlet 2008-04-17 15:06:30 UTC (rev 856)
+++ trunk/axiom/src/input/schaum10.input.pamphlet 2008-04-18 06:42:58 UTC (rev 857)
@@ -15,7 +15,7 @@
)set message auto off
)clear all
---S 1 of 28
+--S 1
aa:=integrate(1/(sqrt(x^2-a^2)),x)
--R
--R
@@ -24,6 +24,39 @@
--R (1) - log(\|x - a - x)
--R Type: Union(Expression Integer,...)
--E
+
+--S 2
+bb:=log(x+sqrt(x^2-a^2))
+--R
+--R +-------+
+--R | 2 2
+--R (2) log(\|x - a + x)
+--R Type: Expression Integer
+--E
+
+--S 3
+cc:=aa-bb
+--R
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R (3) - log(\|x - a + x) - log(\|x - a - x)
+--R Type: Expression Integer
+--E
+
+--S 4
+logmul1:=rule(c*log(a)+c*log(b) == c*log(a*b))
+--R
+--I (4) c log(b) + c log(a) + %I == c log(a b) + %I
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 5 14:210 Schaums and Axiom differ by a constant
+dd:=logmul1 cc
+--R
+--R 2
+--R (5) - log(- a )
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.xxx~~~~~$\displaystyle\int{\frac{x~dx}{\sqrt{x^2-a^2}}}$}
@@ -31,7 +64,7 @@
<<*>>=
)clear all
---S 2 of 28
+--S 6
aa:=integrate(x/(sqrt(x^2-a^2)),x)
--R
--R
@@ -44,6 +77,22 @@
--R \|x - a - x
--R Type: Union(Expression Integer,...)
--E
+
+--S 7
+bb:=sqrt(x^2-a^2)
+--R
+--R +-------+
+--R | 2 2
+--R (2) \|x - a
+--R Type: Expression Integer
+--E
+
+--S 8 14:xxx Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.211~~~~~$\displaystyle
@@ -54,7 +103,7 @@
<<*>>=
)clear all
---S 3 of 28
+--S 9
aa:=integrate(x^2/sqrt(x^2-a^2),x)
--R
--R
@@ -72,6 +121,29 @@
--R 4x\|x - a - 4x + 2a
--R Type: Union(Expression Integer,...)
--E
+
+--S 10
+bb:=(x*sqrt(x^2-a^2))/2+a^2/2*log(x+sqrt(x^2-a^2))
+--R
+--R +-------+ +-------+
+--R 2 | 2 2 | 2 2
+--R a log(\|x - a + x) + x\|x - a
+--R (2) -----------------------------------
+--R 2
+--R Type: Expression Integer
+--E
+
+--S 11 14:211 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +-------+ +-------+
+--R 2 | 2 2 2 | 2 2
+--R - a log(\|x - a + x) - a log(\|x - a - x)
+--R (3) -----------------------------------------------
+--R 2
+--R Type: Expression Integer
+--E
+
@
\section{\cite{1}:14.212~~~~~$\displaystyle
@@ -82,7 +154,7 @@
<<*>>=
)clear all
---S 4 of 28
+--S 12
aa:=integrate(x^3/sqrt(x^2-a^2),x)
--R
--R
@@ -95,6 +167,24 @@
--R (12x - 3a )\|x - a - 12x + 9a x
--R Type: Union(Expression Integer,...)
--E
+
+--S 13
+bb:=(x^2-a^2)^(3/2)/3+a^2*sqrt(x^2-a^2)
+--R
+--R +-------+
+--R 2 2 | 2 2
+--R (x + 2a )\|x - a
+--R (2) --------------------
+--R 3
+--R Type: Expression Integer
+--E
+
+--S 14 14:212 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.213~~~~~$\displaystyle\int{\frac{dx}{x\sqrt{x^2-a^2}}}$}
@@ -104,7 +194,7 @@
<<*>>=
)clear all
---S 5 of 28
+--S 15
aa:=integrate(1/(x*sqrt(x^2-a^2)),x)
--R
--R
@@ -117,6 +207,30 @@
--R a
--R Type: Union(Expression Integer,...)
--E
+
+--S 16
+bb:=1/a*asec(abs(x/a))
+--R
+--R x
+--R asec(abs(-))
+--R a
+--R (2) ------------
+--R a
+--R Type: Expression Integer
+--E
+
+--S 17 14:213 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +-------+
+--R | 2 2
+--R \|x - a - x x
+--R 2atan(--------------) - asec(abs(-))
+--R a a
+--R (3) ------------------------------------
+--R a
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.214~~~~~$\displaystyle
@@ -127,7 +241,7 @@
<<*>>=
)clear all
---S 6 of 28
+--S 18
aa:=integrate(1/(x^2*sqrt(x^2-a^2)),x)
--R
--R
@@ -138,6 +252,28 @@
--R x\|x - a - x
--R Type: Union(Expression Integer,...)
--E
+
+--S 19
+bb:=sqrt(x^2-a^2)/(a^2*x)
+--R
+--R +-------+
+--R | 2 2
+--R \|x - a
+--R (2) ----------
+--R 2
+--R a x
+--R Type: Expression Integer
+--E
+
+--S 20 14:214 Schaums and Axiom differ by a constant
+cc:=aa-bb
+--R
+--R 1
+--R (3) --
+--R 2
+--R a
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.215~~~~~$\displaystyle\int{\frac{dx}{x^3\sqrt{x^2-a^2}}}$}
@@ -148,7 +284,7 @@
<<*>>=
)clear all
---S 7 of 28
+--S 21
aa:=integrate(1/(x^3*sqrt(x^2-a^2)),x)
--R
--R
@@ -168,6 +304,34 @@
--R 4a x \|x - a - 4a x + 2a x
--R Type: Union(Expression Integer,...)
--E
+
+--S 22
+bb:=sqrt(x^2-a^2)/(2*a^2*x^2)+1/(2*a^3)*asec(abs(x/a))
+--R
+--R +-------+
+--R 2 x | 2 2
+--R x asec(abs(-)) + a\|x - a
+--R a
+--R (2) ----------------------------
+--R 3 2
+--R 2a x
+--R Type: Expression Integer
+--E
+
+--S 23 14:215 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +-------+
+--R | 2 2
+--R \|x - a - x x
+--R 2atan(--------------) - asec(abs(-))
+--R a a
+--R (3) ------------------------------------
+--R 3
+--R 2a
+--R Type: Expression Integer
+--E
+
@
\section{\cite{1}:14.216~~~~~$\displaystyle\int{\sqrt{x^2-a^2}}~dx$}
@@ -177,7 +341,7 @@
<<*>>=
)clear all
---S 8 of 28
+--S 24
aa:=integrate(sqrt(x^2-a^2),x)
--R
--R
@@ -195,6 +359,29 @@
--R 4x\|x - a - 4x + 2a
--R Type: Union(Expression Integer,...)
--E
+
+--S 25
+bb:=(x*sqrt(x^2-a^2))/2-a^2/2*log(x+sqrt(x^2-a^2))
+--R
+--R +-------+ +-------+
+--R 2 | 2 2 | 2 2
+--R - a log(\|x - a + x) + x\|x - a
+--R (2) -------------------------------------
+--R 2
+--R Type: Expression Integer
+--E
+
+--S 26 14:216 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +-------+ +-------+
+--R 2 | 2 2 2 | 2 2
+--R a log(\|x - a + x) + a log(\|x - a - x)
+--R (3) ---------------------------------------------
+--R 2
+--R Type: Expression Integer
+--E
+
@
\section{\cite{1}:14.217~~~~~$\displaystyle\int{x\sqrt{x^2-a^2}}~dx$}
@@ -204,7 +391,7 @@
<<*>>=
)clear all
---S 9 of 28
+--S 27
aa:=integrate(x*sqrt(x^2-a^2),x)
--R
--R
@@ -217,6 +404,24 @@
--R (12x - 3a )\|x - a - 12x + 9a x
--R Type: Union(Expression Integer,...)
--E
+
+--S 28
+bb:=(x^2-a^2)^(3/2)/3
+--R
+--R +-------+
+--R 2 2 | 2 2
+--R (x - a )\|x - a
+--R (2) -------------------
+--R 3
+--R Type: Expression Integer
+--E
+
+--S 29 14:217 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.218~~~~~$\displaystyle
@@ -228,7 +433,7 @@
<<*>>=
)clear all
---S 10 of 28
+--S 30
aa:=integrate(x^2*sqrt(x^2-a^2),x)
--R
--R
@@ -246,6 +451,28 @@
--R (64x - 32a x)\|x - a - 64x + 64a x - 8a
--R Type: Union(Expression Integer,...)
--E
+
+--S 31
+bb:=(x*(x^2-a^2)^(3/2))/4+(a^2*x*sqrt(x^2-a^2))/8-a^4/8*log(x+sqrt(x^2-a^2))
+--R
+--R +-------+ +-------+
+--R 4 | 2 2 3 2 | 2 2
+--R - a log(\|x - a + x) + (2x - a x)\|x - a
+--R (2) -----------------------------------------------
+--R 8
+--R Type: Expression Integer
+--E
+
+--S 32 14:218 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +-------+ +-------+
+--R 4 | 2 2 4 | 2 2
+--R a log(\|x - a + x) + a log(\|x - a - x)
+--R (3) ---------------------------------------------
+--R 8
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.219~~~~~$\displaystyle
@@ -256,7 +483,7 @@
<<*>>=
)clear all
---S 11 of 28
+--S 33
aa:=integrate(x^3*sqrt(x^2-a^2),x)
--R
--R
@@ -273,6 +500,24 @@
--R (240x - 180a x + 15a )\|x - a - 240x + 300a x - 75a x
--R Type: Union(Expression Integer,...)
--E
+
+--S 34
+bb:=(x^2-a^2)^(5/2)/5+(a^2*(x^2-a^2)^(3/2))/3
+--R
+--R +-------+
+--R 4 2 2 4 | 2 2
+--R (3x - a x - 2a )\|x - a
+--R (2) ----------------------------
+--R 15
+--R Type: Expression Integer
+--E
+
+--S 35 14:219 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.220~~~~~$\displaystyle
@@ -283,7 +528,7 @@
<<*>>=
)clear all
---S 12 of 28
+--S 36
aa:=integrate(sqrt(x^2-a^2)/x,x)
--R
--R
@@ -298,6 +543,27 @@
--R \|x - a - x
--R Type: Union(Expression Integer,...)
--E
+
+--S 37
+bb:=sqrt(x^2-a^2)-a*asec(abs(x/a))
+--R
+--R +-------+
+--R x | 2 2
+--R (2) - a asec(abs(-)) + \|x - a
+--R a
+--R Type: Expression Integer
+--E
+
+--S 38 14:220 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +-------+
+--R | 2 2
+--R \|x - a - x x
+--R (3) - 2a atan(--------------) + a asec(abs(-))
+--R a a
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.221~~~~~$\displaystyle
@@ -308,7 +574,7 @@
<<*>>=
)clear all
---S 13 of 28
+--S 39
aa:=integrate(sqrt(x^2-a^2)/x^2,x)
--R
--R
@@ -321,6 +587,27 @@
--R x\|x - a - x
--R Type: Union(Expression Integer,...)
--E
+
+--S 40
+bb:=-sqrt(x^2-a^2)/x+log(x+sqrt(x^2-a^2))
+--R
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R x log(\|x - a + x) - \|x - a
+--R (2) ----------------------------------
+--R x
+--R Type: Expression Integer
+--E
+
+--S 41 14:221 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R (3) - log(\|x - a + x) - log(\|x - a - x) - 1
+--R Type: Expression Integer
+--E
+
@
\section{\cite{1}:14.222~~~~~$\displaystyle
@@ -332,7 +619,7 @@
<<*>>=
)clear all
---S 14 of 28
+--S 42
aa:=integrate(sqrt(x^2-a^2)/x^3,x)
--R
--R
@@ -352,6 +639,32 @@
--R 4a x \|x - a - 4a x + 2a x
--R Type: Union(Expression Integer,...)
--E
+
+--S 43
+bb:=-sqrt(x^2-a^2)/(2*x^2)+1/(2*a)*asec(abs(x/a))
+--R
+--R +-------+
+--R 2 x | 2 2
+--R x asec(abs(-)) - a\|x - a
+--R a
+--R (2) ----------------------------
+--R 2
+--R 2a x
+--R Type: Expression Integer
+--E
+
+--S 44 14:222 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +-------+
+--R | 2 2
+--R \|x - a - x x
+--R 2atan(--------------) - asec(abs(-))
+--R a a
+--R (3) ------------------------------------
+--R 2a
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.223~~~~~$\displaystyle\int{\frac{dx}{(x^2-a^2)^{3/2}}}$}
@@ -361,7 +674,7 @@
<<*>>=
)clear all
---S 15 of 28
+--S 45
aa:=integrate(1/(x^2-a^2)^(3/2),x)
--R
--R
@@ -372,6 +685,27 @@
--R x\|x - a - x + a
--R Type: Union(Expression Integer,...)
--E
+
+--S 46
+bb:=-x/(a^2*sqrt(x^2-a^2))
+--R
+--R x
+--R (2) - ------------
+--R +-------+
+--R 2 | 2 2
+--R a \|x - a
+--R Type: Expression Integer
+--E
+
+--S 47 14:223 Schaums and Axiom differ by a constant
+cc:=aa-bb
+--R
+--R 1
+--R (3) - --
+--R 2
+--R a
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.224~~~~~$\displaystyle
@@ -382,7 +716,7 @@
<<*>>=
)clear all
---S 16 of 28
+--S 48
aa:=integrate(x/(x^2-a^2)^(3/2),x)
--R
--R
@@ -395,6 +729,24 @@
--R x\|x - a - x + a
--R Type: Union(Expression Integer,...)
--E
+
+--S 49
+bb:=-1/sqrt(x^2-a^2)
+--R
+--R 1
+--R (2) - ----------
+--R +-------+
+--R | 2 2
+--R \|x - a
+--R Type: Expression Integer
+--E
+
+--S 50 14:224 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.225~~~~~$\displaystyle
@@ -405,7 +757,7 @@
<<*>>=
)clear all
---S 17 of 28
+--S 51
aa:=integrate(x^2/(x^2-a^2)^(3/2),x)
--R
--R
@@ -418,6 +770,28 @@
--R x\|x - a - x + a
--R Type: Union(Expression Integer,...)
--E
+
+--S 52
+bb:=-x/sqrt(x^2-a^2)+log(x+sqrt(x^2-a^2))
+--R
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R \|x - a log(\|x - a + x) - x
+--R (2) ---------------------------------
+--R +-------+
+--R | 2 2
+--R \|x - a
+--R Type: Expression Integer
+--E
+
+--S 53 14:225 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R (3) - log(\|x - a + x) - log(\|x - a - x) - 1
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.226~~~~~$\displaystyle
@@ -428,7 +802,7 @@
<<*>>=
)clear all
---S 18 of 28
+--S 54
aa:=integrate(x^3/(x^2-a^2)^(3/2),x)
--R
--R
@@ -441,6 +815,25 @@
--R (2x - a )\|x - a - 2x + 2a x
--R Type: Union(Expression Integer,...)
--E
+
+--S 55
+bb:=sqrt(x^2-a^2)-a^2/sqrt(x^2-a^2)
+--R
+--R 2 2
+--R x - 2a
+--R (2) ----------
+--R +-------+
+--R | 2 2
+--R \|x - a
+--R Type: Expression Integer
+--E
+
+--S 56 14:226 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.227~~~~~$\displaystyle
@@ -452,7 +845,7 @@
<<*>>=
)clear all
---S 19 of 28
+--S 57
aa:=integrate(1/(x*(x^2-a^2)^(3/2)),x)
--R
--R
@@ -467,6 +860,34 @@
--R a x\|x - a - a x + a
--R Type: Union(Expression Integer,...)
--E
+
+--S 58
+bb:=-1/(a^2*sqrt(x^2-a^2))-1/a^3*asec(abs(x/a))
+--R
+--R +-------+
+--R | 2 2 x
+--R - \|x - a asec(abs(-)) - a
+--R a
+--R (2) ----------------------------
+--R +-------+
+--R 3 | 2 2
+--R a \|x - a
+--R Type: Expression Integer
+--E
+
+--S 59 14:227 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +-------+
+--R | 2 2
+--R \|x - a - x x
+--R - 2atan(--------------) + asec(abs(-))
+--R a a
+--R (3) --------------------------------------
+--R 3
+--R a
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.228~~~~~$\displaystyle
@@ -477,7 +898,7 @@
<<*>>=
)clear all
---S 20 of 28
+--S 60
aa:=integrate(1/(x^2*(x^2-a^2)^(3/2)),x)
--R
--R
@@ -488,6 +909,29 @@
--R (2x - a x)\|x - a - 2x + 2a x
--R Type: Union(Expression Integer,...)
--E
+
+--S 61
+bb:=-sqrt(x^2-a^2)/(a^4*x)-x/(a^4*sqrt(x^2-a^2))
+--R
+--R 2 2
+--R - 2x + a
+--R (2) -------------
+--R +-------+
+--R 4 | 2 2
+--R a x\|x - a
+--R Type: Expression Integer
+--E
+
+--S 62 14:228 Schaums and Axiom differ by a constant
+cc:=aa-bb
+--R
+--R 2
+--R (3) - --
+--R 4
+--R a
+--R Type: Expression Integer
+--E
+
@
\section{\cite{1}:14.229~~~~~$\displaystyle
@@ -500,7 +944,7 @@
<<*>>=
)clear all
---S 21 of 28
+--S 63
aa:=integrate(1/(x^3*(x^2-a^2)^(3/2)),x)
--R
--R
@@ -520,6 +964,34 @@
--R (8a x - 6a x )\|x - a - 8a x + 10a x - 2a x
--R Type: Union(Expression Integer,...)
--E
+
+--S 64
+bb:=1/(2*a^2*x^2*sqrt(x^2-a^2))-3/(2*a^4*sqrt(x^2-a^2))-3/(2*a^5)*asec(abs(x/a))
+--R
+--R +-------+
+--R 2 | 2 2 x 2 3
+--R - 3x \|x - a asec(abs(-)) - 3a x + a
+--R a
+--R (2) ----------------------------------------
+--R +-------+
+--R 5 2 | 2 2
+--R 2a x \|x - a
+--R Type: Expression Integer
+--E
+
+--S 65 14:229 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +-------+
+--R | 2 2
+--R \|x - a - x x
+--R - 6atan(--------------) + 3asec(abs(-))
+--R a a
+--R (3) ---------------------------------------
+--R 5
+--R 2a
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.230~~~~~$\displaystyle\int{(x^2-a^2)^{3/2}}~dx$}
@@ -530,7 +1002,7 @@
<<*>>=
)clear all
---S 22 of 28
+--S 66
aa:=integrate((x^2-a^2)^(3/2),x)
--R
--R
@@ -551,6 +1023,29 @@
--R (64x - 32a x)\|x - a - 64x + 64a x - 8a
--R Type: Union(Expression Integer,...)
--E
+
+--S 67
+bb:=(x*(x^2-a^2)^(3/2))/4-(3*a^2*x*sqrt(x^2-a^2))/8+3/8*a^4*log(x+sqrt(x^2-a^2))
+--R
+--R +-------+ +-------+
+--R 4 | 2 2 3 2 | 2 2
+--R 3a log(\|x - a + x) + (2x - 5a x)\|x - a
+--R (2) -----------------------------------------------
+--R 8
+--R Type: Expression Integer
+--E
+
+--S 68 14:230 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +-------+ +-------+
+--R 4 | 2 2 4 | 2 2
+--R - 3a log(\|x - a + x) - 3a log(\|x - a - x)
+--R (3) -------------------------------------------------
+--R 8
+--R Type: Expression Integer
+--E
+
@
\section{\cite{1}:14.231~~~~~$\displaystyle\int{x(x^2-a^2)^{3/2}}~dx$}
@@ -558,7 +1053,7 @@
<<*>>=
)clear all
---S 23 of 28
+--S 69
aa:=integrate(x*(x^2-a^2)^(3/2),x)
--R
--R
@@ -575,6 +1070,24 @@
--R (80x - 60a x + 5a )\|x - a - 80x + 100a x - 25a x
--R Type: Union(Expression Integer,...)
--E
+
+--S 70
+bb:=(x^2-a^2)^(5/2)/5
+--R
+--R +-------+
+--R 4 2 2 4 | 2 2
+--R (x - 2a x + a )\|x - a
+--R (2) ---------------------------
+--R 5
+--R Type: Expression Integer
+--E
+
+--S 71 14:231 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.232~~~~~$\displaystyle\int{x^2(x^2-a^2)^{3/2}}~dx$}
@@ -586,7 +1099,7 @@
<<*>>=
)clear all
---S 24 of 28
+--S 72
aa:=integrate(x^2*(x^2-a^2)^(3/2),x)
--R
--R
@@ -614,6 +1127,28 @@
--R (1536x - 1536a x + 288a x)\|x - a - 1536x + 2304a x - 864a x + 48a
--R Type: Union(Expression Integer,...)
--E
+
+--S 73
+bb:=(x*(x^2-a^2)^(5/2))/6+(a^2*x*(x^2-a^2)^(3/2))/24-(a^4*x*sqrt(x^2-a^2))/16+a^6/16*log(x+sqrt(x^2-a^2))
+--R
+--R +-------+ +-------+
+--R 6 | 2 2 5 2 3 4 | 2 2
+--R 3a log(\|x - a + x) + (8x - 14a x + 3a x)\|x - a
+--R (2) --------------------------------------------------------
+--R 48
+--R Type: Expression Integer
+--E
+
+--S 74 14:232 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +-------+ +-------+
+--R 6 | 2 2 6 | 2 2
+--R - a log(\|x - a + x) - a log(\|x - a - x)
+--R (3) -----------------------------------------------
+--R 16
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.233~~~~~$\displaystyle\int{x^3(x^2-a^2)^{3/2}}~dx$}
@@ -623,7 +1158,7 @@
<<*>>=
)clear all
---S 25 of 28
+--S 75
aa:=integrate(x^3*(x^2-a^2)^(3/2),x)
--R
--R
@@ -652,6 +1187,24 @@
--R - 1960a x + 245a x
--R Type: Union(Expression Integer,...)
--E
+
+--S 76
+bb:=(x^2-a^2)^(7/2)/7+(a^2*(x^2-a^2)^(5/2))/5
+--R
+--R +-------+
+--R 6 2 4 4 2 6 | 2 2
+--R (5x - 8a x + a x + 2a )\|x - a
+--R (2) ------------------------------------
+--R 35
+--R Type: Expression Integer
+--E
+
+--S 77 14:233 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.234~~~~~$\displaystyle
@@ -663,7 +1216,7 @@
<<*>>=
)clear all
---S 26 of 28
+--S 78
aa:=integrate((x^2-a^2)^(3/2)/x,x)
--R
--R
@@ -683,6 +1236,29 @@
--R (12x - 3a )\|x - a - 12x + 9a x
--R Type: Union(Expression Integer,...)
--E
+
+--S 79
+bb:=(x^2-a^2)^(3/2)/3-a^2*sqrt(x^2-a^2)+a^3*asec(abs(x/a))
+--R
+--R +-------+
+--R 3 x 2 2 | 2 2
+--R 3a asec(abs(-)) + (x - 4a )\|x - a
+--R a
+--R (2) --------------------------------------
+--R 3
+--R Type: Expression Integer
+--E
+
+--S 80 14:234 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +-------+
+--R | 2 2
+--R 3 \|x - a - x 3 x
+--R (3) 2a atan(--------------) - a asec(abs(-))
+--R a a
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.235~~~~~$\displaystyle
@@ -694,7 +1270,7 @@
<<*>>=
)clear all
---S 27 of 28
+--S 81
aa:=integrate((x^2-a^2)^{3/2}/x^2,x)
--R
--R
@@ -712,6 +1288,28 @@
--R (8x - 2a x)\|x - a - 8x + 6a x
--R Type: Union(Expression Integer,...)
--E
+
+--S 82
+bb:=-(x^2-a^2)^(3/2)/x+3*x*sqrt(x^2-a^2)/2-3/2*a^2*log(x+sqrt(x^2-a^2))
+--R
+--R +-------+ +-------+
+--R 2 | 2 2 2 2 | 2 2
+--R - 3a x log(\|x - a + x) + (x + 2a )\|x - a
+--R (2) -------------------------------------------------
+--R 2x
+--R Type: Expression Integer
+--E
+
+--S 83 14:235 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +-------+ +-------+
+--R 2 | 2 2 2 | 2 2 2
+--R 3a log(\|x - a + x) + 3a log(\|x - a - x) + 2a
+--R (3) -----------------------------------------------------
+--R 2
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.236~~~~~$\displaystyle
@@ -723,7 +1321,7 @@
<<*>>=
)clear all
---S 28 of 28
+--S 84
aa:=integrate((x^2-a^2)^(3/2)/x^3,x)
--R
--R
@@ -744,6 +1342,31 @@
--R Type: Union(Expression Integer,...)
--E
+--S 85
+bb:=-(x^2-a^2)^(3/2)/(2*x^2)+(3*sqrt(x^2-a^2))/2-3/2*a*asec(abs(x/a))
+--R
+--R +-------+
+--R 2 x 2 2 | 2 2
+--R - 3a x asec(abs(-)) + (2x + a )\|x - a
+--R a
+--R (2) ------------------------------------------
+--R 2
+--R 2x
+--R Type: Expression Integer
+--E
+
+--S 86 14:236 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +-------+
+--R | 2 2
+--R \|x - a - x x
+--R - 6a atan(--------------) + 3a asec(abs(-))
+--R a a
+--R (3) -------------------------------------------
+--R 2
+--R Type: Expression Integer
+--E
)spool
)lisp (bye)
@
Modified: trunk/axiom/src/input/schaum8.input.pamphlet
===================================================================
--- trunk/axiom/src/input/schaum8.input.pamphlet 2008-04-17 15:06:30 UTC (rev 856)
+++ trunk/axiom/src/input/schaum8.input.pamphlet 2008-04-18 06:42:58 UTC (rev 857)
@@ -16,7 +16,7 @@
)set message auto off
)clear all
---S 1 of 19
+--S 1
aa:=integrate(1/(a^2-x^2),x)
--R
--R
@@ -25,6 +25,63 @@
--R 2a
--R Type: Union(Expression Integer,...)
--E
+
+--S 2
+bb:=1/(2*a)*log((a+x)/(a-x))
+--R
+--R - x - a
+--R log(-------)
+--R x - a
+--R (2) ------------
+--R 2a
+--R Type: Expression Integer
+--E
+
+--S 3
+cc:=aa-bb
+--R
+--R - x - a
+--R log(x + a) - log(x - a) - log(-------)
+--R x - a
+--R (3) --------------------------------------
+--R 2a
+--R Type: Expression Integer
+--E
+
+--S 4
+divlog:=rule(log(a/b) == log(a) - log(b))
+--R
+--R a
+--R (4) log(-) == - log(b) + log(a)
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 5
+dd:=divlog cc
+--R
+--R log(x + a) - log(- x - a)
+--R (5) -------------------------
+--R 2a
+--R Type: Expression Integer
+--E
+
+--S 6
+logminus:=rule(log(x + a) - log(- x - a) == log(-1))
+--R
+--I (6) log(x + a) - log(- x - a) + %I == log(- 1) + %I
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 7 14:163 Schaums and Axiom differ by a constant
+ee:=logminus dd
+--R
+--R log(- 1)
+--R (7) --------
+--R 2a
+--R Type: Expression Integer
+--E
+
@
\section{\cite{1}:14.164~~~~~$\displaystyle\int{\frac{x~dx}{a^2-x^2}}$}
@@ -32,7 +89,7 @@
<<*>>=
)clear all
---S 2 of 19
+--S 8
aa:=integrate(x/(a^2-x^2),x)
--R
--R
@@ -42,6 +99,43 @@
--R 2
--R Type: Union(Expression Integer,...)
--E
+
+--S 9
+bb:=-1/2*log(a^2-x^2)
+--R
+--R 2 2
+--R log(- x + a )
+--R (2) - --------------
+--R 2
+--R Type: Expression Integer
+--E
+
+--S 10
+cc:=aa-bb
+--R
+--R 2 2 2 2
+--R - log(x - a ) + log(- x + a )
+--R (3) -------------------------------
+--R 2
+--R Type: Expression Integer
+--E
+
+--S 11
+logminus1:=rule(-log(x^2-a^2)+log(-x^2+a^2) == log(-1))
+--R
+--R 2 2 2 2
+--I (4) - log(x - a ) + log(- x + a ) + %H == log(- 1) + %H
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 12 14:164 Schaums and Axiom differ by a constant
+dd:=logminus1 cc
+--R
+--R log(- 1)
+--R (5) --------
+--R 2
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.165~~~~~$\displaystyle\int{\frac{x^2~dx}{a^2-x^2}}$}
@@ -49,7 +143,7 @@
<<*>>=
)clear all
---S 3 of 19
+--S 13
aa:=integrate(x^2/(a^2-x^2),x)
--R
--R
@@ -58,6 +152,62 @@
--R 2
--R Type: Union(Expression Integer,...)
--E
+
+--S 14
+bb:=-x+a/2*log((a+x)/(a-x))
+--R
+--R - x - a
+--R a log(-------) - 2x
+--R x - a
+--R (2) -------------------
+--R 2
+--R Type: Expression Integer
+--E
+
+--S 15
+cc:=aa-bb
+--R
+--R - x - a
+--R a log(x + a) - a log(x - a) - a log(-------)
+--R x - a
+--R (3) --------------------------------------------
+--R 2
+--R Type: Expression Integer
+--E
+
+--S 16
+divlog:=rule(log(a/b) == log(a) - log(b))
+--R
+--R a
+--R (4) log(-) == - log(b) + log(a)
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 17
+dd:=divlog cc
+--R
+--R a log(x + a) - a log(- x - a)
+--R (5) -----------------------------
+--R 2
+--R Type: Expression Integer
+--E
+
+--S 18
+logminusa:=rule(b*log(x + a) - b*log(- x - a) == b*log(-1))
+--R
+--I (6) b log(x + a) - b log(- x - a) + %M == b log(- 1) + %M
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 19 14:165 Schaums and Axiom differ by a constant
+ee:=logminusa dd
+--R
+--R a log(- 1)
+--R (7) ----------
+--R 2
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.166~~~~~$\displaystyle\int{\frac{x^3~dx}{a^2-x^2}}$}
@@ -66,7 +216,7 @@
<<*>>=
)clear all
---S 4 of 19
+--S 20
aa:=integrate(x^3/(a^2-x^2),x)
--R
--R
@@ -76,6 +226,45 @@
--R 2
--R Type: Union(Expression Integer,...)
--E
+
+--S 21
+bb:=-x^2/2-a^2/2*log(a^2-x^2)
+--R
+--R 2 2 2 2
+--R - a log(- x + a ) - x
+--R (2) -----------------------
+--R 2
+--R Type: Expression Integer
+--E
+
+--S 22
+cc:=aa-bb
+--R
+--R 2 2 2 2 2 2
+--R - a log(x - a ) + a log(- x + a )
+--R (3) -----------------------------------
+--R 2
+--R Type: Expression Integer
+--E
+
+--S 23
+logminus1b:=rule(-b*log(x^2-a^2)+b*log(-x^2+a^2) == b*log(-1))
+--R
+--R 2 2 2 2
+--I (4) - b log(x - a ) + b log(- x + a ) + %N == b log(- 1) + %N
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 24 14:166 Schaums and Axiom differ by a constant
+dd:=logminus1b cc
+--R
+--R 2
+--R a log(- 1)
+--R (5) ----------
+--R 2
+--R Type: Expression Integer
+--E
+
@
\section{\cite{1}:14.167~~~~~$\displaystyle\int{\frac{dx}{x(a^2-x^2)}}$}
@@ -85,7 +274,7 @@
<<*>>=
)clear all
---S 5 of 19
+--S 25
aa:=integrate(1/(x*(a^2-x^2)),x)
--R
--R
@@ -96,6 +285,72 @@
--R 2a
--R Type: Union(Expression Integer,...)
--E
+
+--S 26
+bb:=1/(2*a^2)*log(x^2/(a^2-x^2))
+--R
+--R 2
+--R x
+--R log(- -------)
+--R 2 2
+--R x - a
+--R (2) --------------
+--R 2
+--R 2a
+--R Type: Expression Integer
+--E
+
+--S 27
+cc:=aa-bb
+--R
+--R 2
+--R 2 2 x
+--R - log(x - a ) + 2log(x) - log(- -------)
+--R 2 2
+--R x - a
+--R (3) -----------------------------------------
+--R 2
+--R 2a
+--R Type: Expression Integer
+--E
+
+--S 28
+divlog:=rule(log(a/b) == log(a) - log(b))
+--R
+--R a
+--R (4) log(-) == - log(b) + log(a)
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 29
+dd:=divlog cc
+--R
+--R 2
+--R 2log(x) - log(- x )
+--R (5) -------------------
+--R 2
+--R 2a
+--R Type: Expression Integer
+--E
+
+--S 30
+logpowminus:=rule(log(-a^n) == n*log(a)+log(-1))
+--R
+--R n
+--R (6) log(- a ) == n log(a) + log(- 1)
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 31 14:167 Schaums and Axiom differ by a constant
+ee:=logpowminus dd
+--R
+--R log(- 1)
+--R (7) - --------
+--R 2
+--R 2a
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.168~~~~~$\displaystyle\int{\frac{dx}{x^2(a^2-x^2)}}$}
@@ -105,7 +360,7 @@
<<*>>=
)clear all
---S 6 of 19
+--S 32
aa:=integrate(1/(x^2*(a^2-x^2)),x)
--R
--R
@@ -115,6 +370,67 @@
--R 2a x
--R Type: Union(Expression Integer,...)
--E
+
+--S 33
+bb:=-1/(a^2*x)+1/(2*a^3)*log((a+x)/(a-x))
+--R
+--R - x - a
+--R x log(-------) - 2a
+--R x - a
+--R (2) -------------------
+--R 3
+--R 2a x
+--R Type: Expression Integer
+--E
+
+--S 34
+cc:=aa-bb
+--R
+--R - x - a
+--R log(x + a) - log(x - a) - log(-------)
+--R x - a
+--R (3) --------------------------------------
+--R 3
+--R 2a
+--R Type: Expression Integer
+--E
+
+--S 35
+divlog:=rule(log(a/b) == log(a) - log(b))
+--R
+--R a
+--R (4) log(-) == - log(b) + log(a)
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 36
+dd:=divlog cc
+--R
+--R log(x + a) - log(- x - a)
+--R (5) -------------------------
+--R 3
+--R 2a
+--R Type: Expression Integer
+--E
+
+--S 37
+logminus:=rule(log(x + a) - log(- x - a) == log(-1))
+--R
+--I (6) log(x + a) - log(- x - a) + %O == log(- 1) + %O
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 38 14:168 Schaums and Axiom differ by a constant
+ee:=logminus dd
+--R
+--R log(- 1)
+--R (7) --------
+--R 3
+--R 2a
+--R Type: Expression Integer
+--E
+
@
\section{\cite{1}:14.169~~~~~$\displaystyle\int{\frac{dx}{x^3(a^2-x^2)}}$}
@@ -124,7 +440,7 @@
<<*>>=
)clear all
---S 7 of 19
+--S 39
aa:=integrate(1/(x^3*(a^2-x^2)),x)
--R
--R
@@ -135,6 +451,70 @@
--R 2a x
--R Type: Union(Expression Integer,...)
--E
+
+--S 40
+bb:=-1/(2*a^2*x^2)+1/(2*a^4)*log(x^2/(a^2-x^2))
+--R
+--R 2
+--R 2 x 2
+--R x log(- -------) - a
+--R 2 2
+--R x - a
+--R (2) ---------------------
+--R 4 2
+--R 2a x
+--R Type: Expression Integer
+--E
+
+--S 41
+divlog:=rule(log(a/b) == log(a) - log(b))
+--R
+--R a
+--R (3) log(-) == - log(b) + log(a)
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 42
+bb1:=divlog bb
+--R
+--R 2 2 2 2 2 2
+--R - x log(x - a ) + x log(- x ) - a
+--R (4) -----------------------------------
+--R 4 2
+--R 2a x
+--R Type: Expression Integer
+--E
+
+--S 43
+cc:=aa-bb1
+--R
+--R 2
+--R 2log(x) - log(- x )
+--R (5) -------------------
+--R 4
+--R 2a
+--R Type: Expression Integer
+--E
+
+--S 44
+logminuspow:=rule(log(-x^n) == n*log(x)+log(-1))
+--R
+--R n
+--R (6) log(- x ) == n log(x) + log(- 1)
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 45 14:169 Schaums and Axiom differ by a constant
+dd:=logminuspow cc
+--R
+--R log(- 1)
+--R (7) - --------
+--R 4
+--R 2a
+--R Type: Expression Integer
+--E
+
@
\section{\cite{1}:14.170~~~~~$\displaystyle\int{\frac{dx}{(a^2-x^2)^2}}$}
@@ -144,7 +524,7 @@
<<*>>=
)clear all
---S 8 of 19
+--S 46
aa:=integrate(1/((a^2-x^2)^2),x)
--R
--R
@@ -155,6 +535,66 @@
--R 4a x - 4a
--R Type: Union(Expression Integer,...)
--E
+
+--S 47
+bb:=x/(2*a^2*(a^2-x^2))+1/(4*a^3)*log((a+x)/(a-x))
+--R
+--R 2 2 - x - a
+--R (x - a )log(-------) - 2a x
+--R x - a
+--R (2) ----------------------------
+--R 3 2 5
+--R 4a x - 4a
+--R Type: Expression Integer
+--E
+
+--S 48
+divlog:=rule(log(a/b) == log(a) - log(b))
+--R
+--R a
+--R (3) log(-) == - log(b) + log(a)
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 49
+bb1:=divlog bb
+--R
+--R 2 2 2 2
+--R (- x + a )log(x - a) + (x - a )log(- x - a) - 2a x
+--R (4) ----------------------------------------------------
+--R 3 2 5
+--R 4a x - 4a
+--R Type: Expression Integer
+--E
+
+--S 50
+cc:=aa-bb1
+--R
+--R log(x + a) - log(- x - a)
+--R (5) -------------------------
+--R 3
+--R 4a
+--R Type: Expression Integer
+--E
+
+--S 51
+logminus:=rule(log(x + a) - log(- x - a) == log(-1))
+--R
+--I (6) log(x + a) - log(- x - a) + %P == log(- 1) + %P
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 52 14:170 Schaums and Axiom differ by a constant
+dd:=logminus cc
+--R
+--R log(- 1)
+--R (7) --------
+--R 3
+--R 4a
+--R Type: Expression Integer
+--E
+
@
\section{\cite{1}:14.171~~~~~$\displaystyle\int{\frac{x~dx}{(a^2-x^2)^2}}$}
@@ -164,7 +604,7 @@
<<*>>=
)clear all
---S 9 of 19
+--S 53
aa:=integrate(x/((a^2-x^2)^2),x)
--R
--R
@@ -174,6 +614,24 @@
--R 2x - 2a
--R Type: Union(Expression Integer,...)
--E
+
+--S 54
+bb:=1/(2*(a^2-x^2))
+--R
+--R 1
+--R (2) - ---------
+--R 2 2
+--R 2x - 2a
+--R Type: Fraction Polynomial Integer
+--E
+
+--S 55 14:171 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression Integer
+--E
+
@
\section{\cite{1}:14.172~~~~~$\displaystyle\int{\frac{x^2dx}{(a^2-x^2)^2}}$}
@@ -183,7 +641,7 @@
<<*>>=
)clear all
---S 10 of 19
+--S 56
aa:=integrate(x^2/((a^2-x^2)^2),x)
--R
--R
@@ -194,6 +652,63 @@
--R 4a x - 4a
--R Type: Union(Expression Integer,...)
--E
+
+--S 57
+bb:=x/(2*(a^2-x^2))-1/(4*a)*log((a+x)/(a-x))
+--R
+--R 2 2 - x - a
+--R (- x + a )log(-------) - 2a x
+--R x - a
+--R (2) ------------------------------
+--R 2 3
+--R 4a x - 4a
+--R Type: Expression Integer
+--E
+
+--S 58
+divlog:=rule(log(a/b) == log(a) - log(b))
+--R
+--R a
+--R (3) log(-) == - log(b) + log(a)
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 59
+bb1:=divlog bb
+--R
+--R 2 2 2 2
+--R (x - a )log(x - a) + (- x + a )log(- x - a) - 2a x
+--R (4) ----------------------------------------------------
+--R 2 3
+--R 4a x - 4a
+--R Type: Expression Integer
+--E
+
+--S 60
+cc:=aa-bb1
+--R
+--R - log(x + a) + log(- x - a)
+--R (5) ---------------------------
+--R 4a
+--R Type: Expression Integer
+--E
+
+--S 61
+logminus2:=rule(-log(x + a) + log(- x - a) == log(-1))
+--R
+--I (6) - log(x + a) + log(- x - a) + %S == log(- 1) + %S
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 62 14:172 Schaums and Axiom differ by a constant
+dd:=logminus2 cc
+--R
+--R log(- 1)
+--R (7) --------
+--R 4a
+--R Type: Expression Integer
+--E
@
\section{\cite{1}:14.173~~~~~$\displaystyle\int{\frac{x^3dx}{(a^2-x^2)^2}}$}
@@ -203,7 +718,7 @@
<<*>>=
)clear all
---S 11 of 19
+--S 63
aa:=integrate(x^3/((a^2-x^2)^2),x)
--R
--R
@@ -214,6 +729,45 @@
--R 2x - 2a
--R Type: Union(Expression Integer,...)
--E
+
+--S 64
+bb:=a^2/(2*(a^2-x^2))+1/2*log(a^2-x^2)
+--R
+--R 2 2 2 2 2
+--R (x - a )log(- x + a ) - a
+--R (2) ----------------------------
+--R 2 2
+--R 2x - 2a
+--R Type: Expression Integer
+--E
+
+--S 65
+cc:=aa-bb
+--R
+--R 2 2 2 2
+--R log(x - a ) - log(- x + a )
+--R (3) -----------------------------
+--R 2
+--R Type: Expression Integer
+--E
+
+--S 66
+logminus3:=rule(log(x^2-a^2)-log(-x^2+a^2) == log(-1))
+--R
+--R 2 2 2 2
+--I (4) log(x - a ) - log(- x + a ) + %T == log(- 1) + %T
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 67 14:173 Schaums and Axiom differ by a constant
+dd:=logminus3 cc
+--R
+--R log(- 1)
+--R (5) --------
+--R 2
+--R Type: Expression Integer
+--E
+
@
\section{\cite{1}:14.174~~~~~$\displaystyle\int{\frac{dx}{x(a^2-x^2)^2}}$}
@@ -223,7 +777,7 @@
<<*>>=
)clear all
---S 12 of 19
+--S 68
aa:=integrate(1/(x*(a^2-x^2)^2),x)
--R
--R
@@ -234,6 +788,70 @@
--R 2a x - 2a
--R Type: Union(Expression Integer,...)
--E
+
+--S 69
+bb:=1/(2*a^2*(a^2-x^2))+1/(2*a^4)*log(x^2/(a^2-x^2))
+--R
+--R 2
+--R 2 2 x 2
+--R (x - a )log(- -------) - a
+--R 2 2
+--R x - a
+--R (2) ----------------------------
+--R 4 2 6
+--R 2a x - 2a
+--R Type: Expression Integer
+--E
+
+--S 70
+divlog:=rule(log(a/b) == log(a) - log(b))
+--R
+--R a
+--R (3) log(-) == - log(b) + log(a)
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 71
+bb1:=divlog bb
+--R
+--R 2 2 2 2 2 2 2 2
+--R (- x + a )log(x - a ) + (x - a )log(- x ) - a
+--R (4) -------------------------------------------------
+--R 4 2 6
+--R 2a x - 2a
+--R Type: Expression Integer
+--E
+
+--S 72
+cc:=aa-bb1
+--R
+--R 2
+--R 2log(x) - log(- x )
+--R (5) -------------------
+--R 4
+--R 2a
+--R Type: Expression Integer
+--E
+
+--S 73
+logpowminus:=rule(log(-a^n) == n*log(a)+log(-1))
+--R
+--R n
+--R (6) log(- a ) == n log(a) + log(- 1)
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 74 14:174 Schaums and Axiom differ by a constant
+dd:=logpowminus cc
+--R
+--R log(- 1)
+--R (7) - --------
+--R 4
+--R 2a
+--R Type: Expression Integer
+--E
+
@
\section{\cite{1}:14.175~~~~~$\displaystyle\int{\frac{dx}{x^2(a^2-x^2)^2}}$}
@@ -244,17 +862,88 @@
<<*>>=
)clear all
---S 13 of 19
-aa:=integrate(1/((a^2-x^2)^2),x)
---R
+--S 75
+aa:=integrate(1/(x^2*(a^2-x^2)^2),x)
--R
---R 2 2 2 2
---R (x - a )log(x + a) + (- x + a )log(x - a) - 2a x
---R (1) --------------------------------------------------
---R 3 2 5
---R 4a x - 4a
+--R 3 2 3 2 2 3
+--R (3x - 3a x)log(x + a) + (- 3x + 3a x)log(x - a) - 6a x + 4a
+--R (1) ---------------------------------------------------------------
+--R 5 3 7
+--R 4a x - 4a x
--R Type: Union(Expression Integer,...)
--E
+
+--S 76
+bb:=-1/(a^4*x)+x/(2*a^4*(a^2-x^2))+3/(4*a^5)*log((a+x)/(a-x))
+--R
+--R 3 2 - x - a 2 3
+--R (3x - 3a x)log(-------) - 6a x + 4a
+--R x - a
+--R (2) --------------------------------------
+--R 5 3 7
+--R 4a x - 4a x
+--R Type: Expression Integer
+--E
+
+--S 77
+divlog:=rule(log(a/b) == log(a) - log(b))
+--R
+--R a
+--R (3) log(-) == - log(b) + log(a)
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 78
+bb1:=divlog bb
+--R
+--R 3 2 3 2 2 3
+--R (- 3x + 3a x)log(x - a) + (3x - 3a x)log(- x - a) - 6a x + 4a
+--R (4) -----------------------------------------------------------------
+--R 5 3 7
+--R 4a x - 4a x
+--R Type: Expression Integer
+--E
+
+--S 79
+cc:=aa-bb
+--R
+--R - x - a
+--R 3log(x + a) - 3log(x - a) - 3log(-------)
+--R x - a
+--R (5) -----------------------------------------
+--R 5
+--R 4a
+--R Type: Expression Integer
+--E
+
+--S 80
+dd:=divlog cc
+--R
+--R 3log(x + a) - 3log(- x - a)
+--R (6) ---------------------------
+--R 5
+--R 4a
+--R Type: Expression Integer
+--E
+
+--S 81
+logminusb:=rule(b*log(x + a) - b*log(- x - a) == b*log(-1))
+--R
+--I (7) b log(x + a) - b log(- x - a) + %U == b log(- 1) + %U
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 82 14:175 Schaums and Axiom differ by a constant
+ee:=logminusb dd
+--R
+--R 3log(- 1)
+--R (8) ---------
+--R 5
+--R 4a
+--R Type: Expression Integer
+--E
+
@
\section{\cite{1}:14.176~~~~~$\displaystyle\int{\frac{dx}{x^3(a^2-x^2)^2}}$}
@@ -265,7 +954,7 @@
<<*>>=
)clear all
---S 14 of 19
+--S 83
aa:=integrate(1/(x^3*(a^2-x^2)^2),x)
--R
--R
@@ -276,6 +965,70 @@
--R 2a x - 2a x
--R Type: Union(Expression Integer,...)
--E
+
+--S 84
+bb:=-1/(2*a^4*x^2)+1/(2*a^4*(a^2-x^2))+1/a^6*log(x^2/(a^2-x^2))
+--R
+--R 2
+--R 4 2 2 x 2 2 4
+--R (2x - 2a x )log(- -------) - 2a x + a
+--R 2 2
+--R x - a
+--R (2) ----------------------------------------
+--R 6 4 8 2
+--R 2a x - 2a x
+--R Type: Expression Integer
+--E
+
+--S 85
+divlog:=rule(log(a/b) == log(a) - log(b))
+--R
+--R a
+--R (3) log(-) == - log(b) + log(a)
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 86
+bb1:=divlog bb
+--R
+--R 4 2 2 2 2 4 2 2 2 2 2 4
+--R (- 2x + 2a x )log(x - a ) + (2x - 2a x )log(- x ) - 2a x + a
+--R (4) -----------------------------------------------------------------
+--R 6 4 8 2
+--R 2a x - 2a x
+--R Type: Expression Integer
+--E
+
+--S 87
+cc:=aa-bb1
+--R
+--R 2
+--R 2log(x) - log(- x )
+--R (5) -------------------
+--R 6
+--R a
+--R Type: Expression Integer
+--E
+
+--S 88
+logpowminus:=rule(log(-a^n) == n*log(a)+log(-1))
+--R
+--R n
+--R (6) log(- a ) == n log(a) + log(- 1)
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 89 14:176 Schaums and Axiom differ by a constant
+dd:=logpowminus cc
+--R
+--R log(- 1)
+--R (7) - --------
+--R 6
+--R a
+--R Type: Expression Integer
+--E
+
@
\section{\cite{1}:14.177~~~~~$\displaystyle\int{\frac{dx}{(a^2-x^2)^n}}$}
@@ -286,7 +1039,7 @@
<<*>>=
)clear all
---S 15 of 19
+--S 90 14:177 Axiom cannot do this integration
aa:=integrate(1/((a^2-x^2)^n),x)
--R
--R
@@ -306,7 +1059,7 @@
<<*>>=
)clear all
---S 16 of 19
+--S 91
aa:=integrate(x/((a^2-x^2)^n),x)
--R
--R
@@ -318,6 +1071,49 @@
--R (2n - 2)%e
--R Type: Union(Expression Integer,...)
--E
+
+--S 92
+bb:=1/(2*(n-1)*(a^2-x^2)^(n-1))
+--R
+--R 1
+--R (2) ------------------------
+--R 2 2 n - 1
+--R (2n - 2)(- x + a )
+--R Type: Expression Integer
+--E
+
+--S 93
+cc:=aa-bb
+--R
+--R 2 2
+--R n log(- x + a ) 2 2 2 2 n - 1
+--R - %e + (- x + a )(- x + a )
+--R (3) --------------------------------------------------
+--R 2 2
+--R 2 2 n - 1 n log(- x + a )
+--R (2n - 2)(- x + a ) %e
+--R Type: Expression Integer
+--E
+
+--S 94
+explog:=rule(%e^(n*log(x)) == x^n)
+--R
+--R n log(x) n
+--R (4) %e == x
+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 95 14:178 Axiom cannot simplify this expression
+dd:=explog cc
+--R
+--R 2 2 n 2 2 2 2 n - 1
+--R - (- x + a ) + (- x + a )(- x + a )
+--R (5) --------------------------------------------
+--R 2 2 n - 1 2 2 n
+--R (2n - 2)(- x + a ) (- x + a )
+--R Type: Expression Integer
+--E
+
@
\section{\cite{1}:14.179~~~~~$\displaystyle\int{\frac{dx}{x(a^2-x^2)^n}}$}
@@ -328,7 +1124,7 @@
<<*>>=
)clear all
---S 17 of 19
+--S 96 14:179 Axiom cannot integrate this expression
aa:=integrate(1/(x*(a^2-x^2)^n),x)
--R
--R
@@ -349,7 +1145,7 @@
<<*>>=
)clear all
---S 18 of 19
+--S 97 14:180 Axiom cannot integrate this expression
aa:=integrate(x^m/((a^2-x^2)^n),x)
--R
--R
@@ -370,7 +1166,7 @@
<<*>>=
)clear all
---S 19 of 19
+--S 98 14:181 Axiom cannot integrate this expression
aa:=integrate(1/(x^m*(a^2-x^2)^n),x)
--R
--R
Modified: trunk/axiom/src/input/schaum9.input.pamphlet
===================================================================
--- trunk/axiom/src/input/schaum9.input.pamphlet 2008-04-17 15:06:30 UTC (rev 856)
+++ trunk/axiom/src/input/schaum9.input.pamphlet 2008-04-18 06:42:58 UTC (rev 857)
@@ -16,7 +16,7 @@
)set message auto off
)clear all
---S 1 of 28
+--S 1
aa:=integrate(1/(sqrt(x^2+a^2)),x)
--R
--R
@@ -25,14 +25,33 @@
--R (1) - log(\|x + a - x)
--R Type: Union(Expression Integer,...)
--E
+
+--S 2
+bb:=log(x+sqrt(x^2+a^2))
+--R
+--R +-------+
+--R | 2 2
+--R (2) log(\|x + a + x)
+--R Type: Expression Integer
+--E
+
+--S 3 14:182 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R (3) - log(\|x + a + x) - log(\|x + a - x)
+--R Type: Expression Integer
+--E
@
+This is equal to $-\log(a^2)$ but Axiom cannot prove it.
\section{\cite{1}:14.183~~~~~$\displaystyle\int{\frac{x~dx}{\sqrt{x^2+a^2}}}$}
$$\int{\frac{x}{\sqrt{x^2+a^2}}}=\sqrt{x^2+a^2}$$
<<*>>=
)clear all
---S 2 of 28
+--S 4
aa:=integrate(x/(sqrt(x^2+a^2)),x)
--R
--R
@@ -45,6 +64,22 @@
--R \|x + a - x
--R Type: Union(Expression Integer,...)
--E
+
+--S 5
+bb:=sqrt(x^2+a^2)
+--R
+--R +-------+
+--R | 2 2
+--R (2) \|x + a
+--R Type: Expression Integer
+--E
+
+--S 6 14:183 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Ex...
[truncated message content] |
|
From: <gi...@ax...> - 2008-04-18 06:40:59
|
changelog | 3 +
src/input/schaum10.input.pamphlet | 679 ++++++++++++++++++++++++++++--
src/input/schaum8.input.pamphlet | 848 +++++++++++++++++++++++++++++++++++--
src/input/schaum9.input.pamphlet | 867 +++++++++++++++++++++++++++++++++++--
4 files changed, 2314 insertions(+), 83 deletions(-)
New commits:
commit b068d6518997e372b370d01deb3d630a51fa345f
Author: Tim Daly <root@localhost.localdomain>
Date: Tue Apr 1 12:21:50 2008 -0400
20080417 tpd src/input/schaum10.input show Schaums-Axiom equivalence
20080417 tpd src/input/schaum9.input show Schaums-Axiom equivalence
20080417 tpd src/input/schaum8.input show Schaums-Axiom equivalence
|
|
From: <da...@us...> - 2008-04-17 15:06:54
|
Revision: 856
http://axiom.svn.sourceforge.net/axiom/?rev=856&view=rev
Author: daly
Date: 2008-04-17 08:06:30 -0700 (Thu, 17 Apr 2008)
Log Message:
-----------
20080417 tpd src/input/schaum7.input fix typo in --S 30
20080417 tpd src/input/schaum5.input fix typo 14:88 -> 14:123
Modified Paths:
--------------
trunk/axiom/changelog
trunk/axiom/src/input/schaum5.input.pamphlet
trunk/axiom/src/input/schaum7.input.pamphlet
Modified: trunk/axiom/changelog
===================================================================
--- trunk/axiom/changelog 2008-04-17 05:46:08 UTC (rev 855)
+++ trunk/axiom/changelog 2008-04-17 15:06:30 UTC (rev 856)
@@ -1,3 +1,5 @@
+20080417 tpd src/input/schaum7.input fix typo in --S 30
+20080417 tpd src/input/schaum5.input fix typo 14:88 -> 14:123
20080416 tpd src/input/schaum7.input show Schaums-Axiom equivalence
20080416 tpd src/input/schaum6.input show Schaums-Axiom equivalence
20080416 tpd src/input/schaum5.input show Schaums-Axiom equivalence
Modified: trunk/axiom/src/input/schaum5.input.pamphlet
===================================================================
--- trunk/axiom/src/input/schaum5.input.pamphlet 2008-04-17 05:46:08 UTC (rev 855)
+++ trunk/axiom/src/input/schaum5.input.pamphlet 2008-04-17 15:06:30 UTC (rev 856)
@@ -1571,7 +1571,7 @@
--R Type: Expression Integer
--E
---S 42 14:88 Axiom cannot simplify these results
+--S 42 14:123 Axiom cannot simplify these results
cc4:=aa2-bb2
--R
--R (14)
Modified: trunk/axiom/src/input/schaum7.input.pamphlet
===================================================================
--- trunk/axiom/src/input/schaum7.input.pamphlet 2008-04-17 05:46:08 UTC (rev 855)
+++ trunk/axiom/src/input/schaum7.input.pamphlet 2008-04-17 15:06:30 UTC (rev 856)
@@ -356,71 +356,64 @@
--E
--S 30
-bb:=1/(2*a^2*x*2)-1/(2*a^4)*log(x^2/(x^2-a^2))
+bb:=1/(2*a^2*x^2)-1/(2*a^4)*log(x^2/(x^2-a^2))
--R
---R 2
---R x 2
---R - 2x log(-------) + a
---R 2 2
---R x - a
---R (2) ----------------------
---R 4
---R 4a x
+--R 2
+--R 2 x 2
+--R - x log(-------) + a
+--R 2 2
+--R x - a
+--R (2) ---------------------
+--R 4 2
+--R 2a x
--R Type: Expression Integer
--E
--S 31
-cc:=aa-bb
---R
---R 2
---R 2 2 2 2 2 x 2 2
---R 2x log(x - a ) - 4x log(x) + 2x log(-------) - a x + 2a
---R 2 2
---R x - a
---R (3) ---------------------------------------------------------
---R 4 2
---R 4a x
---R Type: Expression Integer
---E
-
---S 32
divlog:=rule(log(a/b) == log(a) - log(b))
--R
--R a
---R (4) log(-) == - log(b) + log(a)
+--R (3) log(-) == - log(b) + log(a)
--R b
--R Type: RewriteRule(Integer,Integer,Expression Integer)
--E
---S 33
-dd:=divlog cc
+--S 32
+t1:=divlog bb
--R
---R 2 2 2 2 2
---R 2x log(x ) - 4x log(x) - a x + 2a
---R (5) ----------------------------------
---R 4 2
---R 4a x
+--R 2 2 2 2 2 2
+--R - x log(x ) + x log(x - a ) + a
+--R (4) ---------------------------------
+--R 4 2
+--R 2a x
--R Type: Expression Integer
--E
---S 34
+--S 33
logpow:=rule(log(a^n) == n*log(a))
--R
--R n
---R (6) log(a ) == n log(a)
+--R (5) log(a ) == n log(a)
--R Type: RewriteRule(Integer,Integer,Expression Integer)
--E
---S 35 14:150 Schaums and Axiom DISAGREE BY A NON-CONSTANT
-ee:=logpow dd
+--S 34
+t2:=logpow t1
--R
---R - x + 2
---R (7) -------
---R 2 2
---R 4a x
+--R 2 2 2 2 2
+--R x log(x - a ) - 2x log(x) + a
+--R (6) -------------------------------
+--R 4 2
+--R 2a x
--R Type: Expression Integer
--E
+--S 35 14:150 Schaums and Axiom agree
+cc:=aa-t2
+
+ (7) 0
+ Type: Expression Integer
+--E
@
\section{\cite{1}:14.151~~~~~$\displaystyle\int{\frac{dx}{(x^2-a^2)^2}}$}
This was sent by the SourceForge.net collaborative development platform, the world's largest Open Source development site.
|
|
From: <gi...@ax...> - 2008-04-17 15:04:01
|
changelog | 2 +
src/input/schaum5.input.pamphlet | 2 +-
src/input/schaum7.input.pamphlet | 71 +++++++++++++++++---------------------
3 files changed, 35 insertions(+), 40 deletions(-)
New commits:
commit 60cb1ec69e7a6ce7f8b3a84944229f1dc9c90eb6
Author: Tim Daly <root@localhost.localdomain>
Date: Tue Apr 1 03:06:32 2008 -0400
20080417 tpd src/input/schaum7.input fix typo in --S 30
20080417 tpd src/input/schaum5.input fix typo 14:88 -> 14:123
|