\[\int_{0}^{\infty}J_{0}\left(at\right)\left(J_{0}\left(bt\right)-J_{0}\left(ct%
\right)\right)\frac{\,\mathrm{d}t}{t}=\begin{cases}0,&0\leq b<a,0<c\leq a,\\
\ln\left(c/a\right),&0\leq b<a\leq c.\end{cases}\]