\[\int_{0}^{\infty}e^{-\frac{1}{4}t^{2}}t^{-a-\frac{1}{2}}(x^{2}+t^{2})^{-1}U%
\left(a,t\right)\,\mathrm{d}t=\sqrt{\pi/2}\Gamma\left(\tfrac{1}{2}-a\right)x^{%
-a-\frac{3}{2}}e^{\frac{1}{4}x^{2}}U\left(-a,x\right),\]