I was checking useful integrals in this book. I have found one (6.298) that is what I need, but I don't understand how every step towards the final result works.
$$\int_0^{+\infty}\,\left[2\cosh(ab)-e^{-ab}\Phi\left(\frac{b-2ax^2}{2x}\right)-e^{ab}\Phi\left(\frac{b+2ax^2}{2x}\right)\right]\,x\,e^{-(\mu-a^2)x^2}\,\,dx=\frac{1}{\mu-a^2}e^{-b\sqrt{\mu}}$$
where $\Phi(x)=erf(x)$, $a,b>0$ and $Re\,\mu>0$.
Can anybody help me with the intermediate steps to get the final result? Are there other conditions missing? Like $\mu-a^2>0$?