Fix $a>0$ and $b>0$. Then Gradshteyn and Ryzhik give the following integrals 6.796.4 and 6.796.5: $$ \int_0^\infty \cos(bx) \cosh(\pi x) \left[ K_{ix}(a) \right]^2 \,dx = - \frac{\pi^2}{4} Y_{0}\left( 2 a \sinh\left( \tfrac{b}{2} \right) \right) \\ \int_0^\infty \sin(bx) \sinh(\pi x) \left[ K_{ix}(a) \right]^2\, dx = \frac{\pi^2}{4} J_{0}\left( 2 a \sinh\left( \tfrac{b}{2} \right) \right) \\ $$
I am in the very unfortunate situation where I need a very similar integral which is not listed: $$ \int_0^\infty \cos(bx) \sinh(\pi x) \left[ K_{ix}(a) \right]^2 \,dx = ?? $$
I have tried playing around with the integrals given in G+R (mainly hitting the integrals with $\frac{d}{db}$) but I am not able to do anything useful.
Furthermore, G+R list Erdelyi's Integral Transforms Volume II as the source for these two integrals. But I've gone and checked this reference, and there is no derivation provided - these two integrals are simply listed there.
Is there any way to evaluate $\int_0^\infty \cos(bx) \sinh(\pi x) \left[ K_{ix}(a) \right]^2 \,dx?$