Playing around with some elementary integrals and Mellin transforms, I arrive at the following integral expression
$$(1):~~ I(A,B) = \int_{c-i\infty}^{c+i\infty} dz~(2z+1) e^{-Az} K_z(B)~,$$
where $K_z(B)$ is the modified Bessel function of the second kind. I am interested in real and positive values of $A$ and $B$, and $c$ is some constant (can be chosen to be zero in practice).
I was also able to get the closed expression for $I(A,B)$ of the form
$$(2):~~ I(A,B) = i \pi \big( 1 + 2 B\sinh(A)\big) e^{-\cosh(A)B}~, $$
which I have checked numerically, and it agrees very well with the integral expression (1) above for positive values of $A$ and $B$.
Question: Is it possible to derive (2) by closing the integral contour in (1) and explicitly summing the residues?
Additionally: If I do a change of $A$ variable, I can relate the integral to the expression 10.32.10, i.e. to the inverse Mellin transform of it. The question thus boils down to this: is there some appropriate literature where I could find how to do the inverse Mellin transform of the $K_z(B)$ in $z$ variable?
Edit: The expression (2) can be derived from (1) directly using the integral representation of $K_z(B)$ given in 10.32.9.