I found in an article "Imperfect Bose Gas with Hard-Sphere Interaction", Phys. Rev. 105, 776–784 (1957) the following integral, but I don't know how to solve it. Any hints?
$$\int_0^\infty {\int_0^\infty {\mathrm dp\mathrm dq\frac{\sinh(upq)}{q^2 - p^2}pq} } e^{-vq^2 - wp^2} = \frac{\pi}{4}\frac{u(w - v)}{\left[(w + v)^2-u^2 \right]\left(4wv-u^2\right)^{1/2}}$$
for $u,v,w > 0$.