How to do this integral (appearing in the context of the quantum treatment of phonons in a three-dimensional cubic lattice):
$$\prod_{j=1}^3\int\limits_{0}^{\pi}\mathrm{d}u_j\frac{\sqrt{\sum\limits_{i=1}^3\sin^2(u_i)}}{\exp\left(x\sqrt{\sum\limits_{i=1}^3\sin^2(u_i)}\right)-1}\,,$$
where $x$ is a positive parameter. And a simpler one:
$$\prod_{j=1}^3\int\limits_{0}^{\pi}\mathrm{d}u_j\sqrt{\sum\limits_{i=1}^3\sin^2(u_i)}\,.$$