I'm having problems with this integral
$$\int_{a \sqrt \alpha}^\infty \frac{x^2}{(\sqrt{x^2 - a^2 \alpha})(e^x - 1)} \, dx \qquad \text{with } a, \alpha > 0$$
I tried to solve it with Maple, using the Cauchy Principal Value function, but it doesn't deliver. Any suggestions? Thought also about considering the Fourier series around the $a\sqrt{\alpha}$ value, but I get the worst integrals ever!
The presence of $\sqrt{x^2-u^2}~$ begs for a hyperbolic substitution of the form $x=u\cosh t$. This will inevitably yield a $($non-trivial$)$ integrand containing $e^{\large u\cosh t}$ in its expression. The only functions known to possess such a form are the Bessel and Struve functions. Unfortunately, even they seem unable to help us. The only other solution would be expanding the integrand into a Cauchy product of two binomial series, and switching the order of summation and integration, then rewriting it in terms of some $($rather hideous$)$ hypergeometric functions. But it is very doubtful whether such an approach will yield any $($meaningful$)$ results.