This integral came up in an exercise on the estimation of the specific heat of a 1-D solid and is probably a standard integral, possibly one that can be solved by contour integration:
\begin{equation} \int_0^{+\infty} \frac{x}{e^x-1} dx \end{equation}
I have some rudimentary knowledge of contour integrals, but I can't come up with a proper path, also because of the many singularities along the imaginary axis. Any suggestion?