I came across this integral when helping some friends with a statistical mechanics assignment, Mathematica reports it as $\frac{\pi^2}{3}$. So far I have noticed that the integrand is an even function so the integral is equivalent to, $2\int_{0}^{\infty} \frac{x^2e^x}{(e^x+1)^2}$. My first idea is to run parts with $dv=\frac{e^x}{(e^x+1)^2}dx$ and $u=x^2$ with this the surface term is zero and we get the remaining integral $4\int_{0}^{\infty} \frac{x}{e^x+1}dx$. This surely looks simpler and if I do $x=ln(u)$ then this turns into $4 \int_{1}^{\infty} \frac{ln(u)}{u(u+1)}du$. Unfortunately I've tried for a couple days on the last two integrals and I cannot get either to budge. I was hoping there is a method that avoids contour integration just because I've never really used that technique before but any answers would be appreciated.
Thanks to @LoganMaingi I've realized this is a duplicate. To see that this integral is the same as the one linked make the following substitution that @LoganMaingi suggested in the comments.
