Evaluation of $\int_{-\infty}^{\infty} \frac{x^2}{(e^{2x}+1)(x^2+1)^2} dx$

Question

I am stuck evaluating the integral $$\int_{-\infty}^{\infty} \frac{x^2}{(e^{2x}+1)(x^2+1)^2} dx,$$ but I do know that the answer should be $\frac{\pi}{4}$. I have tried using residue calculus, but I did not get very far. Can anyone help me approach this problem? Thanks in advance.

Answer 1

To elaborate on Quanto's comment, once you see that identity, your integral is simply $\int_0^\infty dx/(x+1/x)^2=\int_0^\infty \frac{dx}{(x-1/x)^2+4}=\int_0^\infty \frac{dx}{x^2+4}=\pi/4$ by Glasser's Master theorem.