Non-Elementary Integration · Medium problem

Glasser's master theorem says $\displaystyle\int_{-\infty}^{\infty}f\!\left(x-\dfrac1x\right)dx=\int_{-\infty}^{\infty}f(u)\,du$. Apply it with $f(u)=\dfrac{1}{u^2+4}$ to evaluate $\displaystyle\int_{-\infty}^{\infty}\dfrac{x^2}{(x^2+1)^2}\,dx$.