Non-Elementary Integration · Challenge problem

Using Glasser's master theorem $\displaystyle\int_{-\infty}^{\infty}f\!\left(x-\dfrac1x\right)dx=\int_{-\infty}^{\infty}f(x)\,dx$, evaluate $\displaystyle\int_{-\infty}^{\infty}\dfrac{x^2}{x^4+x^2+1}\,dx$. Give the answer with a rationalized denominator.