calculus - Finding $int_{-infty}^infty frac{x^2}{x^4+1};dx$

$$\int_{-\infty}^\infty \frac{x^2}{x^4+1}\;dx$$

I'm trying to understand trigonometric substitution better, because I never could get a good handle on it. All I know is that this integral is supposed to reduce to the integral of some power of cosine. I tried $x^2=\tan\theta$, but I ended up with $\sin\theta\cos^3\theta$ as my integrand. Can someone explain how to compute this?