Rational integral $$\int\dfrac{x^2}{1+x^4}dx$$
I tried many standard approaches, but I didn't get too far! Here's the most promising of them:
$$\int\dfrac{dx}{\frac{1}{x^2}+x^2}$$
knowing that $\left(1/x+x\right)^2=\frac{1}{x^2}+x^2+2$ we can change variables $1/x+x=t$. Unluckily this doesn't work either. 

Hint: Express
$$\frac{x^2}{1+x^4}=\frac{ax+b}{x^2-\sqrt{2}x+1}+\frac{cx+d}{x^2+\sqrt{2}x+1}$$
and find $a,b,c,d$.
