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:
Like $\int \frac{x^2}{x^4+x^2+1}\ dx$,
$$\dfrac{2x^2}{x^4+1}=\dfrac{1-1/x^2}{x^2+1/x^2}+\dfrac{1+1/x^2}{x^2+1/x^2}$$
