TITLE: Integrate rational function $\frac{x^2}{1+x^4}$
QUESTION [7 upvotes]: Integrate $$\int\frac{x^2dx}{1+x^4}$$
I've factored the denominator to $(x^2-\sqrt{2}x+1)(x^2+\sqrt{2}x+1)$ and got stuck.

REPLY [6 votes]: Or, use $$\frac{x^2}{1+x^4}=\frac{1}{2} \frac{2x^2}{1+x^4}=\frac{1}{2}\left(\frac{x^2-1}{1+x^4}+\frac{x^2+1}{1+x^4}\right)$$ and this idea.

REPLY [3 votes]: The next step is to perform a partial fraction decomposition:
$$
   \frac{x^2}{(x^2-\sqrt{2}x+1)(x^2+\sqrt{2}x+1)} = \frac{1}{2 \sqrt{2}} \frac{x}{x^2-\sqrt{2}x+1} - \frac{1}{2 \sqrt{2}} \frac{x}{x^2+\sqrt{2}x+1}
$$
And the than use the table anti-derivative for $\int \frac{x}{x^2 + a x+b} \mathrm{d}x$.