Antiderivative of $\arctan(-x^2)$ As I said in the title I'm trying to find an antiderivative of $$f(x)=\arctan(-x^2)$$
I am aware that e.g. WolframAlpha can find one, but I have no clue how to do it by hand. Can anyone give me a hint?

If you integrate by parts, you get
$$
\int\arctan(-x^2)\,dx=x\arctan(-x^2)-\int x\frac{-2x}{1+x^4}\,dx
$$
Now find the partial fraction decomposition
$$
\frac{2x^2}{1+x^4}=
\frac{Ax+B}{x^2+x\sqrt{2}+1}+
\frac{Cx+D}{x^2-x\sqrt{2}+1}
$$
and the rest is standard.
