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?
2026-04-25 02:32:44.1777084364
Antiderivative of $\arctan(-x^2)$
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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.