<s>[INST] Evaluate the intgral $\int{\frac{dx}{x^2(1-x^2)}}$ (solution verification) I have to solve the following integral
$$\int{\frac{dx}{x^2(1-x^2)}}$$
What I've got:
\begin{split}
\int{\frac{dx}{x^2(1-x^2)}} &=\int{\frac{(1-x^2+x^2)dx}{x^2(1-x^2)}}\\
&=\int{\frac{dx}{x^2}}+\int{\frac{dx}{1-x^2}}\\
&=\int{\frac{dx}{x^2}}+\int{\frac{dx}{(1+(xi)^2)}}\\
&=-x^{-1}+\arctan{xi}+C
\end{split}
Is this correct?
Thanks in advance!
 [/INST] Don't forget $\dfrac1i$ before $\arctan$
$$\int{\frac{dx}{x^2(1-x^2)}}=-x^{-1}+\dfrac{1}{i}\arctan{xi}+C$$
Also better to write
$$\int\dfrac{1}{1-x^2}dx=\dfrac12\ln\dfrac{1+x}{1-x}+C$$
</s>