Integrate
$\int (\arctan x)^2 dx $
(in terms of elementary functions , if possible)
Integrate
$\int (\arctan x)^2 dx $
(in terms of elementary functions , if possible)
Here is a result by Maple,
$ 2\,i \left( \arctan \left( x \right) \right) ^{2} \left( {\frac { \left( 1+ix \right) ^{2}}{1+{x}^{2}}}+1 \right) ^{-1}-2\,i \left( \arctan \left( x \right) \right) ^{2}$ $ +2\,\arctan \left( x \right) \ln\left( {\frac{ \left( 1+ix \right) ^{2}}{1+{x}^{2}}}+ 1 \right) - i{Li_{2}} \left( -{\frac { \left( 1+ix \right) ^{2}}{1+{x}^{2}} } \right) \,,$
where the $Li_{s}(z)$ is the polylogarithm function.
You can follow this technique:
Write $\arctan(x)$ in terms of $\ln$ as
$ \frac{1}{2}\,i \left( \ln \left( 1-ix \right) -\ln \left( 1+ix \right) \right)\,,$ then use the binomial theorem to expand $ \arctan(x)^2 \,.$