# Evaluation of $\int (\arctan x)^2 dx$

Integrate

$$\int (\arctan x)^2 dx$$

(in terms of elementary functions , if possible)

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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.

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 $\arctan(x)^2 \,.$