Answer
$\ln \left| {\arctan x} \right| + C$
Work Step by Step
$$\eqalign{
& \int {\frac{1}{{\left( {{x^2} + 1} \right)\arctan x}}} dx \cr
& {\text{Let }}u = \arctan x,{\text{ then }}du = \frac{1}{{{x^2} + 1}}dx,{\text{ then substituting}} \cr
& \int {\frac{1}{{\left( {{x^2} + 1} \right)\arctan x}}} dx = \int {\frac{1}{u}} du \cr
& {\text{Integrate }} \cr
& = \ln \left| u \right| + C \cr
& {\text{Write in terms of }}x,{\text{ substitute }}u = \arctan x \cr
& = \ln \left| {\arctan x} \right| + C \cr} $$
