Evaluate $\displaystyle \int_0^\infty \frac {\tan^{-1}(ax)}{x(1+x^2)}\, dx$ for $\displaystyle a>0, a \ne 1, $.
Need to do this using contour integral. I know how to do it using Leibnitz rule. I cannot figure out poles of the function and how to deal with it at $\displaystyle z = 0 $