Improper integration problem $$ \int \limits^{\infty }_{0}\frac{\tan^{-1}\left( x\right) + \tan^{-1}\left( \frac{\alpha  x +\beta }{\beta  x -\alpha } \right) }{x^{2}+1} dx$$.  For $ \alpha,  \beta >0$... My question is how we can evaluate this improper integration above?. Actually I tried partial fraction and many other ways but I couldn't complete.  How I can find the value of this integration?.  Thanks 

Sketch:
$$\arctan(a) + \arctan(b) = \arctan \bigg(\frac{a+b}{1-ab}\bigg) $$
Plug in our values to get $\arctan \big(-\frac{ \beta}{\alpha}\big   )$.
Hope this helps.
