Evaluate the following integrals : `int(atanx+bcotx)^2dx`

Explanation

`(atanx+bcotx)^2 = a^2tan^2x+b^2cot^2x+2ab tanxcotx`

`=a^2(sec^2x-1)+b^2(cosec^2x-1) +2abtanx*1/tanx`

`=a^2sec^2x -a^2+b^2cosec^2x - b^2 + 2ab`

`=a^2sec^2x+b^2cosec^2x -(a^2-b^2-2ab)`

`=a^2sec^2x+b^2cosec^2x -(a-b)^2`

`=> int (atanx+bcotx)^2 dx = int (a^2sec^2x+b^2cosec^2x -(a-b)^2) dx`

`=> int (atanx+bcotx)^2 dx = a^2tanx-b^2cotx-(a-b)^2x+c`, which is the required solution.