`int(dx)/((sinx-2cosx)(2sinx+cosx))`

Explanation

`int(dx)/((sinx-2cosx)(2sinx+cosx))`

`=int frac{1}{2 sin^2x+sinxcosx-4sinxcosx-3cos^2x}dx`

`=int frac{sec^2x}{2tan^2x-3tanx-2}`

`x=t` and `sec^2xdx=dt`

`=int frac{dt}{2t^2-3t-2}`

`=1/2 int frac{1}{(t-3/4)^2-(5/4)^2}dt`

`=1/5 log abs(frac{2tanx-4}{2tanx+1})+C`