∫csc x cot^2 xdx

∫csc x cot^2 x dx

= ∫csc x (csc^2 x -1) dx
= ∫csc^3 x dx - ∫csc x dx

You probably have handy the form for ∫csc x dx, and you can use the power reduction formulas for ∫csc^3 x dx, or note that

d/dx (cscx cotx) = -csc^3 x - csc x cot^2 x