\[ \int (a+b \cot (x))^n \csc ^2(x) \, dx \]
Optimal antiderivative \[ -\frac {\left (a +b \cot \left (x \right )\right )^{1+n}}{b \left (1+n \right )} \]
command
integrate((a+b*cot(x))^n*csc(x)^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\left (-\frac {b \tan \left (\frac {1}{2} \, x\right )^{2} - 2 \, a \tan \left (\frac {1}{2} \, x\right ) - b}{2 \, \tan \left (\frac {1}{2} \, x\right )}\right )^{n + 1}}{b {\left (n + 1\right )}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int {\left (b \cot \left (x\right ) + a\right )}^{n} \csc \left (x\right )^{2}\,{d x} \]________________________________________________________________________________________