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