\[ \int \cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^2 \, dx \]
Optimal antiderivative \[ -\frac {a^{2} \left (\cot ^{2}\left (f x +e \right )\right )}{2 f}-\frac {\left (a -b \right )^{2} \ln \left (\cos \left (f x +e \right )\right )}{f}-\frac {a \left (a -2 b \right ) \ln \left (\tan \left (f x +e \right )\right )}{f} \]
command
integrate(cot(f*x+e)^3*(a+b*tan(f*x+e)^2)^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {a^{2} {\left (\frac {\cos \left (f x + e\right ) + 1}{\cos \left (f x + e\right ) - 1} + \frac {\cos \left (f x + e\right ) - 1}{\cos \left (f x + e\right ) + 1}\right )} - 4 \, b^{2} \log \left ({\left | -\frac {\cos \left (f x + e\right ) + 1}{\cos \left (f x + e\right ) - 1} - \frac {\cos \left (f x + e\right ) - 1}{\cos \left (f x + e\right ) + 1} - 2 \right |}\right ) + 4 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \log \left ({\left | -\frac {\cos \left (f x + e\right ) + 1}{\cos \left (f x + e\right ) - 1} - \frac {\cos \left (f x + e\right ) - 1}{\cos \left (f x + e\right ) + 1} + 2 \right |}\right )}{8 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________