\[ \int \frac {\cot (e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx \]
Optimal antiderivative \[ \frac {\ln \left (\cos \left (f x +e \right )\right )}{\left (a -b \right )^{3} f}+\frac {\ln \left (\tan \left (f x +e \right )\right )}{a^{3} f}+\frac {b \left (3 a^{2}-3 a b +b^{2}\right ) \ln \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )}{2 a^{3} \left (a -b \right )^{3} f}-\frac {b}{4 a \left (a -b \right ) f \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{2}}-\frac {\left (2 a -b \right ) b}{2 a^{2} \left (a -b \right )^{2} f \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )} \]
command
integrate(cot(f*x+e)/(a+b*tan(f*x+e)^2)^3,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\frac {2 \, {\left (3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \log \left ({\left | -a \sin \left (f x + e\right )^{2} + b \sin \left (f x + e\right )^{2} + a \right |}\right )}{a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}} - \frac {9 \, a^{3} b \sin \left (f x + e\right )^{4} - 18 \, a^{2} b^{2} \sin \left (f x + e\right )^{4} + 12 \, a b^{3} \sin \left (f x + e\right )^{4} - 3 \, b^{4} \sin \left (f x + e\right )^{4} - 18 \, a^{3} b \sin \left (f x + e\right )^{2} + 24 \, a^{2} b^{2} \sin \left (f x + e\right )^{2} - 8 \, a b^{3} \sin \left (f x + e\right )^{2} + 9 \, a^{3} b - 6 \, a^{2} b^{2}}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} {\left (a \sin \left (f x + e\right )^{2} - b \sin \left (f x + e\right )^{2} - a\right )}^{2}} + \frac {2 \, \log \left (\sin \left (f x + e\right )^{2}\right )}{a^{3}}}{4 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________