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