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