\[ \int \frac {\cot ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx \]
Optimal antiderivative \[ \frac {\left (a +2 b \right ) \left (\cot ^{2}\left (f x +e \right )\right )}{2 a^{3} f}-\frac {\cot ^{4}\left (f x +e \right )}{4 a^{2} f}+\frac {\ln \left (\cos \left (f x +e \right )\right )}{\left (a -b \right )^{2} f}+\frac {\left (a^{2}+2 a b +3 b^{2}\right ) \ln \left (\tan \left (f x +e \right )\right )}{a^{4} f}+\frac {\left (4 a -3 b \right ) b^{3} \ln \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )}{2 a^{4} \left (a -b \right )^{2} f}-\frac {b^{3}}{2 a^{3} \left (a -b \right ) f \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )} \]
command
integrate(cot(f*x+e)^5/(a+b*tan(f*x+e)^2)^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\frac {32 \, {\left (4 \, a b^{3} - 3 \, b^{4}\right )} \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^{6} - 2 \, a^{5} b + a^{4} b^{2}} - \frac {64 \, \log \left ({\left | -\frac {\cos \left (f x + e\right ) - 1}{\cos \left (f x + e\right ) + 1} + 1 \right |}\right )}{a^{2} - 2 \, a b + b^{2}} - \frac {32 \, {\left (4 \, a^{2} b^{3} - 3 \, a b^{4} + \frac {8 \, a^{2} b^{3} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - \frac {18 \, a b^{4} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {8 \, b^{5} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {4 \, a^{2} b^{3} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {3 \, a b^{4} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )}}{{\left (a^{6} - 2 \, a^{5} b + a^{4} b^{2}\right )} {\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 )}} + \frac {32 \, {\left (a^{2} + 2 \, a b + 3 \, b^{2}\right )} \log \left (\frac {{\left | -\cos \left (f x + e\right ) + 1 \right |}}{{\left | \cos \left (f x + e\right ) + 1 \right |}}\right )}{a^{4}} - \frac {\frac {12 \, a^{2} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {16 \, a b {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {a^{2} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}}{a^{4}} - \frac {{\left (a^{2} + \frac {12 \, a^{2} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {16 \, a b {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {48 \, a^{2} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {96 \, a b {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {144 \, b^{2} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{2}}{a^{4} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}}{64 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________