\[ \int \frac {\tan ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx \]
Optimal antiderivative \[ -\frac {\ln \left (\cos \left (f x +e \right )\right )}{\left (a -b \right )^{2} f}+\frac {a \left (a -2 b \right ) \ln \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )}{2 \left (a -b \right )^{2} b^{2} f}+\frac {a^{2}}{2 \left (a -b \right ) b^{2} f \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )} \]
command
integrate(tan(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
\[ \frac {\frac {{\left (a^{3} - 2 \, a^{2} b\right )} \log \left ({\left | -a {\left (\frac {\cos \left (f x + e\right ) + 1}{\cos \left (f x + e\right ) - 1} + \frac {\cos \left (f x + e\right ) - 1}{\cos \left (f x + e\right ) + 1}\right )} - 2 \, a + 4 \, b \right |}\right )}{a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}} + \frac {\log \left ({\left | -\frac {\cos \left (f x + e\right ) + 1}{\cos \left (f x + e\right ) - 1} - \frac {\cos \left (f x + e\right ) - 1}{\cos \left (f x + e\right ) + 1} + 2 \right |}\right )}{a^{2} - 2 \, a b + b^{2}} - \frac {a^{3} {\left (\frac {\cos \left (f x + e\right ) + 1}{\cos \left (f x + e\right ) - 1} + \frac {\cos \left (f x + e\right ) - 1}{\cos \left (f x + e\right ) + 1}\right )} - 2 \, a^{2} b {\left (\frac {\cos \left (f x + e\right ) + 1}{\cos \left (f x + e\right ) - 1} + \frac {\cos \left (f x + e\right ) - 1}{\cos \left (f x + e\right ) + 1}\right )} + 2 \, a^{3} - 12 \, a^{2} b + 12 \, a b^{2}}{{\left (a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right )} {\left (a {\left (\frac {\cos \left (f x + e\right ) + 1}{\cos \left (f x + e\right ) - 1} + \frac {\cos \left (f x + e\right ) - 1}{\cos \left (f x + e\right ) + 1}\right )} + 2 \, a - 4 \, b\right )}} - \frac {\log \left ({\left | -\frac {\cos \left (f x + e\right ) + 1}{\cos \left (f x + e\right ) - 1} - \frac {\cos \left (f x + e\right ) - 1}{\cos \left (f x + e\right ) + 1} - 2 \right |}\right )}{b^{2}}}{2 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________