\[ \int \sec (e+f x) (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^5 \, dx \]
Optimal antiderivative \[ \frac {9 a^{2} c^{5} \arctanh \left (\sin \left (f x +e \right )\right )}{16 f}-\frac {3 a^{2} c^{5} \sec \left (f x +e \right ) \tan \left (f x +e \right )}{16 f}-\frac {3 a^{2} c^{5} \left (\sec ^{3}\left (f x +e \right )\right ) \tan \left (f x +e \right )}{8 f}+\frac {a^{2} c^{5} \sec \left (f x +e \right ) \left (\tan ^{3}\left (f x +e \right )\right )}{4 f}+\frac {a^{2} c^{5} \left (\sec ^{3}\left (f x +e \right )\right ) \left (\tan ^{3}\left (f x +e \right )\right )}{2 f}-\frac {4 a^{2} c^{5} \left (\tan ^{5}\left (f x +e \right )\right )}{5 f}-\frac {a^{2} c^{5} \left (\tan ^{7}\left (f x +e \right )\right )}{7 f} \]
command
integrate(sec(f*x+e)*(a+a*sec(f*x+e))^2*(c-c*sec(f*x+e))^5,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {315 \, a^{2} c^{5} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right ) - 315 \, a^{2} c^{5} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right ) - \frac {2 \, {\left (315 \, a^{2} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{13} - 2100 \, a^{2} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{11} - 8393 \, a^{2} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} + 9216 \, a^{2} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 5943 \, a^{2} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 2100 \, a^{2} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 315 \, a^{2} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{7}}}{560 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________