\[ \int \sec (e+f x) (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^3 \, dx \]
Optimal antiderivative \[ \frac {3 a^{2} c^{3} \arctanh \left (\sin \left (f x +e \right )\right )}{8 f}-\frac {3 a^{2} c^{3} \sec \left (f x +e \right ) \tan \left (f x +e \right )}{8 f}+\frac {a^{2} c^{3} \sec \left (f x +e \right ) \left (\tan ^{3}\left (f x +e \right )\right )}{4 f}-\frac {a^{2} c^{3} \left (\tan ^{5}\left (f x +e \right )\right )}{5 f} \]
command
integrate(sec(f*x+e)*(a+a*sec(f*x+e))^2*(c-c*sec(f*x+e))^3,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {15 \, a^{2} c^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right ) - 15 \, a^{2} c^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right ) - \frac {2 \, {\left (15 \, a^{2} c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} - 70 \, a^{2} c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 128 \, a^{2} c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 70 \, a^{2} c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 15 \, a^{2} c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{5}}}{40 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________