\[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^4 \, dx \]
Optimal antiderivative \[ \frac {5 a^{3} c^{4} \arctanh \left (\sin \left (f x +e \right )\right )}{16 f}-\frac {5 a^{3} c^{4} \sec \left (f x +e \right ) \tan \left (f x +e \right )}{16 f}+\frac {5 a^{3} c^{4} \sec \left (f x +e \right ) \left (\tan ^{3}\left (f x +e \right )\right )}{24 f}-\frac {a^{3} c^{4} \sec \left (f x +e \right ) \left (\tan ^{5}\left (f x +e \right )\right )}{6 f}+\frac {a^{3} c^{4} \left (\tan ^{7}\left (f x +e \right )\right )}{7 f} \]
command
integrate(sec(f*x+e)*(a+a*sec(f*x+e))^3*(c-c*sec(f*x+e))^4,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {105 \, a^{3} c^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right ) - 105 \, a^{3} c^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right ) - \frac {2 \, {\left (105 \, a^{3} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{13} - 700 \, a^{3} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{11} + 1981 \, a^{3} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} + 3072 \, a^{3} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 1981 \, a^{3} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 700 \, a^{3} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 105 \, a^{3} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{7}}}{336 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________