\[ \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^4 \, dx \]
Optimal antiderivative \[ a^{3} c^{4} x -\frac {5 a^{3} c^{4} \arctanh \left (\sin \left (f x +e \right )\right )}{16 f}-\frac {a^{3} \left (16 c^{4}-5 c^{4} \sec \left (f x +e \right )\right ) \tan \left (f x +e \right )}{16 f}+\frac {a^{3} \left (8 c^{4}-5 c^{4} \sec \left (f x +e \right )\right ) \left (\tan ^{3}\left (f x +e \right )\right )}{24 f}-\frac {a^{3} \left (6 c^{4}-5 c^{4} \sec \left (f x +e \right )\right ) \left (\tan ^{5}\left (f x +e \right )\right )}{30 f} \]
command
integrate((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
\[ \frac {240 \, {\left (f x + e\right )} a^{3} c^{4} - 75 \, a^{3} c^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right ) + 75 \, a^{3} c^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right ) + \frac {2 \, {\left (315 \, a^{3} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{11} - 1945 \, a^{3} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} + 5118 \, a^{3} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 3138 \, a^{3} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 1095 \, a^{3} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 165 \, a^{3} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{6}}}{240 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________