\[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c+d \sec (e+f x)) \, dx \]
Optimal antiderivative \[ \frac {5 a^{3} \left (4 c +3 d \right ) \arctanh \left (\sin \left (f x +e \right )\right )}{8 f}+\frac {a^{3} \left (4 c +3 d \right ) \tan \left (f x +e \right )}{f}+\frac {3 a^{3} \left (4 c +3 d \right ) \sec \left (f x +e \right ) \tan \left (f x +e \right )}{8 f}+\frac {d \left (a +a \sec \left (f x +e \right )\right )^{3} \tan \left (f x +e \right )}{4 f}+\frac {a^{3} \left (4 c +3 d \right ) \left (\tan ^{3}\left (f x +e \right )\right )}{12 f} \]
command
integrate(sec(f*x+e)*(a+a*sec(f*x+e))^3*(c+d*sec(f*x+e)),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {15 \, {\left (4 \, a^{3} c + 3 \, a^{3} d\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right ) - 15 \, {\left (4 \, a^{3} c + 3 \, a^{3} d\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right ) - \frac {2 \, {\left (60 \, a^{3} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 45 \, a^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 220 \, a^{3} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 165 \, a^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 292 \, a^{3} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 219 \, a^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 132 \, a^{3} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 147 \, a^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{4}}}{24 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________