\[ \int \sec (e+f x) (a+a \sec (e+f x))^2 (c+d \sec (e+f x)) \, dx \]
Optimal antiderivative \[ \frac {a^{2} \left (3 c +2 d \right ) \arctanh \left (\sin \left (f x +e \right )\right )}{2 f}+\frac {2 a^{2} \left (3 c +2 d \right ) \tan \left (f x +e \right )}{3 f}+\frac {a^{2} \left (3 c +2 d \right ) \sec \left (f x +e \right ) \tan \left (f x +e \right )}{6 f}+\frac {d \left (a +a \sec \left (f x +e \right )\right )^{2} \tan \left (f x +e \right )}{3 f} \]
command
integrate(sec(f*x+e)*(a+a*sec(f*x+e))^2*(c+d*sec(f*x+e)),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {3 \, {\left (3 \, a^{2} c + 2 \, a^{2} d\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right ) - 3 \, {\left (3 \, a^{2} c + 2 \, a^{2} d\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right ) - \frac {2 \, {\left (9 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 6 \, a^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 24 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 16 \, a^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 15 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 18 \, a^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{3}}}{6 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________