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