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