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