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