\[ \int \frac {(a+a \cos (e+f x))^2 \sec ^2(e+f x)}{-c+c \cos (e+f x)} \, dx \]
Optimal antiderivative \[ -\frac {3 a^{2} \arctanh \left (\sin \left (f x +e \right )\right )}{c f}+\frac {4 a^{2} \sin \left (f x +e \right )}{c f \left (1-\cos \left (f x +e \right )\right )}-\frac {a^{2} \tan \left (f x +e \right )}{c f} \]
command
integrate((a+a*cos(f*x+e))^2*sec(f*x+e)^2/(-c+c*cos(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
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________