\[ \int \frac {(a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^2} \, dx \]
Optimal antiderivative \[ \frac {a^{3} x}{c^{2}}+\frac {a^{3} \arctanh \left (\sin \left (f x +e \right )\right )}{c^{2} f}-\frac {8 a^{3} \tan \left (f x +e \right )}{3 c^{2} f \left (1-\sec \left (f x +e \right )\right )^{2}}+\frac {4 a^{3} \tan \left (f x +e \right )}{3 c^{2} f \left (1-\sec \left (f x +e \right )\right )} \]
command
integrate((a+a*sec(f*x+e))^3/(c-c*sec(f*x+e))^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\frac {3 \, {\left (f x + e\right )} a^{3}}{c^{2}} + \frac {3 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right )}{c^{2}} - \frac {3 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right )}{c^{2}} - \frac {4 \, a^{3}}{c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3}}}{3 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________