\[ \int \frac {(a+a \sec (e+f x))^3}{c-c \sec (e+f x)} \, dx \]
Optimal antiderivative \[ \frac {a^{3} x}{c}-\frac {4 a^{3} \arctanh \left (\sin \left (f x +e \right )\right )}{c f}+\frac {8 a^{3} \cot \left (f x +e \right )}{c f}+\frac {8 a^{3} \csc \left (f x +e \right )}{c f}-\frac {a^{3} \tan \left (f x +e \right )}{c f} \]
command
integrate((a+a*sec(f*x+e))^3/(c-c*sec(f*x+e)),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\frac {{\left (f x + e\right )} a^{3}}{c} - \frac {4 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right )}{c} + \frac {4 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right )}{c} + \frac {2 \, {\left (5 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 4 \, a^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} c}}{f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________