\[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^3}{c-c \sec (e+f x)} \, dx \]
Optimal antiderivative \[ -\frac {15 a^{3} \arctanh \left (\sin \left (f x +e \right )\right )}{2 c f}-\frac {10 a^{3} \tan \left (f x +e \right )}{c f}-\frac {5 a^{3} \sec \left (f x +e \right ) \tan \left (f x +e \right )}{2 c f}-\frac {2 a \left (a +a \sec \left (f x +e \right )\right )^{2} \tan \left (f x +e \right )}{f \left (c -c \sec \left (f x +e \right )\right )} \]
command
integrate(sec(f*x+e)*(a+a*sec(f*x+e))^3/(c-c*sec(f*x+e)),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\frac {15 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right )}{c} - \frac {15 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right )}{c} - \frac {16 \, a^{3}}{c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )} - \frac {2 \, {\left (7 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 9 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{2} c}}{2 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________