\[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^2} \, dx \]
Optimal antiderivative \[ \frac {5 a^{3} \arctanh \left (\sin \left (f x +e \right )\right )}{c^{2} f}+\frac {5 a^{3} \tan \left (f x +e \right )}{c^{2} f}-\frac {2 a \left (a +a \sec \left (f x +e \right )\right )^{2} \tan \left (f x +e \right )}{3 f \left (c -c \sec \left (f x +e \right )\right )^{2}}+\frac {10 \left (a^{3}+a^{3} \sec \left (f x +e \right )\right ) \tan \left (f x +e \right )}{3 f \left (c^{2}-c^{2} \sec \left (f x +e \right )\right )} \]
command
integrate(sec(f*x+e)*(a+a*sec(f*x+e))^3/(c-c*sec(f*x+e))^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\frac {15 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right )}{c^{2}} - \frac {15 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right )}{c^{2}} - \frac {6 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )} c^{2}} - \frac {4 \, {\left (6 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a^{3}\right )}}{c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3}}}{3 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________