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