\[ \int \frac {\sec (e+f x) (c-c \sec (e+f x))^4}{(a+a \sec (e+f x))^2} \, dx \]
Optimal antiderivative \[ \frac {35 c^{4} \arctanh \left (\sin \left (f x +e \right )\right )}{2 a^{2} f}-\frac {70 c^{4} \tan \left (f x +e \right )}{3 a^{2} f}+\frac {35 c^{4} \sec \left (f x +e \right ) \tan \left (f x +e \right )}{6 a^{2} f}+\frac {2 c \left (c -c \sec \left (f x +e \right )\right )^{3} \tan \left (f x +e \right )}{3 f \left (a +a \sec \left (f x +e \right )\right )^{2}}-\frac {14 \left (c^{2}-c^{2} \sec \left (f x +e \right )\right )^{2} \tan \left (f x +e \right )}{3 f \left (a^{2}+a^{2} \sec \left (f x +e \right )\right )} \]
command
integrate(sec(f*x+e)*(c-c*sec(f*x+e))^4/(a+a*sec(f*x+e))^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\frac {105 \, c^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right )}{a^{2}} - \frac {105 \, c^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right )}{a^{2}} + \frac {6 \, {\left (13 \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 11 \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{2} a^{2}} - \frac {16 \, {\left (a^{4} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 9 \, a^{4} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{a^{6}}}{6 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________