\[ \int \frac {\sec (e+f x) (c-c \sec (e+f x))^5}{(a+a \sec (e+f x))^3} \, dx \]
Optimal antiderivative \[ -\frac {63 c^{5} \arctanh \left (\sin \left (f x +e \right )\right )}{2 a^{3} f}+\frac {42 c^{5} \tan \left (f x +e \right )}{a^{3} f}-\frac {21 c^{5} \sec \left (f x +e \right ) \tan \left (f x +e \right )}{2 a^{3} f}-\frac {6 c^{2} \left (c -c \sec \left (f x +e \right )\right )^{3} \tan \left (f x +e \right )}{5 a f \left (a +a \sec \left (f x +e \right )\right )^{2}}+\frac {2 c \left (c -c \sec \left (f x +e \right )\right )^{4} \tan \left (f x +e \right )}{5 f \left (a +a \sec \left (f x +e \right )\right )^{3}}+\frac {42 c \left (c^{2}-c^{2} \sec \left (f x +e \right )\right )^{2} \tan \left (f x +e \right )}{5 f \left (a^{3}+a^{3} \sec \left (f x +e \right )\right )} \]
command
integrate(sec(f*x+e)*(c-c*sec(f*x+e))^5/(a+a*sec(f*x+e))^3,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\frac {315 \, c^{5} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right )}{a^{3}} - \frac {315 \, c^{5} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right )}{a^{3}} + \frac {10 \, {\left (17 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 15 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{2} a^{3}} - \frac {16 \, {\left (a^{12} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 5 \, a^{12} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 30 \, a^{12} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{a^{15}}}{10 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________