\[ \int \frac {\sec (e+f x) (c-c \sec (e+f x))^6}{(a+a \sec (e+f x))^3} \, dx \]
Optimal antiderivative \[ -\frac {231 c^{6} \arctanh \left (\sin \left (f x +e \right )\right )}{2 a^{3} f}+\frac {924 c^{6} \tan \left (f x +e \right )}{5 a^{3} f}-\frac {693 c^{6} \sec \left (f x +e \right ) \tan \left (f x +e \right )}{10 a^{3} f}-\frac {22 c^{2} \left (c -c \sec \left (f x +e \right )\right )^{4} \tan \left (f x +e \right )}{15 a f \left (a +a \sec \left (f x +e \right )\right )^{2}}+\frac {2 c \left (c -c \sec \left (f x +e \right )\right )^{5} \tan \left (f x +e \right )}{5 f \left (a +a \sec \left (f x +e \right )\right )^{3}}+\frac {66 \left (c^{2}-c^{2} \sec \left (f x +e \right )\right )^{3} \tan \left (f x +e \right )}{5 f \left (a^{3}+a^{3} \sec \left (f x +e \right )\right )}+\frac {77 c^{6} \left (\tan ^{3}\left (f x +e \right )\right )}{5 a^{3} f} \]
command
integrate(sec(f*x+e)*(c-c*sec(f*x+e))^6/(a+a*sec(f*x+e))^3,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\frac {3465 \, c^{6} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right )}{a^{3}} - \frac {3465 \, c^{6} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right )}{a^{3}} + \frac {10 \, {\left (267 \, c^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 472 \, c^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 213 \, c^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{3} a^{3}} - \frac {32 \, {\left (3 \, a^{12} c^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 20 \, a^{12} c^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 150 \, a^{12} c^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{a^{15}}}{30 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________