\[ \int \frac {\sec (e+f x) (c-c \sec (e+f x))^5}{(a+a \sec (e+f x))^2} \, dx \]
Optimal antiderivative \[ \frac {105 c^{5} \arctanh \left (\sin \left (f x +e \right )\right )}{2 a^{2} f}-\frac {84 c^{5} \tan \left (f x +e \right )}{a^{2} f}+\frac {63 c^{5} \sec \left (f x +e \right ) \tan \left (f x +e \right )}{2 a^{2} f}-\frac {6 c^{2} \left (c -c \sec \left (f x +e \right )\right )^{3} \tan \left (f x +e \right )}{f \left (a^{2}+a^{2} \sec \left (f x +e \right )\right )}+\frac {2 c \left (c -c \sec \left (f x +e \right )\right )^{4} \tan \left (f x +e \right )}{3 f \left (a +a \sec \left (f x +e \right )\right )^{2}}-\frac {7 c^{5} \left (\tan ^{3}\left (f x +e \right )\right )}{a^{2} f} \]
command
integrate(sec(f*x+e)*(c-c*sec(f*x+e))^5/(a+a*sec(f*x+e))^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\frac {315 \, c^{5} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right )}{a^{2}} - \frac {315 \, c^{5} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right )}{a^{2}} + \frac {2 \, {\left (165 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 280 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 123 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{3} a^{2}} - \frac {32 \, {\left (a^{4} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 12 \, a^{4} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{a^{6}}}{6 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________