\[ \int \frac {(c-c \sec (e+f x))^5}{(a+a \sec (e+f x))^2} \, dx \]
Optimal antiderivative \[ \frac {c^{5} x}{a^{2}}-\frac {47 c^{5} \arctanh \left (\sin \left (f x +e \right )\right )}{2 a^{2} f}+\frac {13 c^{5} \tan \left (f x +e \right )}{2 a^{2} f}+\frac {112 c^{5} \tan \left (f x +e \right )}{3 a^{2} f \left (1+\sec \left (f x +e \right )\right )}-\frac {32 c^{5} \tan \left (f x +e \right )}{3 f \left (a +a \sec \left (f x +e \right )\right )^{2}}+\frac {\left (c^{5}-c^{5} \sec \left (f x +e \right )\right ) \tan \left (f x +e \right )}{2 a^{2} f} \]
command
integrate((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
\[ \frac {\frac {6 \, {\left (f x + e\right )} c^{5}}{a^{2}} - \frac {141 \, c^{5} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right )}{a^{2}} + \frac {141 \, c^{5} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right )}{a^{2}} - \frac {6 \, {\left (15 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 13 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{2} a^{2}} + \frac {32 \, {\left (a^{4} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 6 \, a^{4} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{a^{6}}}{6 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________