\[ \int \frac {(c-c \sec (e+f x))^5}{(a+a \sec (e+f x))^3} \, dx \]
Optimal antiderivative \[ \frac {c^{5} x}{a^{3}}+\frac {8 c^{5} \arctanh \left (\sin \left (f x +e \right )\right )}{a^{3} f}+\frac {32 c^{5} \cot \left (f x +e \right )}{a^{3} f}+\frac {128 c^{5} \left (\cot ^{3}\left (f x +e \right )\right )}{3 a^{3} f}+\frac {128 c^{5} \left (\cot ^{5}\left (f x +e \right )\right )}{5 a^{3} f}-\frac {16 c^{5} \csc \left (f x +e \right )}{a^{3} f}+\frac {64 c^{5} \left (\csc ^{3}\left (f x +e \right )\right )}{3 a^{3} f}-\frac {128 c^{5} \left (\csc ^{5}\left (f x +e \right )\right )}{5 a^{3} f}-\frac {c^{5} \tan \left (f x +e \right )}{a^{3} f} \]
command
integrate((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
\[ \frac {\frac {15 \, {\left (f x + e\right )} c^{5}}{a^{3}} + \frac {120 \, c^{5} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right )}{a^{3}} - \frac {120 \, c^{5} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right )}{a^{3}} + \frac {30 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )} a^{3}} - \frac {8 \, {\left (3 \, a^{12} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 5 \, a^{12} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 30 \, a^{12} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{a^{15}}}{15 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________