\[ \int \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^4 \, dx \]
Optimal antiderivative \[ \frac {2 c^{4} \arctan \left (\frac {\sqrt {a}\, \tan \left (f x +e \right )}{\sqrt {a +a \sec \left (f x +e \right )}}\right ) \sqrt {a}}{f}-\frac {2 a \,c^{4} \tan \left (f x +e \right )}{f \sqrt {a +a \sec \left (f x +e \right )}}+\frac {2 a^{2} c^{4} \left (\tan ^{3}\left (f x +e \right )\right )}{3 f \left (a +a \sec \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {2 a^{3} c^{4} \left (\tan ^{5}\left (f x +e \right )\right )}{5 f \left (a +a \sec \left (f x +e \right )\right )^{\frac {5}{2}}}+\frac {2 a^{4} c^{4} \left (\tan ^{7}\left (f x +e \right )\right )}{7 f \left (a +a \sec \left (f x +e \right )\right )^{\frac {7}{2}}} \]
command
integrate((c-c*sec(f*x+e))^4*(a+a*sec(f*x+e))^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\frac {105 \, \sqrt {-a} a c^{4} \log \left (\frac {{\left | 2 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{2} - 4 \, \sqrt {2} {\left | a \right |} - 6 \, a \right |}}{{\left | 2 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{2} + 4 \, \sqrt {2} {\left | a \right |} - 6 \, a \right |}}\right ) \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}{{\left | a \right |}} - \frac {2 \, {\left (105 \, \sqrt {2} a^{4} c^{4} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - {\left (385 \, \sqrt {2} a^{4} c^{4} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + {\left (379 \, \sqrt {2} a^{4} c^{4} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 539 \, \sqrt {2} a^{4} c^{4} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )\right )} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}\right )} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}\right )} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{{\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a\right )}^{3} \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}}}{105 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________