\[ \int \frac {\sqrt {a+a \sec (e+f x)}}{(c-c \sec (e+f x))^4} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (\cot ^{3}\left (f x +e \right )\right ) \left (a +a \sec \left (f x +e \right )\right )^{\frac {3}{2}}}{3 a \,c^{4} f}+\frac {2 \left (\cot ^{5}\left (f x +e \right )\right ) \left (a +a \sec \left (f x +e \right )\right )^{\frac {5}{2}}}{5 a^{2} c^{4} f}-\frac {2 \left (\cot ^{7}\left (f x +e \right )\right ) \left (a +a \sec \left (f x +e \right )\right )^{\frac {7}{2}}}{7 a^{3} c^{4} f}+\frac {2 \arctan \left (\frac {\sqrt {a}\, \tan \left (f x +e \right )}{\sqrt {a +a \sec \left (f x +e \right )}}\right ) \sqrt {a}}{c^{4} f}+\frac {2 \cot \left (f x +e \right ) \sqrt {a +a \sec \left (f x +e \right )}}{c^{4} f} \]
command
integrate((a+a*sec(f*x+e))^(1/2)/(c-c*sec(f*x+e))^4,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\sqrt {2} {\left (\frac {210 \, \sqrt {2} \sqrt {-a} a \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 )}{c^{4} {\left | a \right |}} + \frac {1575 \, {\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 )}^{12} \sqrt {-a} a - 7140 \, {\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 )}^{10} \sqrt {-a} a^{2} + 16415 \, {\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 )}^{8} \sqrt {-a} a^{3} - 19880 \, {\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 )}^{6} \sqrt {-a} a^{4} + 14637 \, {\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 )}^{4} \sqrt {-a} a^{5} - 5684 \, {\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} \sqrt {-a} a^{6} + 1037 \, \sqrt {-a} a^{7}}{{\left ({\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} - a\right )}^{7} c^{4}}\right )} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}{420 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________