\[ \int \sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))^4 \, dx \]
Optimal antiderivative \[ \frac {2 a d \left (2 c +d \right ) \left (2 c^{2}+2 c d +d^{2}\right ) \tan \left (f x +e \right )}{f \sqrt {a +a \sec \left (f x +e \right )}}-\frac {2 d^{2} \left (6 c^{2}+8 c d +3 d^{2}\right ) \left (a -a \sec \left (f x +e \right )\right ) \tan \left (f x +e \right )}{3 f \sqrt {a +a \sec \left (f x +e \right )}}+\frac {2 d^{3} \left (4 c +3 d \right ) \left (a -a \sec \left (f x +e \right )\right )^{2} \tan \left (f x +e \right )}{5 a f \sqrt {a +a \sec \left (f x +e \right )}}-\frac {2 d^{4} \left (a -a \sec \left (f x +e \right )\right )^{3} \tan \left (f x +e \right )}{7 a^{2} f \sqrt {a +a \sec \left (f x +e \right )}}+\frac {2 a^{\frac {3}{2}} c^{4} \arctanh \left (\frac {\sqrt {a -a \sec \left (f x +e \right )}}{\sqrt {a}}\right ) \tan \left (f x +e \right )}{f \sqrt {a -a \sec \left (f x +e \right )}\, \sqrt {a +a \sec \left (f x +e \right )}} \]
command
integrate((c+d*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 (420 \, \sqrt {2} a^{4} c^{3} d \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 630 \, \sqrt {2} a^{4} c^{2} d^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 420 \, \sqrt {2} a^{4} c d^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 105 \, \sqrt {2} a^{4} d^{4} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - {\left (1260 \, \sqrt {2} a^{4} c^{3} d \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 1470 \, \sqrt {2} a^{4} c^{2} d^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 700 \, \sqrt {2} a^{4} c d^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 105 \, \sqrt {2} a^{4} d^{4} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - {\left (1260 \, \sqrt {2} a^{4} c^{3} d \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 1050 \, \sqrt {2} a^{4} c^{2} d^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 476 \, \sqrt {2} a^{4} c d^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 147 \, \sqrt {2} a^{4} d^{4} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - {\left (420 \, \sqrt {2} a^{4} c^{3} d \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 210 \, \sqrt {2} a^{4} c^{2} d^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 196 \, \sqrt {2} a^{4} c d^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 27 \, \sqrt {2} a^{4} d^{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 )^{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
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________