\[ \int \frac {1}{(c \sec (a+b x))^{7/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \sin \left (b x +a \right )}{7 b c \left (c \sec \left (b x +a \right )\right )^{\frac {5}{2}}}+\frac {10 \sin \left (b x +a \right )}{21 b \,c^{3} \sqrt {c \sec \left (b x +a \right )}}+\frac {10 \sqrt {\frac {\cos \left (b x +a \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {a}{2}+\frac {b x}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (b x +a \right )\right ) \sqrt {c \sec \left (b x +a \right )}}{21 \cos \left (\frac {a}{2}+\frac {b x}{2}\right ) b \,c^{4}} \]
command
integrate(1/(c*sec(b*x+a))^(7/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (3 \, \cos \left (b x + a\right )^{3} + 5 \, \cos \left (b x + a\right )\right )} \sqrt {\frac {c}{\cos \left (b x + a\right )}} \sin \left (b x + a\right ) - 5 i \, \sqrt {2} \sqrt {c} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + 5 i \, \sqrt {2} \sqrt {c} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )}{21 \, b c^{4}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {c \sec \left (b x + a\right )}}{c^{4} \sec \left (b x + a\right )^{4}}, x\right ) \]