\[ \int \frac {1}{(c \cos (a+b x))^{7/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \sin \left (b x +a \right )}{5 b c \left (c \cos \left (b x +a \right )\right )^{\frac {5}{2}}}+\frac {6 \sin \left (b x +a \right )}{5 b \,c^{3} \sqrt {c \cos \left (b x +a \right )}}-\frac {6 \sqrt {\frac {\cos \left (b x +a \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {a}{2}+\frac {b x}{2}\right ), \sqrt {2}\right ) \sqrt {c \cos \left (b x +a \right )}}{5 \cos \left (\frac {a}{2}+\frac {b x}{2}\right ) b \,c^{4} \sqrt {\cos \left (b x +a \right )}} \]
command
integrate(1/(c*cos(b*x+a))^(7/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {-3 i \, \sqrt {2} \sqrt {c} \cos \left (b x + a\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) + 3 i \, \sqrt {2} \sqrt {c} \cos \left (b x + a\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right ) + 2 \, \sqrt {c \cos \left (b x + a\right )} {\left (3 \, \cos \left (b x + a\right )^{2} + 1\right )} \sin \left (b x + a\right )}{5 \, b c^{4} \cos \left (b x + a\right )^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {c \cos \left (b x + a\right )}}{c^{4} \cos \left (b x + a\right )^{4}}, x\right ) \]