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