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