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