\[ \int \frac {1}{\sqrt {\sec (c+d x)} \sqrt {-2+3 \sec (c+d x)}} \, dx \]
Optimal antiderivative \[ \frac {3 \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), 2 i\right ) \sqrt {3-2 \cos \left (d x +c \right )}\, \left (\sqrt {\sec }\left (d x +c \right )\right )}{\cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d \sqrt {-2+3 \sec \left (d x +c \right )}}-\frac {\sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), 2 i\right ) \sqrt {-2+3 \sec \left (d x +c \right )}}{\cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d \sqrt {3-2 \cos \left (d x +c \right )}\, \sqrt {\sec \left (d x +c \right )}} \]
command
integrate(1/sec(d*x+c)^(1/2)/(-2+3*sec(d*x+c))^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {{\rm weierstrassPInverse}\left (8, 4, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) - 1\right ) + {\rm weierstrassPInverse}\left (8, 4, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) - 1\right ) - {\rm weierstrassZeta}\left (8, 4, {\rm weierstrassPInverse}\left (8, 4, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) - 1\right )\right ) - {\rm weierstrassZeta}\left (8, 4, {\rm weierstrassPInverse}\left (8, 4, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) - 1\right )\right )}{d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {3 \, \sec \left (d x + c\right ) - 2} \sqrt {\sec \left (d x + c\right )}}{3 \, \sec \left (d x + c\right )^{2} - 2 \, \sec \left (d x + c\right )}, x\right ) \]