\[ \int \frac {1}{\sqrt {\sec (c+d x)} \sqrt {-3+2 \sec (c+d x)}} \, dx \]
Optimal antiderivative \[ -\frac {4 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \frac {\sqrt {30}}{5}\right ) \sqrt {2-3 \cos \left (d x +c \right )}\, \left (\sqrt {\sec }\left (d x +c \right )\right ) \sqrt {5}}{15 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) d \sqrt {-3+2 \sec \left (d x +c \right )}}+\frac {2 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \frac {\sqrt {30}}{5}\right ) \sqrt {5}\, \sqrt {-3+2 \sec \left (d x +c \right )}}{3 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) d \sqrt {2-3 \cos \left (d x +c \right )}\, \sqrt {\sec \left (d x +c \right )}} \]
command
integrate(1/sec(d*x+c)^(1/2)/(-3+2*sec(d*x+c))^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {4 \, \sqrt {6} {\rm weierstrassPInverse}\left (-\frac {44}{27}, -\frac {784}{729}, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) - \frac {4}{9}\right ) + 4 \, \sqrt {6} {\rm weierstrassPInverse}\left (-\frac {44}{27}, -\frac {784}{729}, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) - \frac {4}{9}\right ) - 9 \, \sqrt {6} {\rm weierstrassZeta}\left (-\frac {44}{27}, -\frac {784}{729}, {\rm weierstrassPInverse}\left (-\frac {44}{27}, -\frac {784}{729}, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) - \frac {4}{9}\right )\right ) - 9 \, \sqrt {6} {\rm weierstrassZeta}\left (-\frac {44}{27}, -\frac {784}{729}, {\rm weierstrassPInverse}\left (-\frac {44}{27}, -\frac {784}{729}, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) - \frac {4}{9}\right )\right )}{27 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {2 \, \sec \left (d x + c\right ) - 3} \sqrt {\sec \left (d x + c\right )}}{2 \, \sec \left (d x + c\right )^{2} - 3 \, \sec \left (d x + c\right )}, x\right ) \]