\[ \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {-3-2 \sec (c+d x)}} \, dx \]
Optimal antiderivative \[ -\frac {2 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {6}\right ) \sqrt {-2-3 \cos \left (d x +c \right )}\, \left (\sqrt {\sec }\left (d x +c \right )\right )}{\sin \left (\frac {d x}{2}+\frac {c}{2}\right ) d \sqrt {-3-2 \sec \left (d x +c \right )}} \]
command
integrate(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
\[ -\frac {\sqrt {6} {\rm weierstrassPInverse}\left (-\frac {44}{27}, \frac {784}{729}, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + \frac {4}{9}\right ) + \sqrt {6} {\rm weierstrassPInverse}\left (-\frac {44}{27}, \frac {784}{729}, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) + \frac {4}{9}\right )}{3 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {-2 \, \sec \left (d x + c\right ) - 3} \sqrt {\sec \left (d x + c\right )}}{2 \, \sec \left (d x + c\right ) + 3}, x\right ) \]