\[ \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {-2-3 \sec (c+d x)}} \, dx \]
Optimal antiderivative \[ \frac {2 \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \frac {2 \sqrt {5}}{5}\right ) \sqrt {3+2 \cos \left (d x +c \right )}\, \left (\sqrt {\sec }\left (d x +c \right )\right ) \sqrt {5}}{5 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d \sqrt {-2-3 \sec \left (d x +c \right )}} \]
command
integrate(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
\[ -\frac {{\rm weierstrassPInverse}\left (8, -4, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + 1\right ) + {\rm weierstrassPInverse}\left (8, -4, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) + 1\right )}{d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {-3 \, \sec \left (d x + c\right ) - 2} \sqrt {\sec \left (d x + c\right )}}{3 \, \sec \left (d x + c\right ) + 2}, x\right ) \]