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