\[ \int \frac {(a+b \csc (x)) \left (A+B \csc (x)+C \csc ^2(x)\right )}{\sqrt {\csc (x)}} \, dx \]
Optimal antiderivative \[ -\frac {2 b C \cos \left (x \right ) \left (\csc ^{\frac {3}{2}}\left (x \right )\right )}{3}-2 \left (b B +a C \right ) \cos \left (x \right ) \left (\sqrt {\csc }\left (x \right )\right )+\frac {2 \left (b B -a \left (A -C \right )\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (x \right )}{2}}\, \EllipticE \left (\cos \left (\frac {\pi }{4}+\frac {x}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\csc }\left (x \right )\right ) \left (\sqrt {\sin }\left (x \right )\right )}{\sin \left (\frac {\pi }{4}+\frac {x}{2}\right )}-\frac {2 \left (3 A b +3 B a +C b \right ) \sqrt {\frac {1}{2}+\frac {\sin \left (x \right )}{2}}\, \EllipticF \left (\cos \left (\frac {\pi }{4}+\frac {x}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\csc }\left (x \right )\right ) \left (\sqrt {\sin }\left (x \right )\right )}{3 \sin \left (\frac {\pi }{4}+\frac {x}{2}\right )} \]
command
integrate((a+b*csc(x))*(A+B*csc(x)+C*csc(x)^2)/csc(x)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {\sqrt {2 i} {\left (-3 i \, B a - i \, {\left (3 \, A + C\right )} b\right )} \sin \left (x\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right ) + \sqrt {-2 i} {\left (3 i \, B a + i \, {\left (3 \, A + C\right )} b\right )} \sin \left (x\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right ) + 3 \, \sqrt {2 i} {\left ({\left (A - C\right )} a - B b\right )} \sin \left (x\right ) {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right )\right ) + 3 \, \sqrt {-2 i} {\left ({\left (A - C\right )} a - B b\right )} \sin \left (x\right ) {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right )\right ) - \frac {2 \, {\left (C b \cos \left (x\right ) + 3 \, {\left (C a + B b\right )} \cos \left (x\right ) \sin \left (x\right )\right )}}{\sqrt {\sin \left (x\right )}}}{3 \, \sin \left (x\right )} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {C b \csc \left (x\right )^{3} + {\left (C a + B b\right )} \csc \left (x\right )^{2} + A a + {\left (B a + A b\right )} \csc \left (x\right )}{\sqrt {\csc \left (x\right )}}, x\right ) \]