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