\[ \int \frac {\cos ^2(a+b x)}{\sqrt {\csc (a+b x)}} \, dx \]
Optimal antiderivative \[ \frac {2 \cos \left (b x +a \right )}{5 b \csc \left (b x +a \right )^{\frac {3}{2}}}-\frac {4 \sqrt {\frac {1}{2}+\frac {\sin \left (b x +a \right )}{2}}\, \EllipticE \left (\cos \left (\frac {a}{2}+\frac {\pi }{4}+\frac {b x}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\csc }\left (b x +a \right )\right ) \left (\sqrt {\sin }\left (b x +a \right )\right )}{5 \sin \left (\frac {a}{2}+\frac {\pi }{4}+\frac {b x}{2}\right ) b} \]
command
integrate(cos(b*x+a)^2/csc(b*x+a)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (\sqrt {2 i} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) + \sqrt {-2 i} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right ) - \frac {\cos \left (b x + a\right )^{3} - \cos \left (b x + a\right )}{\sqrt {\sin \left (b x + a\right )}}\right )}}{5 \, b} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\cos \left (b x + a\right )^{2}}{\sqrt {\csc \left (b x + a\right )}}, x\right ) \]