\[ \int \cos ^4(a+b x) \sqrt {\csc (a+b x)} \, dx \]
Optimal antiderivative \[ \frac {4 \cos \left (b x +a \right )}{7 b \sqrt {\csc \left (b x +a \right )}}+\frac {2 \left (\cos ^{3}\left (b x +a \right )\right )}{7 b \sqrt {\csc \left (b x +a \right )}}-\frac {8 \sqrt {\frac {1}{2}+\frac {\sin \left (b x +a \right )}{2}}\, \EllipticF \left (\cos \left (\frac {a}{2}+\frac {\pi }{4}+\frac {b x}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\csc }\left (b x +a \right )\right ) \left (\sqrt {\sin }\left (b x +a \right )\right )}{7 \sin \left (\frac {a}{2}+\frac {\pi }{4}+\frac {b x}{2}\right ) b} \]
command
integrate(cos(b*x+a)^4*csc(b*x+a)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left ({\left (\cos \left (b x + a\right )^{3} + 2 \, \cos \left (b x + a\right )\right )} \sqrt {\sin \left (b x + a\right )} - 2 i \, \sqrt {2 i} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + 2 i \, \sqrt {-2 i} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right )}}{7 \, b} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\cos \left (b x + a\right )^{4} \sqrt {\csc \left (b x + a\right )}, x\right ) \]