\[ \int \sqrt {\csc (a+b x)} \sec ^4(a+b x) \, dx \]
Optimal antiderivative \[ \frac {5 \sec \left (b x +a \right )}{6 b \sqrt {\csc \left (b x +a \right )}}+\frac {\sec ^{3}\left (b x +a \right )}{3 b \sqrt {\csc \left (b x +a \right )}}-\frac {5 \sqrt {\frac {1}{2}+\frac {\sin \left (b x +a \right )}{2}}\, \EllipticF \left (\cos \left (\frac {a}{2}+\frac {\pi }{4}+\frac {b x}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\csc }\left (b x +a \right )\right ) \left (\sqrt {\sin }\left (b x +a \right )\right )}{6 \sin \left (\frac {a}{2}+\frac {\pi }{4}+\frac {b x}{2}\right ) b} \]
command
integrate(csc(b*x+a)^(1/2)*sec(b*x+a)^4,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {-5 i \, \sqrt {2 i} \cos \left (b x + a\right )^{3} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + 5 i \, \sqrt {-2 i} \cos \left (b x + a\right )^{3} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) + 2 \, {\left (5 \, \cos \left (b x + a\right )^{2} + 2\right )} \sqrt {\sin \left (b x + a\right )}}{12 \, b \cos \left (b x + a\right )^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\sqrt {\csc \left (b x + a\right )} \sec \left (b x + a\right )^{4}, x\right ) \]