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