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