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