\[ \int \sin ^{\frac {7}{2}}(b x) \, dx \]
Optimal antiderivative \[ -\frac {10 \sqrt {\frac {1}{2}+\frac {\sin \left (b x \right )}{2}}\, \EllipticF \left (\cos \left (\frac {\pi }{4}+\frac {b x}{2}\right ), \sqrt {2}\right )}{21 \sin \left (\frac {\pi }{4}+\frac {b x}{2}\right ) b}-\frac {2 \cos \left (b x \right ) \left (\sin ^{\frac {5}{2}}\left (b x \right )\right )}{7 b}-\frac {10 \cos \left (b x \right ) \left (\sqrt {\sin }\left (b x \right )\right )}{21 b} \]
command
integrate(sin(b*x)^(7/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {5 \, \sqrt {2} \sqrt {-i} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x\right ) + i \, \sin \left (b x\right )\right ) + 5 \, \sqrt {2} \sqrt {i} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x\right ) - i \, \sin \left (b x\right )\right ) + 2 \, {\left (3 \, \cos \left (b x\right )^{3} - 8 \, \cos \left (b x\right )\right )} \sqrt {\sin \left (b x\right )}}{21 \, b} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-{\left (\cos \left (b x\right )^{2} - 1\right )} \sin \left (b x\right )^{\frac {3}{2}}, x\right ) \]