\[ \int \frac {1}{\sin ^{\frac {7}{2}}(a+b x)} \, dx \]
Optimal antiderivative \[ \frac {6 \sqrt {\frac {1}{2}+\frac {\sin \left (b x +a \right )}{2}}\, \EllipticE \left (\cos \left (\frac {a}{2}+\frac {\pi }{4}+\frac {b x}{2}\right ), \sqrt {2}\right )}{5 \sin \left (\frac {a}{2}+\frac {\pi }{4}+\frac {b x}{2}\right ) b}-\frac {2 \cos \left (b x +a \right )}{5 b \sin \left (b x +a \right )^{\frac {5}{2}}}-\frac {6 \cos \left (b x +a \right )}{5 b \sqrt {\sin \left (b x +a \right )}} \]
command
integrate(1/sin(b*x+a)^(7/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {3 \, \sqrt {-i} {\left (i \, \sqrt {2} \cos \left (b x + a\right )^{2} - i \, \sqrt {2}\right )} \sin \left (b x + a\right ) {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) + 3 \, \sqrt {i} {\left (-i \, \sqrt {2} \cos \left (b x + a\right )^{2} + i \, \sqrt {2}\right )} \sin \left (b x + a\right ) {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right ) + 2 \, {\left (3 \, \cos \left (b x + a\right )^{3} - 4 \, \cos \left (b x + a\right )\right )} \sqrt {\sin \left (b x + a\right )}}{5 \, {\left (b \cos \left (b x + a\right )^{2} - b\right )} \sin \left (b x + a\right )} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {\sin \left (b x + a\right )}}{\cos \left (b x + a\right )^{4} - 2 \, \cos \left (b x + a\right )^{2} + 1}, x\right ) \]