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