\[ \int \frac {\csc ^2(a+b x)}{\sin ^{\frac {9}{2}}(2 a+2 b x)} \, dx \]
Optimal antiderivative \[ -\frac {30 \sqrt {\frac {1}{2}+\frac {\sin \left (2 b x +2 a \right )}{2}}\, \EllipticF \left (\cos \left (a +\frac {\pi }{4}+b x \right ), \sqrt {2}\right )}{77 \sin \left (a +\frac {\pi }{4}+b x \right ) b}-\frac {18 \cos \left (2 b x +2 a \right )}{77 b \sin \left (2 b x +2 a \right )^{\frac {7}{2}}}-\frac {\csc ^{2}\left (b x +a \right )}{11 b \sin \left (2 b x +2 a \right )^{\frac {7}{2}}}-\frac {30 \cos \left (2 b x +2 a \right )}{77 b \sin \left (2 b x +2 a \right )^{\frac {3}{2}}} \]
command
integrate(csc(b*x+a)^2/sin(2*b*x+2*a)^(9/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {240 \, \sqrt {2 i} {\left (\cos \left (b x + a\right )^{10} - 3 \, \cos \left (b x + a\right )^{8} + 3 \, \cos \left (b x + a\right )^{6} - \cos \left (b x + a\right )^{4}\right )} {\rm ellipticF}\left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ), -1\right ) + 240 \, \sqrt {-2 i} {\left (\cos \left (b x + a\right )^{10} - 3 \, \cos \left (b x + a\right )^{8} + 3 \, \cos \left (b x + a\right )^{6} - \cos \left (b x + a\right )^{4}\right )} {\rm ellipticF}\left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ), -1\right ) - \sqrt {2} {\left (240 \, \cos \left (b x + a\right )^{8} - 600 \, \cos \left (b x + a\right )^{6} + 444 \, \cos \left (b x + a\right )^{4} - 66 \, \cos \left (b x + a\right )^{2} - 11\right )} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )}}{1232 \, {\left (b \cos \left (b x + a\right )^{10} - 3 \, b \cos \left (b x + a\right )^{8} + 3 \, b \cos \left (b x + a\right )^{6} - b \cos \left (b x + a\right )^{4}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\csc \left (b x + a\right )^{2}}{{\left (\cos \left (2 \, b x + 2 \, a\right )^{4} - 2 \, \cos \left (2 \, b x + 2 \, a\right )^{2} + 1\right )} \sqrt {\sin \left (2 \, b x + 2 \, a\right )}}, x\right ) \]