\[ \int \frac {\csc ^2(a+b x)}{\sin ^{\frac {5}{2}}(2 a+2 b x)} \, dx \]
Optimal antiderivative \[ -\frac {10 \sqrt {\frac {1}{2}+\frac {\sin \left (2 b x +2 a \right )}{2}}\, \EllipticF \left (\cos \left (a +\frac {\pi }{4}+b x \right ), \sqrt {2}\right )}{21 \sin \left (a +\frac {\pi }{4}+b x \right ) b}-\frac {10 \cos \left (2 b x +2 a \right )}{21 b \sin \left (2 b x +2 a \right )^{\frac {3}{2}}}-\frac {\csc ^{2}\left (b x +a \right )}{7 b \sin \left (2 b x +2 a \right )^{\frac {3}{2}}} \]
command
integrate(csc(b*x+a)^2/sin(2*b*x+2*a)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {20 \, \sqrt {2 i} {\left (\cos \left (b x + a\right )^{6} - 2 \, \cos \left (b x + a\right )^{4} + \cos \left (b x + a\right )^{2}\right )} {\rm ellipticF}\left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ), -1\right ) + 20 \, \sqrt {-2 i} {\left (\cos \left (b x + a\right )^{6} - 2 \, \cos \left (b x + a\right )^{4} + \cos \left (b x + a\right )^{2}\right )} {\rm ellipticF}\left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ), -1\right ) - \sqrt {2} {\left (20 \, \cos \left (b x + a\right )^{4} - 30 \, \cos \left (b x + a\right )^{2} + 7\right )} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )}}{84 \, {\left (b \cos \left (b x + a\right )^{6} - 2 \, b \cos \left (b x + a\right )^{4} + b \cos \left (b x + a\right )^{2}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\csc \left (b x + a\right )^{2}}{{\left (\cos \left (2 \, b x + 2 \, a\right )^{2} - 1\right )} \sqrt {\sin \left (2 \, b x + 2 \, a\right )}}, x\right ) \]