\[ \int (d \csc (a+b x))^{9/2} \sqrt {c \sec (a+b x)} \, dx \]
Optimal antiderivative \[ -\frac {4 c \,d^{3} \left (d \csc \left (b x +a \right )\right )^{\frac {3}{2}}}{7 b \sqrt {c \sec \left (b x +a \right )}}-\frac {2 c d \left (d \csc \left (b x +a \right )\right )^{\frac {7}{2}}}{7 b \sqrt {c \sec \left (b x +a \right )}}-\frac {4 d^{4} \sqrt {\frac {1}{2}+\frac {\sin \left (2 b x +2 a \right )}{2}}\, \EllipticF \left (\cos \left (a +\frac {\pi }{4}+b x \right ), \sqrt {2}\right ) \sqrt {d \csc \left (b x +a \right )}\, \sqrt {c \sec \left (b x +a \right )}\, \left (\sqrt {\sin }\left (2 b x +2 a \right )\right )}{7 \sin \left (a +\frac {\pi }{4}+b x \right ) b} \]
command
integrate((d*csc(b*x+a))^(9/2)*(c*sec(b*x+a))^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left ({\left (i \, d^{4} \cos \left (b x + a\right )^{2} - i \, d^{4}\right )} \sqrt {-4 i \, c d} {\rm ellipticF}\left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ), -1\right ) \sin \left (b x + a\right ) + {\left (-i \, d^{4} \cos \left (b x + a\right )^{2} + i \, d^{4}\right )} \sqrt {4 i \, c d} {\rm ellipticF}\left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ), -1\right ) \sin \left (b x + a\right ) + {\left (2 \, d^{4} \cos \left (b x + a\right )^{3} - 3 \, d^{4} \cos \left (b x + a\right )\right )} \sqrt {\frac {c}{\cos \left (b x + a\right )}} \sqrt {\frac {d}{\sin \left (b x + a\right )}}\right )}}{7 \, {\left (b \cos \left (b x + a\right )^{2} - b\right )} \sin \left (b x + a\right )} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\sqrt {d \csc \left (b x + a\right )} \sqrt {c \sec \left (b x + a\right )} d^{4} \csc \left (b x + a\right )^{4}, x\right ) \]