\[ \int (d \cos (a+b x))^{11/2} \csc ^2(a+b x) \, dx \]
Optimal antiderivative \[ -\frac {d \left (d \cos \left (b x +a \right )\right )^{\frac {9}{2}} \csc \left (b x +a \right )}{b}-\frac {9 d^{3} \left (d \cos \left (b x +a \right )\right )^{\frac {5}{2}} \sin \left (b x +a \right )}{7 b}-\frac {15 d^{6} \sqrt {\frac {\cos \left (b x +a \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {a}{2}+\frac {b x}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (b x +a \right )\right )}{7 \cos \left (\frac {a}{2}+\frac {b x}{2}\right ) b \sqrt {d \cos \left (b x +a \right )}}-\frac {15 d^{5} \sin \left (b x +a \right ) \sqrt {d \cos \left (b x +a \right )}}{7 b} \]
command
integrate((d*cos(b*x+a))^(11/2)*csc(b*x+a)^2,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {15 i \, \sqrt {2} d^{\frac {11}{2}} \sin \left (b x + a\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) - 15 i \, \sqrt {2} d^{\frac {11}{2}} \sin \left (b x + a\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) + 2 \, {\left (2 \, d^{5} \cos \left (b x + a\right )^{4} + 6 \, d^{5} \cos \left (b x + a\right )^{2} - 15 \, d^{5}\right )} \sqrt {d \cos \left (b x + a\right )}}{14 \, b \sin \left (b x + a\right )} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\sqrt {d \cos \left (b x + a\right )} d^{5} \cos \left (b x + a\right )^{5} \csc \left (b x + a\right )^{2}, x\right ) \]