\[ \int (d \cos (a+b x))^{9/2} \sin ^4(a+b x) \, dx \]
Optimal antiderivative \[ \frac {56 d^{3} \left (d \cos \left (b x +a \right )\right )^{\frac {3}{2}} \sin \left (b x +a \right )}{3315 b}+\frac {8 d \left (d \cos \left (b x +a \right )\right )^{\frac {7}{2}} \sin \left (b x +a \right )}{663 b}-\frac {12 \left (d \cos \left (b x +a \right )\right )^{\frac {11}{2}} \sin \left (b x +a \right )}{221 b d}-\frac {2 \left (d \cos \left (b x +a \right )\right )^{\frac {11}{2}} \left (\sin ^{3}\left (b x +a \right )\right )}{17 b d}+\frac {56 d^{4} \sqrt {\frac {\cos \left (b x +a \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {a}{2}+\frac {b x}{2}\right ), \sqrt {2}\right ) \sqrt {d \cos \left (b x +a \right )}}{1105 \cos \left (\frac {a}{2}+\frac {b x}{2}\right ) b \sqrt {\cos \left (b x +a \right )}} \]
command
integrate((d*cos(b*x+a))^(9/2)*sin(b*x+a)^4,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (-42 i \, \sqrt {2} d^{\frac {9}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) + 42 i \, \sqrt {2} d^{\frac {9}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right ) - {\left (195 \, d^{4} \cos \left (b x + a\right )^{7} - 285 \, d^{4} \cos \left (b x + a\right )^{5} + 20 \, d^{4} \cos \left (b x + a\right )^{3} + 28 \, d^{4} \cos \left (b x + a\right )\right )} \sqrt {d \cos \left (b x + a\right )} \sin \left (b x + a\right )\right )}}{3315 \, b} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (d^{4} \cos \left (b x + a\right )^{8} - 2 \, d^{4} \cos \left (b x + a\right )^{6} + d^{4} \cos \left (b x + a\right )^{4}\right )} \sqrt {d \cos \left (b x + a\right )}, x\right ) \]