\[ \int \sqrt {d \cos (a+b x)} \sin ^4(a+b x) \, dx \]
Optimal antiderivative \[ -\frac {4 \left (d \cos \left (b x +a \right )\right )^{\frac {3}{2}} \sin \left (b x +a \right )}{15 b d}-\frac {2 \left (d \cos \left (b x +a \right )\right )^{\frac {3}{2}} \left (\sin ^{3}\left (b x +a \right )\right )}{9 b d}+\frac {8 \sqrt {\frac {\cos \left (b x +a \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {a}{2}+\frac {b x}{2}\right ), \sqrt {2}\right ) \sqrt {d \cos \left (b x +a \right )}}{15 \cos \left (\frac {a}{2}+\frac {b x}{2}\right ) b \sqrt {\cos \left (b x +a \right )}} \]
command
integrate((d*cos(b*x+a))^(1/2)*sin(b*x+a)^4,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left ({\left (5 \, \cos \left (b x + a\right )^{3} - 11 \, \cos \left (b x + a\right )\right )} \sqrt {d \cos \left (b x + a\right )} \sin \left (b x + a\right ) + 6 i \, \sqrt {2} \sqrt {d} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) - 6 i \, \sqrt {2} \sqrt {d} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right )\right )}}{45 \, b} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (\cos \left (b x + a\right )^{4} - 2 \, \cos \left (b x + a\right )^{2} + 1\right )} \sqrt {d \cos \left (b x + a\right )}, x\right ) \]