\[ \int \frac {\sin ^4(a+b x)}{\sqrt {d \cos (a+b x)}} \, dx \]
Optimal antiderivative \[ \frac {8 \sqrt {\frac {\cos \left (b x +a \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {a}{2}+\frac {b x}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (b x +a \right )\right )}{7 \cos \left (\frac {a}{2}+\frac {b x}{2}\right ) b \sqrt {d \cos \left (b x +a \right )}}-\frac {4 \sin \left (b x +a \right ) \sqrt {d \cos \left (b x +a \right )}}{7 b d}-\frac {2 \left (\sin ^{3}\left (b x +a \right )\right ) \sqrt {d \cos \left (b x +a \right )}}{7 b d} \]
command
integrate(sin(b*x+a)^4/(d*cos(b*x+a))^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (\sqrt {d \cos \left (b x + a\right )} {\left (\cos \left (b x + a\right )^{2} - 3\right )} \sin \left (b x + a\right ) - 2 i \, \sqrt {2} \sqrt {d} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + 2 i \, \sqrt {2} \sqrt {d} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right )}}{7 \, b d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (\cos \left (b x + a\right )^{4} - 2 \, \cos \left (b x + a\right )^{2} + 1\right )} \sqrt {d \cos \left (b x + a\right )}}{d \cos \left (b x + a\right )}, x\right ) \]