\[ \int (d \cos (a+b x))^{7/2} \sin ^4(a+b x) \, dx \]
Optimal antiderivative \[ \frac {8 d \left (d \cos \left (b x +a \right )\right )^{\frac {5}{2}} \sin \left (b x +a \right )}{385 b}-\frac {4 \left (d \cos \left (b x +a \right )\right )^{\frac {9}{2}} \sin \left (b x +a \right )}{55 b d}-\frac {2 \left (d \cos \left (b x +a \right )\right )^{\frac {9}{2}} \left (\sin ^{3}\left (b x +a \right )\right )}{15 b d}+\frac {8 d^{4} \sqrt {\frac {\cos \left (b x +a \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {a}{2}+\frac {b x}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (b x +a \right )\right )}{231 \cos \left (\frac {a}{2}+\frac {b x}{2}\right ) b \sqrt {d \cos \left (b x +a \right )}}+\frac {8 d^{3} \sin \left (b x +a \right ) \sqrt {d \cos \left (b x +a \right )}}{231 b} \]
command
integrate((d*cos(b*x+a))^(7/2)*sin(b*x+a)^4,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (10 i \, \sqrt {2} d^{\frac {7}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) - 10 i \, \sqrt {2} d^{\frac {7}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) - {\left (77 \, d^{3} \cos \left (b x + a\right )^{6} - 119 \, d^{3} \cos \left (b x + a\right )^{4} + 12 \, d^{3} \cos \left (b x + a\right )^{2} + 20 \, d^{3}\right )} \sqrt {d \cos \left (b x + a\right )} \sin \left (b x + a\right )\right )}}{1155 \, b} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (d^{3} \cos \left (b x + a\right )^{7} - 2 \, d^{3} \cos \left (b x + a\right )^{5} + d^{3} \cos \left (b x + a\right )^{3}\right )} \sqrt {d \cos \left (b x + a\right )}, x\right ) \]