\[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx \]
Optimal antiderivative \[ \frac {64 a \left (80 a^{4}-118 a^{2} b^{2}+17 b^{4}\right ) \cos \left (d x +c \right ) \sqrt {a +b \sin \left (d x +c \right )}}{15015 b^{6} d}-\frac {8 \left (480 a^{4}-683 a^{2} b^{2}+77 b^{4}\right ) \cos \left (d x +c \right ) \sin \left (d x +c \right ) \sqrt {a +b \sin \left (d x +c \right )}}{15015 b^{5} d}+\frac {4 a \left (160 a^{2}-223 b^{2}\right ) \cos \left (d x +c \right ) \left (\sin ^{2}\left (d x +c \right )\right ) \sqrt {a +b \sin \left (d x +c \right )}}{3003 b^{4} d}-\frac {10 \left (8 a^{2}-11 b^{2}\right ) \cos \left (d x +c \right ) \left (\sin ^{3}\left (d x +c \right )\right ) \sqrt {a +b \sin \left (d x +c \right )}}{429 b^{3} d}+\frac {24 a \cos \left (d x +c \right ) \left (\sin ^{4}\left (d x +c \right )\right ) \sqrt {a +b \sin \left (d x +c \right )}}{143 b^{2} d}-\frac {2 \cos \left (d x +c \right ) \left (\sin ^{5}\left (d x +c \right )\right ) \sqrt {a +b \sin \left (d x +c \right )}}{13 b d}-\frac {8 \left (1280 a^{6}-2048 a^{4} b^{2}+453 a^{2} b^{4}+231 b^{6}\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (d x +c \right )}{2}}\, \EllipticE \left (\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {a +b \sin \left (d x +c \right )}}{15015 \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) b^{7} d \sqrt {\frac {a +b \sin \left (d x +c \right )}{a +b}}}+\frac {8 a \left (1280 a^{6}-2368 a^{4} b^{2}+875 a^{2} b^{4}+213 b^{6}\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (d x +c \right )}{2}}\, \EllipticF \left (\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {\frac {a +b \sin \left (d x +c \right )}{a +b}}}{15015 \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) b^{7} d \sqrt {a +b \sin \left (d x +c \right )}} \]
command
integrate(cos(d*x+c)^4*sin(d*x+c)^3/(a+b*sin(d*x+c))^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (8 \, \sqrt {2} {\left (640 \, a^{7} - 1264 \, a^{5} b^{2} + 543 \, a^{3} b^{4} + 102 \, a b^{6}\right )} \sqrt {i \, b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) - 2 i \, a}{3 \, b}\right ) + 8 \, \sqrt {2} {\left (640 \, a^{7} - 1264 \, a^{5} b^{2} + 543 \, a^{3} b^{4} + 102 \, a b^{6}\right )} \sqrt {-i \, b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 i \, a}{3 \, b}\right ) + 6 \, \sqrt {2} {\left (1280 i \, a^{6} b - 2048 i \, a^{4} b^{3} + 453 i \, a^{2} b^{5} + 231 i \, b^{7}\right )} \sqrt {i \, b} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) - 2 i \, a}{3 \, b}\right )\right ) + 6 \, \sqrt {2} {\left (-1280 i \, a^{6} b + 2048 i \, a^{4} b^{3} - 453 i \, a^{2} b^{5} - 231 i \, b^{7}\right )} \sqrt {-i \, b} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 i \, a}{3 \, b}\right )\right ) - 3 \, {\left (1260 \, a b^{6} \cos \left (d x + c\right )^{5} - 10 \, {\left (160 \, a^{3} b^{4} + 29 \, a b^{6}\right )} \cos \left (d x + c\right )^{3} + 2 \, {\left (1280 \, a^{5} b^{2} - 1088 \, a^{3} b^{4} - 213 \, a b^{6}\right )} \cos \left (d x + c\right ) - {\left (1155 \, b^{7} \cos \left (d x + c\right )^{5} - 35 \, {\left (40 \, a^{2} b^{5} + 11 \, b^{7}\right )} \cos \left (d x + c\right )^{3} + 6 \, {\left (320 \, a^{4} b^{3} - 222 \, a^{2} b^{5} - 77 \, b^{7}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )} \sqrt {b \sin \left (d x + c\right ) + a}\right )}}{45045 \, b^{8} d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {{\left (\cos \left (d x + c\right )^{6} - \cos \left (d x + c\right )^{4}\right )} \sin \left (d x + c\right )}{\sqrt {b \sin \left (d x + c\right ) + a}}, x\right ) \]