\[ \int \frac {\cos ^6(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{b d \sqrt {a +b \sin \left (d x +c \right )}}+\frac {20 \left (\cos ^{3}\left (d x +c \right )\right ) \left (8 a -7 b \sin \left (d x +c \right )\right ) \sqrt {a +b \sin \left (d x +c \right )}}{63 b^{3} d}-\frac {8 \cos \left (d x +c \right ) \left (a \left (32 a^{2}-33 b^{2}\right )-3 b \left (8 a^{2}-7 b^{2}\right ) \sin \left (d x +c \right )\right ) \sqrt {a +b \sin \left (d x +c \right )}}{63 b^{5} d}+\frac {16 \left (32 a^{4}-57 a^{2} b^{2}+21 b^{4}\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (d x +c \right )}{2}}\, \EllipticE \left (\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {a +b \sin \left (d x +c \right )}}{63 \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) b^{6} d \sqrt {\frac {a +b \sin \left (d x +c \right )}{a +b}}}-\frac {16 a \left (32 a^{4}-65 a^{2} b^{2}+33 b^{4}\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (d x +c \right )}{2}}\, \EllipticF \left (\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {\frac {a +b \sin \left (d x +c \right )}{a +b}}}{63 \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) b^{6} d \sqrt {a +b \sin \left (d x +c \right )}} \]
command
integrate(cos(d*x+c)^6/(a+b*sin(d*x+c))^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (8 \, {\left (\sqrt {2} {\left (32 \, a^{5} b - 69 \, a^{3} b^{3} + 39 \, a b^{5}\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (32 \, a^{6} - 69 \, a^{4} b^{2} + 39 \, a^{2} b^{4}\right )}\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 \, {\left (\sqrt {2} {\left (32 \, a^{5} b - 69 \, a^{3} b^{3} + 39 \, a b^{5}\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (32 \, a^{6} - 69 \, a^{4} b^{2} + 39 \, a^{2} b^{4}\right )}\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 ) - 12 \, {\left (\sqrt {2} {\left (-32 i \, a^{4} b^{2} + 57 i \, a^{2} b^{4} - 21 i \, b^{6}\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (-32 i \, a^{5} b + 57 i \, a^{3} b^{3} - 21 i \, a b^{5}\right )}\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 ) - 12 \, {\left (\sqrt {2} {\left (32 i \, a^{4} b^{2} - 57 i \, a^{2} b^{4} + 21 i \, b^{6}\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (32 i \, a^{5} b - 57 i \, a^{3} b^{3} + 21 i \, a b^{5}\right )}\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 (7 \, b^{6} \cos \left (d x + c\right )^{5} - 2 \, {\left (8 \, a^{2} b^{4} - 7 \, b^{6}\right )} \cos \left (d x + c\right )^{3} - 4 \, {\left (32 \, a^{4} b^{2} - 57 \, a^{2} b^{4} + 21 \, b^{6}\right )} \cos \left (d x + c\right ) + 2 \, {\left (5 \, a b^{5} \cos \left (d x + c\right )^{3} - 8 \, {\left (2 \, a^{3} b^{3} - 3 \, a b^{5}\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 )}}{189 \, {\left (b^{8} d \sin \left (d x + c\right ) + a b^{7} d\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {b \sin \left (d x + c\right ) + a} \cos \left (d x + c\right )^{6}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}, x\right ) \]