\[ \int \frac {\cos ^8(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{3 b d \left (a +b \sin \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {28 \left (\cos ^{5}\left (d x +c \right )\right ) \left (12 a +b \sin \left (d x +c \right )\right )}{33 b^{3} d \sqrt {a +b \sin \left (d x +c \right )}}+\frac {40 \left (\cos ^{3}\left (d x +c \right )\right ) \left (32 a^{2}-3 b^{2}-28 a b \sin \left (d x +c \right )\right ) \sqrt {a +b \sin \left (d x +c \right )}}{99 b^{5} d}-\frac {16 \cos \left (d x +c \right ) \left (128 a^{4}-144 a^{2} b^{2}+15 b^{4}-3 a b \left (32 a^{2}-31 b^{2}\right ) \sin \left (d x +c \right )\right ) \sqrt {a +b \sin \left (d x +c \right )}}{99 b^{7} d}+\frac {128 a \left (8 a^{2}-9 b^{2}\right ) \left (4 a^{2}-3 b^{2}\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (d x +c \right )}{2}}\, \EllipticE \left (\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {a +b \sin \left (d x +c \right )}}{99 \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) b^{8} d \sqrt {\frac {a +b \sin \left (d x +c \right )}{a +b}}}-\frac {32 \left (128 a^{6}-272 a^{4} b^{2}+159 a^{2} b^{4}-15 b^{6}\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (d x +c \right )}{2}}\, \EllipticF \left (\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {\frac {a +b \sin \left (d x +c \right )}{a +b}}}{99 \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) b^{8} d \sqrt {a +b \sin \left (d x +c \right )}} \]
command
integrate(cos(d*x+c)^8/(a+b*sin(d*x+c))^(5/2),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 (256 \, a^{6} b^{2} - 576 \, a^{4} b^{4} + 369 \, a^{2} b^{6} - 45 \, b^{8}\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {2} {\left (256 \, a^{7} b - 576 \, a^{5} b^{3} + 369 \, a^{3} b^{5} - 45 \, a b^{7}\right )} \sin \left (d x + c\right ) - \sqrt {2} {\left (256 \, a^{8} - 320 \, a^{6} b^{2} - 207 \, a^{4} b^{4} + 324 \, a^{2} b^{6} - 45 \, b^{8}\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 (256 \, a^{6} b^{2} - 576 \, a^{4} b^{4} + 369 \, a^{2} b^{6} - 45 \, b^{8}\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {2} {\left (256 \, a^{7} b - 576 \, a^{5} b^{3} + 369 \, a^{3} b^{5} - 45 \, a b^{7}\right )} \sin \left (d x + c\right ) - \sqrt {2} {\left (256 \, a^{8} - 320 \, a^{6} b^{2} - 207 \, a^{4} b^{4} + 324 \, a^{2} b^{6} - 45 \, b^{8}\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 ) + 96 \, {\left (\sqrt {2} {\left (32 i \, a^{5} b^{3} - 60 i \, a^{3} b^{5} + 27 i \, a b^{7}\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {2} {\left (-32 i \, a^{6} b^{2} + 60 i \, a^{4} b^{4} - 27 i \, a^{2} b^{6}\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (-32 i \, a^{7} b + 28 i \, a^{5} b^{3} + 33 i \, a^{3} b^{5} - 27 i \, a b^{7}\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 ) + 96 \, {\left (\sqrt {2} {\left (-32 i \, a^{5} b^{3} + 60 i \, a^{3} b^{5} - 27 i \, a b^{7}\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {2} {\left (32 i \, a^{6} b^{2} - 60 i \, a^{4} b^{4} + 27 i \, a^{2} b^{6}\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (32 i \, a^{7} b - 28 i \, a^{5} b^{3} - 33 i \, a^{3} b^{5} + 27 i \, a b^{7}\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 (9 \, b^{8} \cos \left (d x + c\right )^{7} - 6 \, {\left (4 \, a^{2} b^{6} - 3 \, b^{8}\right )} \cos \left (d x + c\right )^{5} + 4 \, {\left (32 \, a^{4} b^{4} - 51 \, a^{2} b^{6} + 15 \, b^{8}\right )} \cos \left (d x + c\right )^{3} - 8 \, {\left (128 \, a^{6} b^{2} - 208 \, a^{4} b^{4} + 57 \, a^{2} b^{6} + 15 \, b^{8}\right )} \cos \left (d x + c\right ) + 2 \, {\left (7 \, a b^{7} \cos \left (d x + c\right )^{5} - 8 \, {\left (3 \, a^{3} b^{5} - 4 \, a b^{7}\right )} \cos \left (d x + c\right )^{3} - 4 \, {\left (160 \, a^{5} b^{3} - 291 \, a^{3} b^{5} + 123 \, a 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 )}}{297 \, {\left (b^{11} d \cos \left (d x + c\right )^{2} - 2 \, a b^{10} d \sin \left (d x + c\right ) - {\left (a^{2} b^{9} + b^{11}\right )} d\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {b \sin \left (d x + c\right ) + a} \cos \left (d x + c\right )^{8}}{3 \, a b^{2} \cos \left (d x + c\right )^{2} - a^{3} - 3 \, a b^{2} + {\left (b^{3} \cos \left (d x + c\right )^{2} - 3 \, a^{2} b - b^{3}\right )} \sin \left (d x + c\right )}, x\right ) \]