\[ \int \frac {\cos ^4(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 a^{2} \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 b \left (a^{2}-b^{2}\right ) d \left (a +b \cos \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {4 a^{3} \left (3 a^{2}-5 b^{2}\right ) \sin \left (d x +c \right )}{3 b^{3} \left (a^{2}-b^{2}\right )^{2} d \sqrt {a +b \cos \left (d x +c \right )}}+\frac {2 \left (2 a^{2}-b^{2}\right ) \sin \left (d x +c \right ) \sqrt {a +b \cos \left (d x +c \right )}}{3 b^{3} \left (a^{2}-b^{2}\right ) d}-\frac {8 a \left (4 a^{4}-7 a^{2} b^{2}+2 b^{4}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {a +b \cos \left (d x +c \right )}}{3 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{4} \left (a^{2}-b^{2}\right )^{2} d \sqrt {\frac {a +b \cos \left (d x +c \right )}{a +b}}}+\frac {2 \left (16 a^{4}-16 a^{2} b^{2}-b^{4}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {\frac {a +b \cos \left (d x +c \right )}{a +b}}}{3 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{4} \left (a^{2}-b^{2}\right ) d \sqrt {a +b \cos \left (d x +c \right )}} \]
command
integrate(cos(d*x+c)^4/(a+b*cos(d*x+c))^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {6 \, {\left (8 \, a^{6} b^{2} - 13 \, a^{4} b^{4} + a^{2} b^{6} + {\left (a^{4} b^{4} - 2 \, a^{2} b^{6} + b^{8}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (5 \, a^{5} b^{3} - 8 \, a^{3} b^{5} + a b^{7}\right )} \cos \left (d x + c\right )\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sin \left (d x + c\right ) + {\left (\sqrt {2} {\left (-32 i \, a^{6} b^{2} + 68 i \, a^{4} b^{4} - 37 i \, a^{2} b^{6} - 3 i \, b^{8}\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {2} {\left (32 i \, a^{7} b - 68 i \, a^{5} b^{3} + 37 i \, a^{3} b^{5} + 3 i \, a b^{7}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-32 i \, a^{8} + 68 i \, a^{6} b^{2} - 37 i \, a^{4} b^{4} - 3 i \, a^{2} b^{6}\right )}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, 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 \, a}{3 \, b}\right ) + {\left (\sqrt {2} {\left (32 i \, a^{6} b^{2} - 68 i \, a^{4} b^{4} + 37 i \, a^{2} b^{6} + 3 i \, b^{8}\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {2} {\left (-32 i \, a^{7} b + 68 i \, a^{5} b^{3} - 37 i \, a^{3} b^{5} - 3 i \, a b^{7}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (32 i \, a^{8} - 68 i \, a^{6} b^{2} + 37 i \, a^{4} b^{4} + 3 i \, a^{2} b^{6}\right )}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, 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 \, a}{3 \, b}\right ) - 12 \, {\left (\sqrt {2} {\left (4 i \, a^{5} b^{3} - 7 i \, a^{3} b^{5} + 2 i \, a b^{7}\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {2} {\left (4 i \, a^{6} b^{2} - 7 i \, a^{4} b^{4} + 2 i \, a^{2} b^{6}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (4 i \, a^{7} b - 7 i \, a^{5} b^{3} + 2 i \, a^{3} b^{5}\right )}\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, 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 \, a^{3} - 9 \, 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 \, a}{3 \, b}\right )\right ) - 12 \, {\left (\sqrt {2} {\left (-4 i \, a^{5} b^{3} + 7 i \, a^{3} b^{5} - 2 i \, a b^{7}\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {2} {\left (-4 i \, a^{6} b^{2} + 7 i \, a^{4} b^{4} - 2 i \, a^{2} b^{6}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-4 i \, a^{7} b + 7 i \, a^{5} b^{3} - 2 i \, a^{3} b^{5}\right )}\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, 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 \, a^{3} - 9 \, 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 \, a}{3 \, b}\right )\right )}{9 \, {\left ({\left (a^{4} b^{7} - 2 \, a^{2} b^{9} + b^{11}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{5} b^{6} - 2 \, a^{3} b^{8} + a b^{10}\right )} d \cos \left (d x + c\right ) + {\left (a^{6} b^{5} - 2 \, a^{4} b^{7} + a^{2} b^{9}\right )} d\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {b \cos \left (d x + c\right ) + a} \cos \left (d x + c\right )^{4}}{b^{3} \cos \left (d x + c\right )^{3} + 3 \, a b^{2} \cos \left (d x + c\right )^{2} + 3 \, a^{2} b \cos \left (d x + c\right ) + a^{3}}, x\right ) \]