\[ \int \frac {\cos ^4(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 a^{2} \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{b \left (a^{2}-b^{2}\right ) d \sqrt {a +b \cos \left (d x +c \right )}}-\frac {2 a \left (8 a^{2}-3 b^{2}\right ) \sin \left (d x +c \right ) \sqrt {a +b \cos \left (d x +c \right )}}{5 b^{3} \left (a^{2}-b^{2}\right ) d}+\frac {2 \left (6 a^{2}-b^{2}\right ) \cos \left (d x +c \right ) \sin \left (d x +c \right ) \sqrt {a +b \cos \left (d x +c \right )}}{5 b^{2} \left (a^{2}-b^{2}\right ) d}+\frac {2 \left (16 a^{4}-8 a^{2} b^{2}-3 b^{4}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {a +b \cos \left (d x +c \right )}}{5 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{4} \left (a^{2}-b^{2}\right ) d \sqrt {\frac {a +b \cos \left (d x +c \right )}{a +b}}}-\frac {8 a \left (4 a^{2}+b^{2}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {\frac {a +b \cos \left (d x +c \right )}{a +b}}}{5 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{4} d \sqrt {a +b \cos \left (d x +c \right )}} \]
command
integrate(cos(d*x+c)^4/(a+b*cos(d*x+c))^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {6 \, {\left (8 \, a^{4} b^{2} - 3 \, a^{2} b^{4} - {\left (a^{2} b^{4} - b^{6}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{3} b^{3} - a b^{5}\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^{5} b + 28 i \, a^{3} b^{3} + 9 i \, a b^{5}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-32 i \, a^{6} + 28 i \, a^{4} b^{2} + 9 i \, a^{2} b^{4}\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^{5} b - 28 i \, a^{3} b^{3} - 9 i \, a b^{5}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (32 i \, a^{6} - 28 i \, a^{4} b^{2} - 9 i \, a^{2} b^{4}\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 ) - 3 \, {\left (\sqrt {2} {\left (16 i \, a^{4} b^{2} - 8 i \, a^{2} b^{4} - 3 i \, b^{6}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (16 i \, a^{5} b - 8 i \, a^{3} b^{3} - 3 i \, a 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 ) - 3 \, {\left (\sqrt {2} {\left (-16 i \, a^{4} b^{2} + 8 i \, a^{2} b^{4} + 3 i \, b^{6}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-16 i \, a^{5} b + 8 i \, a^{3} b^{3} + 3 i \, a 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 )}{15 \, {\left ({\left (a^{2} b^{6} - b^{8}\right )} d \cos \left (d x + c\right ) + {\left (a^{3} b^{5} - a b^{7}\right )} d\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {b \cos \left (d x + c\right ) + a} \cos \left (d x + c\right )^{4}}{b^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + a^{2}}, x\right ) \]