\[ \int \sqrt {3-4 \cos (c+d x)} \cos ^3(c+d x) \, dx \]
Optimal antiderivative \[ -\frac {3 \left (3-4 \cos \left (d x +c \right )\right )^{\frac {3}{2}} \sin \left (d x +c \right )}{70 d}-\frac {\left (3-4 \cos \left (d x +c \right )\right )^{\frac {3}{2}} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{14 d}+\frac {47 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \frac {2 \sqrt {14}}{7}\right ) \sqrt {7}}{140 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {59 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \frac {2 \sqrt {14}}{7}\right ) \sqrt {7}}{420 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {59 \sin \left (d x +c \right ) \sqrt {3-4 \cos \left (d x +c \right )}}{105 d} \]
command
integrate(cos(d*x+c)^3*(3-4*cos(d*x+c))^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {4 \, {\left (60 \, \cos \left (d x + c\right )^{2} - 9 \, \cos \left (d x + c\right ) + 91\right )} \sqrt {-4 \, \cos \left (d x + c\right ) + 3} \sin \left (d x + c\right ) + 277 \, \sqrt {2} {\rm weierstrassPInverse}\left (-1, -1, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) - \frac {1}{2}\right ) + 277 \, \sqrt {2} {\rm weierstrassPInverse}\left (-1, -1, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) - \frac {1}{2}\right ) + 282 \, \sqrt {2} {\rm weierstrassZeta}\left (-1, -1, {\rm weierstrassPInverse}\left (-1, -1, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) - \frac {1}{2}\right )\right ) + 282 \, \sqrt {2} {\rm weierstrassZeta}\left (-1, -1, {\rm weierstrassPInverse}\left (-1, -1, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) - \frac {1}{2}\right )\right )}{840 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\sqrt {-4 \, \cos \left (d x + c\right ) + 3} \cos \left (d x + c\right )^{3}, x\right ) \]