\[ \int \cos ^3(c+d x) \sqrt {3+4 \cos (c+d x)} \, dx \]
Optimal antiderivative \[ -\frac {3 \left (3+4 \cos \left (d x +c \right )\right )^{\frac {3}{2}} \sin \left (d x +c \right )}{70 d}+\frac {\cos \left (d x +c \right ) \left (3+4 \cos \left (d x +c \right )\right )^{\frac {3}{2}} \sin \left (d x +c \right )}{14 d}+\frac {47 \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \frac {2 \sqrt {14}}{7}\right ) \sqrt {7}}{140 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {59 \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \frac {2 \sqrt {14}}{7}\right ) \sqrt {7}}{420 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {59 \sin \left (d x +c \right ) \sqrt {3+4 \cos \left (d x +c \right )}}{105 d} \]
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
\[ \frac {4 \, {\left (60 \, \cos \left (d x + c\right )^{2} + 9 \, \cos \left (d x + c\right ) + 91\right )} \sqrt {4 \, \cos \left (d x + c\right ) + 3} \sin \left (d x + c\right ) - 277 i \, \sqrt {2} {\rm weierstrassPInverse}\left (-1, 1, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + \frac {1}{2}\right ) + 277 i \, \sqrt {2} {\rm weierstrassPInverse}\left (-1, 1, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) + \frac {1}{2}\right ) + 282 i \, \sqrt {2} {\rm weierstrassZeta}\left (-1, 1, {\rm weierstrassPInverse}\left (-1, 1, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + \frac {1}{2}\right )\right ) - 282 i \, \sqrt {2} {\rm weierstrassZeta}\left (-1, 1, {\rm weierstrassPInverse}\left (-1, 1, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) + \frac {1}{2}\right )\right )}{840 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\sqrt {4 \, \cos \left (d x + c\right ) + 3} \cos \left (d x + c\right )^{3}, x\right ) \]