\[ \int \frac {\cos ^3(c+d x)}{\sqrt {3+4 \cos (c+d x)}} \, dx \]
Optimal antiderivative \[ -\frac {23 \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \frac {2 \sqrt {14}}{7}\right ) \sqrt {7}}{140 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {9 \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \frac {2 \sqrt {14}}{7}\right ) \sqrt {7}}{20 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}-\frac {\sin \left (d x +c \right ) \sqrt {3+4 \cos \left (d x +c \right )}}{10 d}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right ) \sqrt {3+4 \cos \left (d x +c \right )}}{10 d} \]
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
\[ \frac {4 \, \sqrt {4 \, \cos \left (d x + c\right ) + 3} {\left (\cos \left (d x + c\right ) - 1\right )} \sin \left (d x + c\right ) + 7 i \, \sqrt {2} {\rm weierstrassPInverse}\left (-1, 1, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + \frac {1}{2}\right ) - 7 i \, \sqrt {2} {\rm weierstrassPInverse}\left (-1, 1, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) + \frac {1}{2}\right ) + 18 i \, \sqrt {2} {\rm weierstrassZeta}\left (-1, 1, {\rm weierstrassPInverse}\left (-1, 1, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + \frac {1}{2}\right )\right ) - 18 i \, \sqrt {2} {\rm weierstrassZeta}\left (-1, 1, {\rm weierstrassPInverse}\left (-1, 1, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) + \frac {1}{2}\right )\right )}{40 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\cos \left (d x + c\right )^{3}}{\sqrt {4 \, \cos \left (d x + c\right ) + 3}}, x\right ) \]