\[ \int \frac {-1+\sqrt {3}-x}{\sqrt {1+x^3}} \, dx \]
Optimal antiderivative \[ -\frac {2 \sqrt {x^{3}+1}}{1+x +\sqrt {3}}+\frac {3^{\frac {1}{4}} \left (1+x \right ) \EllipticE \left (\frac {1+x -\sqrt {3}}{1+x +\sqrt {3}}, i \sqrt {3}+2 i\right ) \left (\frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {x^{2}-x +1}{\left (1+x +\sqrt {3}\right )^{2}}}}{\sqrt {x^{3}+1}\, \sqrt {\frac {1+x}{\left (1+x +\sqrt {3}\right )^{2}}}} \]
command
integrate((-1-x+3^(1/2))/(x^3+1)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ 2 \, {\left (\sqrt {3} - 1\right )} {\rm weierstrassPInverse}\left (0, -4, x\right ) + 2 \, {\rm weierstrassZeta}\left (0, -4, {\rm weierstrassPInverse}\left (0, -4, x\right )\right ) \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {x - \sqrt {3} + 1}{\sqrt {x^{3} + 1}}, x\right ) \]