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