\[ \int \frac {x}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx \]
Optimal antiderivative \[ -\frac {\ln \left (x^{2}+3\right ) 2^{\frac {1}{3}}}{8}+\frac {3 \ln \left (2^{\frac {2}{3}}-\left (-x^{2}+1\right )^{\frac {1}{3}}\right ) 2^{\frac {1}{3}}}{8}+\frac {\arctan \left (\frac {\left (1+\left (-2 x^{2}+2\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3}\right ) \sqrt {3}\, 2^{\frac {1}{3}}}{4} \]
command
integrate(x/(-x**2+1)**(1/3)/(x**2+3),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} \sqrt [3]{2} \left (\frac {\log {\left (\sqrt [3]{2 - 2 x^{2}} - 2 \right )}}{4} - \frac {\log {\left (\left (2 - 2 x^{2}\right )^{\frac {2}{3}} + 2 \sqrt [3]{2 - 2 x^{2}} + 4 \right )}}{8} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {\sqrt {3} \left (\sqrt [3]{2 - 2 x^{2}} + 1\right )}{3} \right )}}{4}\right ) & \text {for}\: x > -1 \wedge x < 1 \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \int \frac {x}{\sqrt [3]{- \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + 3\right )}\, dx \]________________________________________________________________________________________