\[ \int \frac {1+3 x}{(-1+3 x) \sqrt [3]{-x+x^3}} \, dx \]
Optimal antiderivative \[ -\sqrt {3}\, \arctan \! \left (\frac {\sqrt {3}\, \left (x^{3}-x \right )^{\frac {1}{3}}}{-2+2 x +\left (x^{3}-x \right )^{\frac {1}{3}}}\right )-\ln \! \left (1-x +\left (x^{3}-x \right )^{\frac {1}{3}}\right )+\frac {\ln \! \left (1-2 x +x^{2}+\left (-1+x \right ) \left (x^{3}-x \right )^{\frac {1}{3}}+\left (x^{3}-x \right )^{\frac {2}{3}}\right )}{2} \]
command
Int[(1 + 3*x)/((-1 + 3*x)*(-x + x^3)^(1/3)),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \int \frac {1+3 x}{(-1+3 x) \sqrt [3]{-x+x^3}} \, dx \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ -\frac {18 \sqrt [3]{1-x^2} x^2 \operatorname {AppellF1}\left (\frac {5}{6},\frac {1}{3},1,\frac {11}{6},x^2,9 x^2\right )}{5 \sqrt [3]{x^3-x}}+\frac {\sqrt {3} \sqrt [3]{x^2-1} \sqrt [3]{x} \arctan \left (\frac {1-\frac {4 x^{2/3}}{\sqrt [3]{x^2-1}}}{\sqrt {3}}\right )}{2 \sqrt [3]{x^3-x}}+\frac {\sqrt {3} \sqrt [3]{x^2-1} \sqrt [3]{x} \arctan \left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2-1}}+1}{\sqrt {3}}\right )}{2 \sqrt [3]{x^3-x}}+\frac {\sqrt [3]{x^2-1} \sqrt [3]{x} \log \left (1-9 x^2\right )}{4 \sqrt [3]{x^3-x}}-\frac {3 \sqrt [3]{x^2-1} \sqrt [3]{x} \log \left (x^{2/3}-\sqrt [3]{x^2-1}\right )}{4 \sqrt [3]{x^3-x}}-\frac {3 \sqrt [3]{x^2-1} \sqrt [3]{x} \log \left (2 x^{2/3}+\sqrt [3]{x^2-1}\right )}{4 \sqrt [3]{x^3-x}} \]