\[ \int \frac {1}{\sqrt [3]{-8+12 x+54 x^2-135 x^3+81 x^4}} \, dx \]
Optimal antiderivative \[ -\frac {\arctan \! \left (\frac {-2 \,3^{\frac {5}{6}}+3 x 3^{\frac {5}{6}}}{-2 \,3^{\frac {1}{3}}+3 \,3^{\frac {1}{3}} x +2 \left (81 x^{4}-135 x^{3}+54 x^{2}+12 x -8\right )^{\frac {1}{3}}}\right ) 3^{\frac {1}{6}}}{3}+\frac {\ln \! \left (6-9 x +3^{\frac {2}{3}} \left (81 x^{4}-135 x^{3}+54 x^{2}+12 x -8\right )^{\frac {1}{3}}\right ) 3^{\frac {2}{3}}}{9}-\frac {\ln \! \left (12-36 x +27 x^{2}+\left (-2 \,3^{\frac {2}{3}}+3 \,3^{\frac {2}{3}} x \right ) \left (81 x^{4}-135 x^{3}+54 x^{2}+12 x -8\right )^{\frac {1}{3}}+3^{\frac {1}{3}} \left (81 x^{4}-135 x^{3}+54 x^{2}+12 x -8\right )^{\frac {2}{3}}\right ) 3^{\frac {2}{3}}}{18} \]
command
Int[(-8 + 12*x + 54*x^2 - 135*x^3 + 81*x^4)^(-1/3),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \int \frac {1}{\sqrt [3]{-8+12 x+54 x^2-135 x^3+81 x^4}} \, dx \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ -\frac {(2-3 x) \sqrt [3]{3 x+1} \arctan \left (\frac {2 \sqrt [3]{3 x+1}+\sqrt [3]{3}}{3^{5/6}}\right )}{3^{5/6} \sqrt [3]{-(2-3 x)^3 (3 x+1)}}+\frac {(2-3 x) \sqrt [3]{3 x+1} \log (2-3 x)}{6 \sqrt [3]{3} \sqrt [3]{-(2-3 x)^3 (3 x+1)}}-\frac {(2-3 x) \sqrt [3]{3 x+1} \log \left (\sqrt [3]{3}-\sqrt [3]{3 x+1}\right )}{2 \sqrt [3]{3} \sqrt [3]{-(2-3 x)^3 (3 x+1)}} \]