\[ \int \frac {-4 x^2+2 e^{1+x} x^2-2 x^4+\left (4 x^2+6 x^4+e^{1+x} \left (-2 x^2-2 x^3\right )\right ) \log (x)+\left (4-2 x^2+2 x^3+e^{1+x} \left (-2+2 x-x^2\right )\right ) \log ^2(x)}{6 x^2 \log ^2(x)} \, dx \]
Optimal antiderivative \[ \left (x -\frac {{\mathrm e}^{1+x}-2}{x}\right ) \left (\frac {x^{2}}{3 \ln \left (x \right )}-\frac {1}{3}+\frac {x}{6}\right ) \]
command
Int[(-4*x^2 + 2*E^(1 + x)*x^2 - 2*x^4 + (4*x^2 + 6*x^4 + E^(1 + x)*(-2*x^2 - 2*x^3))*Log[x] + (4 - 2*x^2 + 2*x^3 + E^(1 + x)*(-2 + 2*x - x^2))*Log[x]^2)/(6*x^2*Log[x]^2),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \frac {x^3}{3 \log (x)}+\frac {x^2}{6}-\frac {e^{x+1} \left (2 x^3 \log (x)+x^2 \log ^2(x)-2 x \log ^2(x)\right )}{6 x^2 \log ^2(x)}-\frac {x}{3}-\frac {2}{3 x}+\frac {2 x}{3 \log (x)} \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \int \frac {-4 x^2+2 e^{1+x} x^2-2 x^4+\left (4 x^2+6 x^4+e^{1+x} \left (-2 x^2-2 x^3\right )\right ) \log (x)+\left (4-2 x^2+2 x^3+e^{1+x} \left (-2+2 x-x^2\right )\right ) \log ^2(x)}{6 x^2 \log ^2(x)} \, dx \]________________________________________________________________________________________