\[ \int \frac {-11 x+e^{\sqrt [4]{e}} \left (11 x+12 x^2\right )+\left (2 x+e^{\sqrt [4]{e}} \left (-2 x-2 x^2\right )\right ) \log \left (x+e^{\sqrt [4]{e}} \left (-x-x^2\right )\right )}{-25-5 x^2+e^{\sqrt [4]{e}} \left (25+25 x+5 x^2+5 x^3\right )+\left (10+x^2+e^{\sqrt [4]{e}} \left (-10-10 x-x^2-x^3\right )\right ) \log \left (x+e^{\sqrt [4]{e}} \left (-x-x^2\right )\right )+\left (-1+e^{\sqrt [4]{e}} (1+x)\right ) \log ^2\left (x+e^{\sqrt [4]{e}} \left (-x-x^2\right )\right )} \, dx \]
Optimal antiderivative \[ \ln \! \left (\frac {x^{2}}{\ln \! \left (x -x \,{\mathrm e}^{{\mathrm e}^{\frac {1}{4}}} \left (1+x \right )\right )-5}-1\right ) \]
command
Int[(-11*x + E^E^(1/4)*(11*x + 12*x^2) + (2*x + E^E^(1/4)*(-2*x - 2*x^2))*Log[x + E^E^(1/4)*(-x - x^2)])/(-25 - 5*x^2 + E^E^(1/4)*(25 + 25*x + 5*x^2 + 5*x^3) + (10 + x^2 + E^E^(1/4)*(-10 - 10*x - x^2 - x^3))*Log[x + E^E^(1/4)*(-x - x^2)] + (-1 + E^E^(1/4)*(1 + x))*Log[x + E^E^(1/4)*(-x - x^2)]^2),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \int \frac {-11 x+e^{\sqrt [4]{e}} \left (11 x+12 x^2\right )+\left (2 x+e^{\sqrt [4]{e}} \left (-2 x-2 x^2\right )\right ) \log \left (x+e^{\sqrt [4]{e}} \left (-x-x^2\right )\right )}{-25-5 x^2+e^{\sqrt [4]{e}} \left (25+25 x+5 x^2+5 x^3\right )+\left (10+x^2+e^{\sqrt [4]{e}} \left (-10-10 x-x^2-x^3\right )\right ) \log \left (x+e^{\sqrt [4]{e}} \left (-x-x^2\right )\right )+\left (-1+e^{\sqrt [4]{e}} (1+x)\right ) \log ^2\left (x+e^{\sqrt [4]{e}} \left (-x-x^2\right )\right )} \, dx \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \log \left (x^2-\log \left (x-e^{\sqrt [4]{e}} x (x+1)\right )+5\right )-\log \left (5-\log \left (x-e^{\sqrt [4]{e}} x (x+1)\right )\right ) \]