\[ \int \frac {-x^3 \log (x)+8 e^{e^x+x} x^3 \log (x)+8 e^{2 e^x+x} x^3 \log (x)+\left (x^2-x^3+\left (-2 x^2+2 x^3\right ) \log (x)+e^{e^x} \left (8 x^2-16 x^2 \log (x)\right )+e^{2 e^x} \left (4 x^2-8 x^2 \log (x)\right )\right ) \log \left (\frac {1}{4} \left (-1-8 e^{e^x}-4 e^{2 e^x}+x\right )\right )+\left (2+16 e^{e^x}+8 e^{2 e^x}-2 x\right ) \log ^2\left (\frac {1}{4} \left (-1-8 e^{e^x}-4 e^{2 e^x}+x\right )\right )}{x^5+8 e^{e^x} x^5+4 e^{2 e^x} x^5-x^6+\left (4 x^3+32 e^{e^x} x^3+16 e^{2 e^x} x^3-4 x^4\right ) \log \left (\frac {1}{4} \left (-1-8 e^{e^x}-4 e^{2 e^x}+x\right )\right )+\left (4 x+32 e^{e^x} x+16 e^{2 e^x} x-4 x^2\right ) \log ^2\left (\frac {1}{4} \left (-1-8 e^{e^x}-4 e^{2 e^x}+x\right )\right )} \, dx \]
Optimal antiderivative \[ \frac {\ln \! \left (x \right )}{2+\frac {x^{2}}{\ln \left (\frac {3}{4}+\frac {x}{4}-\left ({\mathrm e}^{{\mathrm e}^{x}}+1\right )^{2}\right )}} \]
command
Integrate[(-(x^3*Log[x]) + 8*E^(E^x + x)*x^3*Log[x] + 8*E^(2*E^x + x)*x^3*Log[x] + (x^2 - x^3 + (-2*x^2 + 2*x^3)*Log[x] + E^E^x*(8*x^2 - 16*x^2*Log[x]) + E^(2*E^x)*(4*x^2 - 8*x^2*Log[x]))*Log[(-1 - 8*E^E^x - 4*E^(2*E^x) + x)/4] + (2 + 16*E^E^x + 8*E^(2*E^x) - 2*x)*Log[(-1 - 8*E^E^x - 4*E^(2*E^x) + x)/4]^2)/(x^5 + 8*E^E^x*x^5 + 4*E^(2*E^x)*x^5 - x^6 + (4*x^3 + 32*E^E^x*x^3 + 16*E^(2*E^x)*x^3 - 4*x^4)*Log[(-1 - 8*E^E^x - 4*E^(2*E^x) + x)/4] + (4*x + 32*E^E^x*x + 16*E^(2*E^x)*x - 4*x^2)*Log[(-1 - 8*E^E^x - 4*E^(2*E^x) + x)/4]^2),x]
Mathematica 13.1 output
\[ \int \frac {-x^3 \log (x)+8 e^{e^x+x} x^3 \log (x)+8 e^{2 e^x+x} x^3 \log (x)+\left (x^2-x^3+\left (-2 x^2+2 x^3\right ) \log (x)+e^{e^x} \left (8 x^2-16 x^2 \log (x)\right )+e^{2 e^x} \left (4 x^2-8 x^2 \log (x)\right )\right ) \log \left (\frac {1}{4} \left (-1-8 e^{e^x}-4 e^{2 e^x}+x\right )\right )+\left (2+16 e^{e^x}+8 e^{2 e^x}-2 x\right ) \log ^2\left (\frac {1}{4} \left (-1-8 e^{e^x}-4 e^{2 e^x}+x\right )\right )}{x^5+8 e^{e^x} x^5+4 e^{2 e^x} x^5-x^6+\left (4 x^3+32 e^{e^x} x^3+16 e^{2 e^x} x^3-4 x^4\right ) \log \left (\frac {1}{4} \left (-1-8 e^{e^x}-4 e^{2 e^x}+x\right )\right )+\left (4 x+32 e^{e^x} x+16 e^{2 e^x} x-4 x^2\right ) \log ^2\left (\frac {1}{4} \left (-1-8 e^{e^x}-4 e^{2 e^x}+x\right )\right )} \, dx \]
Mathematica 12.3 output
\[ \frac {\log (x) \left (2 \left (8 e^{e^x+x}+8 e^{2 e^x+x}+8 e^{e^x} x+4 e^{2 e^x} x-x^2\right ) \log \left (\frac {1}{4} \left (-1-8 e^{e^x}-4 e^{2 e^x}+x\right )\right )-(-1+x) \left (\log (16)-2 \log \left (-1-8 e^{e^x}-4 e^{2 e^x}+x\right )\right )\right )}{2 \left (-1+8 e^{e^x+x}+8 e^{2 e^x+x}+x+8 e^{e^x} x+4 e^{2 e^x} x-x^2\right ) \left (x^2+2 \log \left (\frac {1}{4} \left (-1-8 e^{e^x}-4 e^{2 e^x}+x\right )\right )\right )} \]