\[ \int \frac {4 e^{e^5} x+2 x^2+\left (-10 x^2-2 x^3-5 x^4-x^5+\left (-2 x^2-x^4\right ) \log (3)+e^{2 e^5} \left (-20 x^2-4 x^3-4 x^2 \log (3)\right )+e^{e^5} \left (-20 x^3-4 x^4-4 x^3 \log (3)\right )\right ) \log (5+x+\log (3))+\left (10 x^3+2 x^4+2 x^3 \log (3)+e^{2 e^5} \left (40 x+8 x^2+8 x \log (3)\right )+e^{e^5} \left (-20-4 x+40 x^2+8 x^3+\left (-4+8 x^2\right ) \log (3)\right )\right ) \log (5+x+\log (3)) \log (\log (5+x+\log (3)))+\left (-5 x^2-x^3+e^{2 e^5} (-20-4 x-4 \log (3))-x^2 \log (3)+e^{e^5} \left (-20 x-4 x^2-4 x \log (3)\right )\right ) \log (5+x+\log (3)) \log ^2(\log (5+x+\log (3)))}{\left (5 x^4+x^5+x^4 \log (3)+e^{2 e^5} \left (20 x^2+4 x^3+4 x^2 \log (3)\right )+e^{e^5} \left (20 x^3+4 x^4+4 x^3 \log (3)\right )\right ) \log (5+x+\log (3))+\left (-10 x^3-2 x^4-2 x^3 \log (3)+e^{2 e^5} \left (-40 x-8 x^2-8 x \log (3)\right )+e^{e^5} \left (-40 x^2-8 x^3-8 x^2 \log (3)\right )\right ) \log (5+x+\log (3)) \log (\log (5+x+\log (3)))+\left (5 x^2+x^3+x^2 \log (3)+e^{2 e^5} (20+4 x+4 \log (3))+e^{e^5} \left (20 x+4 x^2+4 x \log (3)\right )\right ) \log (5+x+\log (3)) \log ^2(\log (5+x+\log (3)))} \, dx \]
Optimal antiderivative \[ -1+\frac {x}{\left ({\mathrm e}^{{\mathrm e}^{5}}+\frac {x}{2}\right ) \left (x -\ln \! \left (\ln \! \left (\ln \! \left (3\right )+5+x \right )\right )\right )}-x \]
command
Integrate[(4*E^E^5*x + 2*x^2 + (-10*x^2 - 2*x^3 - 5*x^4 - x^5 + (-2*x^2 - x^4)*Log[3] + E^(2*E^5)*(-20*x^2 - 4*x^3 - 4*x^2*Log[3]) + E^E^5*(-20*x^3 - 4*x^4 - 4*x^3*Log[3]))*Log[5 + x + Log[3]] + (10*x^3 + 2*x^4 + 2*x^3*Log[3] + E^(2*E^5)*(40*x + 8*x^2 + 8*x*Log[3]) + E^E^5*(-20 - 4*x + 40*x^2 + 8*x^3 + (-4 + 8*x^2)*Log[3]))*Log[5 + x + Log[3]]*Log[Log[5 + x + Log[3]]] + (-5*x^2 - x^3 + E^(2*E^5)*(-20 - 4*x - 4*Log[3]) - x^2*Log[3] + E^E^5*(-20*x - 4*x^2 - 4*x*Log[3]))*Log[5 + x + Log[3]]*Log[Log[5 + x + Log[3]]]^2)/((5*x^4 + x^5 + x^4*Log[3] + E^(2*E^5)*(20*x^2 + 4*x^3 + 4*x^2*Log[3]) + E^E^5*(20*x^3 + 4*x^4 + 4*x^3*Log[3]))*Log[5 + x + Log[3]] + (-10*x^3 - 2*x^4 - 2*x^3*Log[3] + E^(2*E^5)*(-40*x - 8*x^2 - 8*x*Log[3]) + E^E^5*(-40*x^2 - 8*x^3 - 8*x^2*Log[3]))*Log[5 + x + Log[3]]*Log[Log[5 + x + Log[3]]] + (5*x^2 + x^3 + x^2*Log[3] + E^(2*E^5)*(20 + 4*x + 4*Log[3]) + E^E^5*(20*x + 4*x^2 + 4*x*Log[3]))*Log[5 + x + Log[3]]*Log[Log[5 + x + Log[3]]]^2),x]
Mathematica 13.1 output
\[ \int \frac {4 e^{e^5} x+2 x^2+\left (-10 x^2-2 x^3-5 x^4-x^5+\left (-2 x^2-x^4\right ) \log (3)+e^{2 e^5} \left (-20 x^2-4 x^3-4 x^2 \log (3)\right )+e^{e^5} \left (-20 x^3-4 x^4-4 x^3 \log (3)\right )\right ) \log (5+x+\log (3))+\left (10 x^3+2 x^4+2 x^3 \log (3)+e^{2 e^5} \left (40 x+8 x^2+8 x \log (3)\right )+e^{e^5} \left (-20-4 x+40 x^2+8 x^3+\left (-4+8 x^2\right ) \log (3)\right )\right ) \log (5+x+\log (3)) \log (\log (5+x+\log (3)))+\left (-5 x^2-x^3+e^{2 e^5} (-20-4 x-4 \log (3))-x^2 \log (3)+e^{e^5} \left (-20 x-4 x^2-4 x \log (3)\right )\right ) \log (5+x+\log (3)) \log ^2(\log (5+x+\log (3)))}{\left (5 x^4+x^5+x^4 \log (3)+e^{2 e^5} \left (20 x^2+4 x^3+4 x^2 \log (3)\right )+e^{e^5} \left (20 x^3+4 x^4+4 x^3 \log (3)\right )\right ) \log (5+x+\log (3))+\left (-10 x^3-2 x^4-2 x^3 \log (3)+e^{2 e^5} \left (-40 x-8 x^2-8 x \log (3)\right )+e^{e^5} \left (-40 x^2-8 x^3-8 x^2 \log (3)\right )\right ) \log (5+x+\log (3)) \log (\log (5+x+\log (3)))+\left (5 x^2+x^3+x^2 \log (3)+e^{2 e^5} (20+4 x+4 \log (3))+e^{e^5} \left (20 x+4 x^2+4 x \log (3)\right )\right ) \log (5+x+\log (3)) \log ^2(\log (5+x+\log (3)))} \, dx \]
Mathematica 12.3 output
\[ x \left (-1+\frac {-2 \left (2 e^{e^5}+x\right )+\left (x (10+2 x+\log (9))+e^{e^5} (20+4 x+\log (81))\right ) \log (5+x+\log (3))}{\left (2 e^{e^5}+x\right )^2 (-1+(5+x+\log (3)) \log (5+x+\log (3))) (x-\log (\log (5+x+\log (3))))}\right ) \]