\[ \int \frac {-4 x-2 e^3 x-6 x^2+\left (10+4 e^3\right ) \log (3)+\left (-6 x-2 e^3 x+6 x^2+\left (6+2 e^3-6 x\right ) \log (3)\right ) \log (x-\log (3))+\log (2 x) (-x+2 \log (3)+(-x+\log (3)) \log (x-\log (3)))+\left (-6 x-2 e^3 x+6 x^2+\left (6+2 e^3-6 x\right ) \log (3)+(-x+\log (3)) \log (2 x)\right ) \log \left (\frac {36+4 e^6+e^3 (24-24 x)-72 x+36 x^2+\left (12+4 e^3-12 x\right ) \log (2 x)+\log ^2(2 x)}{4 x^2}\right )}{\left (-6 x-2 e^3 x+6 x^2+\left (6+2 e^3-6 x\right ) \log (3)\right ) \log ^2(x-\log (3))+(-x+\log (3)) \log (2 x) \log ^2(x-\log (3))+\left (\left (-12 x-4 e^3 x+12 x^2+\left (12+4 e^3-12 x\right ) \log (3)\right ) \log (x-\log (3))+(-2 x+2 \log (3)) \log (2 x) \log (x-\log (3))\right ) \log \left (\frac {36+4 e^6+e^3 (24-24 x)-72 x+36 x^2+\left (12+4 e^3-12 x\right ) \log (2 x)+\log ^2(2 x)}{4 x^2}\right )+\left (-6 x-2 e^3 x+6 x^2+\left (6+2 e^3-6 x\right ) \log (3)+(-x+\log (3)) \log (2 x)\right ) \log ^2\left (\frac {36+4 e^6+e^3 (24-24 x)-72 x+36 x^2+\left (12+4 e^3-12 x\right ) \log (2 x)+\log ^2(2 x)}{4 x^2}\right )} \, dx \]
Optimal antiderivative \[ \frac {x}{\ln \! \left (-\ln \! \left (3\right )+x \right )+\ln \! \left (\left (\frac {{\mathrm e}^{3}+\frac {\ln \left (2 x \right )}{2}+3}{x}-3\right )^{2}\right )} \]
command
Integrate[(-4*x - 2*E^3*x - 6*x^2 + (10 + 4*E^3)*Log[3] + (-6*x - 2*E^3*x + 6*x^2 + (6 + 2*E^3 - 6*x)*Log[3])*Log[x - Log[3]] + Log[2*x]*(-x + 2*Log[3] + (-x + Log[3])*Log[x - Log[3]]) + (-6*x - 2*E^3*x + 6*x^2 + (6 + 2*E^3 - 6*x)*Log[3] + (-x + Log[3])*Log[2*x])*Log[(36 + 4*E^6 + E^3*(24 - 24*x) - 72*x + 36*x^2 + (12 + 4*E^3 - 12*x)*Log[2*x] + Log[2*x]^2)/(4*x^2)])/((-6*x - 2*E^3*x + 6*x^2 + (6 + 2*E^3 - 6*x)*Log[3])*Log[x - Log[3]]^2 + (-x + Log[3])*Log[2*x]*Log[x - Log[3]]^2 + ((-12*x - 4*E^3*x + 12*x^2 + (12 + 4*E^3 - 12*x)*Log[3])*Log[x - Log[3]] + (-2*x + 2*Log[3])*Log[2*x]*Log[x - Log[3]])*Log[(36 + 4*E^6 + E^3*(24 - 24*x) - 72*x + 36*x^2 + (12 + 4*E^3 - 12*x)*Log[2*x] + Log[2*x]^2)/(4*x^2)] + (-6*x - 2*E^3*x + 6*x^2 + (6 + 2*E^3 - 6*x)*Log[3] + (-x + Log[3])*Log[2*x])*Log[(36 + 4*E^6 + E^3*(24 - 24*x) - 72*x + 36*x^2 + (12 + 4*E^3 - 12*x)*Log[2*x] + Log[2*x]^2)/(4*x^2)]^2),x]
Mathematica 13.1 output
\[ \int \frac {-4 x-2 e^3 x-6 x^2+\left (10+4 e^3\right ) \log (3)+\left (-6 x-2 e^3 x+6 x^2+\left (6+2 e^3-6 x\right ) \log (3)\right ) \log (x-\log (3))+\log (2 x) (-x+2 \log (3)+(-x+\log (3)) \log (x-\log (3)))+\left (-6 x-2 e^3 x+6 x^2+\left (6+2 e^3-6 x\right ) \log (3)+(-x+\log (3)) \log (2 x)\right ) \log \left (\frac {36+4 e^6+e^3 (24-24 x)-72 x+36 x^2+\left (12+4 e^3-12 x\right ) \log (2 x)+\log ^2(2 x)}{4 x^2}\right )}{\left (-6 x-2 e^3 x+6 x^2+\left (6+2 e^3-6 x\right ) \log (3)\right ) \log ^2(x-\log (3))+(-x+\log (3)) \log (2 x) \log ^2(x-\log (3))+\left (\left (-12 x-4 e^3 x+12 x^2+\left (12+4 e^3-12 x\right ) \log (3)\right ) \log (x-\log (3))+(-2 x+2 \log (3)) \log (2 x) \log (x-\log (3))\right ) \log \left (\frac {36+4 e^6+e^3 (24-24 x)-72 x+36 x^2+\left (12+4 e^3-12 x\right ) \log (2 x)+\log ^2(2 x)}{4 x^2}\right )+\left (-6 x-2 e^3 x+6 x^2+\left (6+2 e^3-6 x\right ) \log (3)+(-x+\log (3)) \log (2 x)\right ) \log ^2\left (\frac {36+4 e^6+e^3 (24-24 x)-72 x+36 x^2+\left (12+4 e^3-12 x\right ) \log (2 x)+\log ^2(2 x)}{4 x^2}\right )} \, dx \]
Mathematica 12.3 output
\[ \frac {x}{\log \left (\frac {1}{4} (x-\log (3))\right )+\log \left (\frac {\left (6+2 e^3-6 x+\log (2 x)\right )^2}{x^2}\right )} \]