\[ \int \frac {-32 \log (3)+32 \log (3) \log \left (\frac {x}{3}\right )+(-8+16 x) \log (3) \log ^2\left (\frac {x}{3}\right )-8 \log (3) \log ^2\left (\frac {x}{3}\right ) \log (x)}{\left (-4 x \log \left (\frac {x}{3}\right )-x^2 \log ^2\left (\frac {x}{3}\right )+x \log ^2\left (\frac {x}{3}\right ) \log (x)\right ) \log ^2\left (\frac {16 x^2+8 x^3 \log \left (\frac {x}{3}\right )+x^4 \log ^2\left (\frac {x}{3}\right )+\left (-8 x^2 \log \left (\frac {x}{3}\right )-2 x^3 \log ^2\left (\frac {x}{3}\right )\right ) \log (x)+x^2 \log ^2\left (\frac {x}{3}\right ) \log ^2(x)}{\log ^2\left (\frac {x}{3}\right )}\right )} \, dx \]
Optimal antiderivative \[ \frac {4 \ln \! \left (3\right )}{\ln \! \left (x^{2} \left (\frac {4}{\ln \left (\frac {x}{3}\right )}-\ln \! \left (x \right )+x \right )^{2}\right )} \]
command
Integrate[(-32*Log[3] + 32*Log[3]*Log[x/3] + (-8 + 16*x)*Log[3]*Log[x/3]^2 - 8*Log[3]*Log[x/3]^2*Log[x])/((-4*x*Log[x/3] - x^2*Log[x/3]^2 + x*Log[x/3]^2*Log[x])*Log[(16*x^2 + 8*x^3*Log[x/3] + x^4*Log[x/3]^2 + (-8*x^2*Log[x/3] - 2*x^3*Log[x/3]^2)*Log[x] + x^2*Log[x/3]^2*Log[x]^2)/Log[x/3]^2]^2),x]
Mathematica 13.1 output
\[ \int \frac {-32 \log (3)+32 \log (3) \log \left (\frac {x}{3}\right )+(-8+16 x) \log (3) \log ^2\left (\frac {x}{3}\right )-8 \log (3) \log ^2\left (\frac {x}{3}\right ) \log (x)}{\left (-4 x \log \left (\frac {x}{3}\right )-x^2 \log ^2\left (\frac {x}{3}\right )+x \log ^2\left (\frac {x}{3}\right ) \log (x)\right ) \log ^2\left (\frac {16 x^2+8 x^3 \log \left (\frac {x}{3}\right )+x^4 \log ^2\left (\frac {x}{3}\right )+\left (-8 x^2 \log \left (\frac {x}{3}\right )-2 x^3 \log ^2\left (\frac {x}{3}\right )\right ) \log (x)+x^2 \log ^2\left (\frac {x}{3}\right ) \log ^2(x)}{\log ^2\left (\frac {x}{3}\right )}\right )} \, dx \]
Mathematica 12.3 output
\[ \frac {4 \log (3) \left (-4-\log (81)+\log ^2\left (\frac {x}{3}\right ) (-1+2 x-\log (x))+4 \log (x)\right )}{\left (-4+4 \log \left (\frac {x}{3}\right )+\log ^2\left (\frac {x}{3}\right ) (-1+2 x-\log (x))\right ) \log \left (\frac {x^2 \left (4+\log \left (\frac {x}{3}\right ) (x-\log (x))\right )^2}{\log ^2\left (\frac {x}{3}\right )}\right )} \]