\[ \int \frac {-32 \log (3)+32 \log (3) \log \left (\frac {x}{3}\right )+(-8+16 x) \log (3) \log ^2\left (\frac {x}{3}\right )-8 \log (3) \log ^2\left (\frac {x}{3}\right ) \log (x)}{\left (-4 x \log \left (\frac {x}{3}\right )-x^2 \log ^2\left (\frac {x}{3}\right )+x \log ^2\left (\frac {x}{3}\right ) \log (x)\right ) \log ^2\left (\frac {16 x^2+8 x^3 \log \left (\frac {x}{3}\right )+x^4 \log ^2\left (\frac {x}{3}\right )+\left (-8 x^2 \log \left (\frac {x}{3}\right )-2 x^3 \log ^2\left (\frac {x}{3}\right )\right ) \log (x)+x^2 \log ^2\left (\frac {x}{3}\right ) \log ^2(x)}{\log ^2\left (\frac {x}{3}\right )}\right )} \, dx \]
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((-8*log(3)*log(1/3*x)^2*log(x)+(16*x-8)*log(3)*log(1/3*x)^2+32*log(3)*log(1/3*x)-32*log(3))/(x*log(1/3*x)^2*log(x)-x^2*log(1/3*x)^2-4*x*log(1/3*x))/log((x^2*log(1/3*x)^2*log(x)^2+(-2*x^3*log(1/3*x)^2-8*x^2*log(1/3*x))*log(x)+x^4*log(1/3*x)^2+8*x^3*log(1/3*x)+16*x^2)/log(1/3*x)^2)^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {4 \, \log \left (3\right )}{\log \left (x^{2} \log \left (3\right )^{2} - 2 \, x^{2} \log \left (3\right ) \log \left (x\right ) - 2 \, x \log \left (3\right )^{2} \log \left (x\right ) + x^{2} \log \left (x\right )^{2} + 4 \, x \log \left (3\right ) \log \left (x\right )^{2} + \log \left (3\right )^{2} \log \left (x\right )^{2} - 2 \, x \log \left (x\right )^{3} - 2 \, \log \left (3\right ) \log \left (x\right )^{3} + \log \left (x\right )^{4} - 8 \, x \log \left (3\right ) + 8 \, x \log \left (x\right ) + 8 \, \log \left (3\right ) \log \left (x\right ) - 8 \, \log \left (x\right )^{2} + 16\right ) - \log \left (\log \left (3\right )^{2} - 2 \, \log \left (3\right ) \log \left (x\right ) + \log \left (x\right )^{2}\right ) + 2 \, \log \left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________