\[ \int \frac {3^{2/x} \left (x^4\right )^{-2/x} \left (-2 x^2+\left (-8 x+2 x^2\right ) \log (x)+2 x \log (x) \log \left (\frac {x^4}{3}\right )+3^{-1/x} \left (x^4\right )^{\frac {1}{x}} \left (40 x \log (x)+(160-40 x) \log ^2(x)-40 \log ^2(x) \log \left (\frac {x^4}{3}\right )\right )\right )}{x \log ^3(x)} \, dx \]
Optimal antiderivative \[ \left (20-\frac {x \,{\mathrm e}^{-\frac {\ln \left (\frac {x^{4}}{3}\right )}{x}}}{\ln \! \left (x \right )}\right )^{2} \]
command
Integrate[(3^(2/x)*(-2*x^2 + (-8*x + 2*x^2)*Log[x] + 2*x*Log[x]*Log[x^4/3] + ((x^4)^x^(-1)*(40*x*Log[x] + (160 - 40*x)*Log[x]^2 - 40*Log[x]^2*Log[x^4/3]))/3^x^(-1)))/(x*(x^4)^(2/x)*Log[x]^3),x]
Mathematica 13.1 output
\[ \int \frac {3^{2/x} \left (x^4\right )^{-2/x} \left (-2 x^2+\left (-8 x+2 x^2\right ) \log (x)+2 x \log (x) \log \left (\frac {x^4}{3}\right )+3^{-1/x} \left (x^4\right )^{\frac {1}{x}} \left (40 x \log (x)+(160-40 x) \log ^2(x)-40 \log ^2(x) \log \left (\frac {x^4}{3}\right )\right )\right )}{x \log ^3(x)} \, dx \]
Mathematica 12.3 output
\[ \frac {3^{\frac {1}{x}} x \left (x^4\right )^{-2/x} \left (3^{\frac {1}{x}} x-40 \left (x^4\right )^{\frac {1}{x}} \log (x)\right )}{\log ^2(x)} \]