\[ \int \frac {\left (-2 x^3+10 x^4\right ) \log ^2\left (\frac {4}{x}\right )-8 x^3 \log ^2\left (\frac {4}{x}\right ) \log (x)+e^{\frac {1}{2} \left (-10+e^{\frac {3}{\log \left (\frac {4}{x}\right )}}\right )} \left (\left (-2 x+6 x^2\right ) \log ^2\left (\frac {4}{x}\right )-4 x \log ^2\left (\frac {4}{x}\right ) \log (x)+e^{\frac {3}{\log \left (\frac {4}{x}\right )}} \left (3 x^2-3 x \log (x)\right )\right )}{2 \log ^2\left (\frac {4}{x}\right )} \, dx \]
Optimal antiderivative \[ x^{2} \left (x -\ln \! \left (x \right )\right ) \left ({\mathrm e}^{\frac {{\mathrm e}^{\frac {3}{\ln \left (\frac {4}{x}\right )}}}{2}-5}+x^{2}\right ) \]
command
Int[((-2*x^3 + 10*x^4)*Log[4/x]^2 - 8*x^3*Log[4/x]^2*Log[x] + E^((-10 + E^(3/Log[4/x]))/2)*((-2*x + 6*x^2)*Log[4/x]^2 - 4*x*Log[4/x]^2*Log[x] + E^(3/Log[4/x])*(3*x^2 - 3*x*Log[x])))/(2*Log[4/x]^2),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \int \frac {\left (-2 x^3+10 x^4\right ) \log ^2\left (\frac {4}{x}\right )-8 x^3 \log ^2\left (\frac {4}{x}\right ) \log (x)+e^{\frac {1}{2} \left (-10+e^{\frac {3}{\log \left (\frac {4}{x}\right )}}\right )} \left (\left (-2 x+6 x^2\right ) \log ^2\left (\frac {4}{x}\right )-4 x \log ^2\left (\frac {4}{x}\right ) \log (x)+e^{\frac {3}{\log \left (\frac {4}{x}\right )}} \left (3 x^2-3 x \log (x)\right )\right )}{2 \log ^2\left (\frac {4}{x}\right )} \, dx \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ x^5-x^4 \log (x)+x^2 e^{\frac {1}{2} \left (e^{\frac {3}{\log \left (\frac {4}{x}\right )}}-10\right )} (x-\log (x)) \]