\[ \int \frac {-10 e^{\frac {1}{x}} x \log \left (\frac {1}{x^2}\right )+\left (-20 e^{\frac {1}{x}} x+e^{\frac {1}{x}} (-10-10 x) \log \left (\frac {1}{x^2}\right )\right ) \log (x)}{x^3 \log ^2(x)} \, dx \]
Optimal antiderivative \[ \frac {10 \,{\mathrm e}^{\frac {1}{x}} \ln \left (\frac {1}{x^{2}}\right )}{x \ln \left (x \right )} \]
command
Int[(-10*E^x^(-1)*x*Log[x^(-2)] + (-20*E^x^(-1)*x + E^x^(-1)*(-10 - 10*x)*Log[x^(-2)])*Log[x])/(x^3*Log[x]^2),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \frac {10 e^{\frac {1}{x}} \log \left (\frac {1}{x^2}\right )}{x \log (x)} \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \int \frac {-10 e^{\frac {1}{x}} x \log \left (\frac {1}{x^2}\right )+\left (-20 e^{\frac {1}{x}} x+e^{\frac {1}{x}} (-10-10 x) \log \left (\frac {1}{x^2}\right )\right ) \log (x)}{x^3 \log ^2(x)} \, dx \]________________________________________________________________________________________