\[ \int \frac {e^{\frac {1}{x}} \left (-104 x+26 x^2-2 x^4+2 x^5+\left (-52 x-x^4\right ) \log (3)\right )+e^{\frac {1}{x}} \left (52-26 x-2 x^3+x^4+\left (26-x^3\right ) \log (3)\right ) \log \left (\frac {52-26 x-2 x^3+x^4+\left (26-x^3\right ) \log (3)}{x^2}\right )}{\left (-52 x^2+26 x^3+2 x^5-x^6+\left (-26 x^2+x^5\right ) \log (3)\right ) \log ^2\left (\frac {52-26 x-2 x^3+x^4+\left (26-x^3\right ) \log (3)}{x^2}\right )} \, dx \]
Optimal antiderivative \[ \frac {{\mathrm e}^{\frac {1}{x}}}{\ln \! \left (\left (\frac {26}{x^{2}}-x \right ) \left (\ln \! \left (3\right )-x +2\right )\right )} \]
command
Int[(E^x^(-1)*(-104*x + 26*x^2 - 2*x^4 + 2*x^5 + (-52*x - x^4)*Log[3]) + E^x^(-1)*(52 - 26*x - 2*x^3 + x^4 + (26 - x^3)*Log[3])*Log[(52 - 26*x - 2*x^3 + x^4 + (26 - x^3)*Log[3])/x^2])/((-52*x^2 + 26*x^3 + 2*x^5 - x^6 + (-26*x^2 + x^5)*Log[3])*Log[(52 - 26*x - 2*x^3 + x^4 + (26 - x^3)*Log[3])/x^2]^2),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \int \frac {e^{\frac {1}{x}} \left (-104 x+26 x^2-2 x^4+2 x^5+\left (-52 x-x^4\right ) \log (3)\right )+e^{\frac {1}{x}} \left (52-26 x-2 x^3+x^4+\left (26-x^3\right ) \log (3)\right ) \log \left (\frac {52-26 x-2 x^3+x^4+\left (26-x^3\right ) \log (3)}{x^2}\right )}{\left (-52 x^2+26 x^3+2 x^5-x^6+\left (-26 x^2+x^5\right ) \log (3)\right ) \log ^2\left (\frac {52-26 x-2 x^3+x^4+\left (26-x^3\right ) \log (3)}{x^2}\right )} \, dx \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \frac {e^{\frac {1}{x}}}{\log \left (\frac {\left (26-x^3\right ) (-x+2+\log (3))}{x^2}\right )} \]