\[ \int \frac {2 e^x x-4 x^3+\left (-4 x+4 x^2\right ) \log (16)+\left (-4 x^2+e^x (2 x-2 \log (16))+4 x \log (16)\right ) \log \left (\frac {e^{-x} \left (e^x-2 x\right )}{-2 x+2 \log (16)}\right )}{2 x^4-2 x^3 \log (16)+e^x \left (-x^3+x^2 \log (16)\right )} \, dx \]
Optimal antiderivative \[ \frac {2 \ln \! \left (\frac {\left (\frac {{\mathrm e}^{x}}{2}-x \right ) {\mathrm e}^{-x}}{4 \ln \left (2\right )-x}\right )}{x} \]
command
Integrate[(2*E^x*x - 4*x^3 + (-4*x + 4*x^2)*Log[16] + (-4*x^2 + E^x*(2*x - 2*Log[16]) + 4*x*Log[16])*Log[(E^x - 2*x)/(E^x*(-2*x + 2*Log[16]))])/(2*x^4 - 2*x^3*Log[16] + E^x*(-x^3 + x^2*Log[16])),x]
Mathematica 13.1 output
\[ \int \frac {2 e^x x-4 x^3+\left (-4 x+4 x^2\right ) \log (16)+\left (-4 x^2+e^x (2 x-2 \log (16))+4 x \log (16)\right ) \log \left (\frac {e^{-x} \left (e^x-2 x\right )}{-2 x+2 \log (16)}\right )}{2 x^4-2 x^3 \log (16)+e^x \left (-x^3+x^2 \log (16)\right )} \, dx \]
Mathematica 12.3 output
\[ \frac {2 \log (x)}{\log (16)}-\frac {4 \log (x)}{\log (256)}-\frac {2 \log (x-\log (16))}{\log (16)}+\frac {2 \log \left (-\frac {e^{-x} \left (e^x-2 x\right )}{2 x-\log (256)}\right )}{x}+\frac {4 \log (2 x-\log (256))}{\log (256)} \]