\[ \int \frac {\frac {e^{4-2 e^x} \left (2 x+\left (2-2 x^2\right ) \log (4)+e^x \left (2 x^2+\left (2 x-2 x^3\right ) \log (4)\right )\right )}{x^2}+\left (-2 x+4 x^2 \log (4)\right ) \log \left (-x+\left (-1+x^2\right ) \log (4)\right )+\frac {e^{2-e^x} \left (2 x-4 x^2 \log (4)+\left (-2 x+\left (-2+2 x^2\right ) \log (4)+e^x \left (-2 x^2+\left (-2 x+2 x^3\right ) \log (4)\right )\right ) \log \left (-x+\left (-1+x^2\right ) \log (4)\right )\right )}{x}}{-9 x^2+\left (-9 x+9 x^3\right ) \log (4)} \, dx \]
Optimal antiderivative \[ \frac {\left ({\mathrm e}^{-\ln \left (x \right )-{\mathrm e}^{x}+2}-\ln \left (2 \left (x^{2}-1\right ) \ln \left (2\right )-x \right )\right ) \left (\frac {{\mathrm e}^{-\ln \left (x \right )-{\mathrm e}^{x}+2}}{3}-\frac {\ln \left (2 \left (x^{2}-1\right ) \ln \left (2\right )-x \right )}{3}\right )}{3} \]
command
integrate((((2*(-2*x^3+2*x)*log(2)+2*x^2)*exp(x)+2*(-2*x^2+2)*log(2)+2*x)*exp(-log(x)-exp(x)+2)^2+(((2*(2*x^3-2*x)*log(2)-2*x^2)*exp(x)+2*(2*x^2-2)*log(2)-2*x)*log(2*(x^2-1)*log(2)-x)-8*x^2*log(2)+2*x)*exp(-log(x)-exp(x)+2)+(8*x^2*log(2)-2*x)*log(2*(x^2-1)*log(2)-x))/(2*(9*x^3-9*x)*log(2)-9*x^2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {x^{2} \log \left (2 \, x^{2} \log \left (2\right ) - x - 2 \, \log \left (2\right )\right )^{2} - 2 \, x e^{\left (-e^{x} + 2\right )} \log \left (2 \, x^{2} \log \left (2\right ) - x - 2 \, \log \left (2\right )\right ) + e^{\left (-2 \, e^{x} + 4\right )}}{9 \, x^{2}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {2 \, {\left ({\left (4 \, x^{2} \log \left (2\right ) + {\left ({\left (x^{2} - 2 \, {\left (x^{3} - x\right )} \log \left (2\right )\right )} e^{x} - 2 \, {\left (x^{2} - 1\right )} \log \left (2\right ) + x\right )} \log \left (2 \, {\left (x^{2} - 1\right )} \log \left (2\right ) - x\right ) - x\right )} e^{\left (-e^{x} - \log \left (x\right ) + 2\right )} - {\left ({\left (x^{2} - 2 \, {\left (x^{3} - x\right )} \log \left (2\right )\right )} e^{x} - 2 \, {\left (x^{2} - 1\right )} \log \left (2\right ) + x\right )} e^{\left (-2 \, e^{x} - 2 \, \log \left (x\right ) + 4\right )} - {\left (4 \, x^{2} \log \left (2\right ) - x\right )} \log \left (2 \, {\left (x^{2} - 1\right )} \log \left (2\right ) - x\right )\right )}}{9 \, {\left (x^{2} - 2 \, {\left (x^{3} - x\right )} \log \left (2\right )\right )}}\,{d x} \]________________________________________________________________________________________