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