\[ \int \frac {-e^{2-2 x} x^2+e^{1-x+\frac {1}{4} (4+\log (4))} \left (-5+5 x-2 x^2\right )+e^{\frac {1}{2} (4+\log (4))} \left (-5-x^2\right )}{e^{2-2 x} x^2+e^{\frac {1}{2} (4+\log (4))} x^2+2 e^{1-x+\frac {1}{4} (4+\log (4))} x^2} \, dx \]
Optimal antiderivative \[ \frac {5}{x +x \,{\mathrm e}^{-1-\frac {\ln \left (2\right )}{2}} {\mathrm e}^{1-x}}-x \]
command
integrate(((-x^2-5)*exp(1+1/2*log(2))^2+(-2*x^2+5*x-5)*exp(1-x)*exp(1+1/2*log(2))-x^2*exp(1-x)^2)/(x^2*exp(1+1/2*log(2))^2+2*x^2*exp(1-x)*exp(1+1/2*log(2))+x^2*exp(1-x)^2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {2 \, {\left (2 \, x - \log \left (2\right ) - 4\right )}^{2} e^{2} + {\left (2 \, x - \log \left (2\right ) - 4\right )}^{2} e^{\left (-x + \frac {1}{2} \, \log \left (2\right ) + 2\right )} + 2 \, {\left (2 \, x - \log \left (2\right ) - 4\right )} e^{2} \log \left (2\right ) + {\left (2 \, x - \log \left (2\right ) - 4\right )} e^{\left (-x + \frac {1}{2} \, \log \left (2\right ) + 2\right )} \log \left (2\right ) - 4 \, {\left (2 \, x - \log \left (2\right ) - 4\right )} e^{2} \log \left (2 \, e^{2} + e^{\left (-x + \frac {1}{2} \, \log \left (2\right ) + 2\right )}\right ) - 2 \, {\left (2 \, x - \log \left (2\right ) - 4\right )} e^{\left (-x + \frac {1}{2} \, \log \left (2\right ) + 2\right )} \log \left (2 \, e^{2} + e^{\left (-x + \frac {1}{2} \, \log \left (2\right ) + 2\right )}\right ) - 4 \, e^{2} \log \left (2\right ) \log \left (2 \, e^{2} + e^{\left (-x + \frac {1}{2} \, \log \left (2\right ) + 2\right )}\right ) - 2 \, e^{\left (-x + \frac {1}{2} \, \log \left (2\right ) + 2\right )} \log \left (2\right ) \log \left (2 \, e^{2} + e^{\left (-x + \frac {1}{2} \, \log \left (2\right ) + 2\right )}\right ) + 4 \, {\left (2 \, x - \log \left (2\right ) - 4\right )} e^{2} \log \left (-2 \, e^{2} - e^{\left (-x + \frac {1}{2} \, \log \left (2\right ) + 2\right )}\right ) + 2 \, {\left (2 \, x - \log \left (2\right ) - 4\right )} e^{\left (-x + \frac {1}{2} \, \log \left (2\right ) + 2\right )} \log \left (-2 \, e^{2} - e^{\left (-x + \frac {1}{2} \, \log \left (2\right ) + 2\right )}\right ) + 4 \, e^{2} \log \left (2\right ) \log \left (-2 \, e^{2} - e^{\left (-x + \frac {1}{2} \, \log \left (2\right ) + 2\right )}\right ) + 2 \, e^{\left (-x + \frac {1}{2} \, \log \left (2\right ) + 2\right )} \log \left (2\right ) \log \left (-2 \, e^{2} - e^{\left (-x + \frac {1}{2} \, \log \left (2\right ) + 2\right )}\right ) + 8 \, {\left (2 \, x - \log \left (2\right ) - 4\right )} e^{2} + 4 \, {\left (2 \, x - \log \left (2\right ) - 4\right )} e^{\left (-x + \frac {1}{2} \, \log \left (2\right ) + 2\right )} - 16 \, e^{2} \log \left (2 \, e^{2} + e^{\left (-x + \frac {1}{2} \, \log \left (2\right ) + 2\right )}\right ) - 8 \, e^{\left (-x + \frac {1}{2} \, \log \left (2\right ) + 2\right )} \log \left (2 \, e^{2} + e^{\left (-x + \frac {1}{2} \, \log \left (2\right ) + 2\right )}\right ) + 16 \, e^{2} \log \left (-2 \, e^{2} - e^{\left (-x + \frac {1}{2} \, \log \left (2\right ) + 2\right )}\right ) + 8 \, e^{\left (-x + \frac {1}{2} \, \log \left (2\right ) + 2\right )} \log \left (-2 \, e^{2} - e^{\left (-x + \frac {1}{2} \, \log \left (2\right ) + 2\right )}\right ) - 40 \, e^{2}}{2 \, {\left (2 \, {\left (2 \, x - \log \left (2\right ) - 4\right )} e^{2} + {\left (2 \, x - \log \left (2\right ) - 4\right )} e^{\left (-x + \frac {1}{2} \, \log \left (2\right ) + 2\right )} + 2 \, e^{2} \log \left (2\right ) + e^{\left (-x + \frac {1}{2} \, \log \left (2\right ) + 2\right )} \log \left (2\right ) + 8 \, e^{2} + 4 \, e^{\left (-x + \frac {1}{2} \, \log \left (2\right ) + 2\right )}\right )}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int -\frac {x^{2} e^{\left (-2 \, x + 2\right )} + {\left (2 \, x^{2} - 5 \, x + 5\right )} e^{\left (-x + \frac {1}{2} \, \log \left (2\right ) + 2\right )} + {\left (x^{2} + 5\right )} e^{\left (\log \left (2\right ) + 2\right )}}{2 \, x^{2} e^{\left (-x + \frac {1}{2} \, \log \left (2\right ) + 2\right )} + x^{2} e^{\left (-2 \, x + 2\right )} + x^{2} e^{\left (\log \left (2\right ) + 2\right )}}\,{d x} \]________________________________________________________________________________________