\[ \int -\frac {2 e^{\frac {2 x}{e^{10} (25+5 x)+e^5 (-50-10 x) \log (4)+(25+5 x) \log ^2(4)}}}{e^{10} \left (25+10 x+x^2\right )+e^5 \left (-50-20 x-2 x^2\right ) \log (4)+\left (25+10 x+x^2\right ) \log ^2(4)} \, dx \]
Optimal antiderivative \[ 5-{\mathrm e}^{\frac {x}{2 \left ({\mathrm e}^{5}-2 \ln \left (2\right )\right )^{2} \left (\frac {25}{4}+\frac {5 x}{4}\right )}} \]
command
integrate(-2*exp(2*x/(4*(25+5*x)*log(2)^2+2*(-10*x-50)*exp(5)*log(2)+(25+5*x)*exp(5)^2))/(4*(x^2+10*x+25)*log(2)^2+2*(-2*x^2-20*x-50)*exp(5)*log(2)+(x^2+10*x+25)*exp(5)^2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {4 \, e^{\left (-\frac {2 \, x}{5 \, {\left (4 \, x e^{5} \log \left (2\right ) - 4 \, x \log \left (2\right )^{2} - x e^{10} + 20 \, e^{5} \log \left (2\right ) - 20 \, \log \left (2\right )^{2} - 5 \, e^{10}\right )}}\right )} \log \left (2\right )^{2} - 4 \, e^{\left (-\frac {2 \, x}{5 \, {\left (4 \, x e^{5} \log \left (2\right ) - 4 \, x \log \left (2\right )^{2} - x e^{10} + 20 \, e^{5} \log \left (2\right ) - 20 \, \log \left (2\right )^{2} - 5 \, e^{10}\right )}} + 5\right )} \log \left (2\right ) + e^{\left (-\frac {2 \, x}{5 \, {\left (4 \, x e^{5} \log \left (2\right ) - 4 \, x \log \left (2\right )^{2} - x e^{10} + 20 \, e^{5} \log \left (2\right ) - 20 \, \log \left (2\right )^{2} - 5 \, e^{10}\right )}} + 10\right )}}{4 \, e^{5} \log \left (2\right ) - 4 \, \log \left (2\right )^{2} - e^{10}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {2 \, e^{\left (-\frac {2 \, x}{5 \, {\left (4 \, {\left (x + 5\right )} e^{5} \log \left (2\right ) - 4 \, {\left (x + 5\right )} \log \left (2\right )^{2} - {\left (x + 5\right )} e^{10}\right )}}\right )}}{4 \, {\left (x^{2} + 10 \, x + 25\right )} e^{5} \log \left (2\right ) - 4 \, {\left (x^{2} + 10 \, x + 25\right )} \log \left (2\right )^{2} - {\left (x^{2} + 10 \, x + 25\right )} e^{10}}\,{d x} \]________________________________________________________________________________________