\[ \int \frac {e^{\log ^2\left (\frac {-25+e^{2+2 x}+e^{2+x} (-40-8 x)-10 x-x^2+e^2 \left (525+185 x+11 x^2-x^3\right )}{e^2 \left (25+10 x+x^2\right )}\right )} \left (e^{2+2 x} (16+4 x)+e^{2+x} \left (-320-144 x-16 x^2\right )+e^2 \left (-250-150 x-30 x^2-2 x^3\right )\right ) \log \left (\frac {-25+e^{2+2 x}+e^{2+x} (-40-8 x)-10 x-x^2+e^2 \left (525+185 x+11 x^2-x^3\right )}{e^2 \left (25+10 x+x^2\right )}\right )}{-125-75 x-15 x^2-x^3+e^{2+2 x} (5+x)+e^{2+x} \left (-200-80 x-8 x^2\right )+e^2 \left (2625+1450 x+240 x^2+6 x^3-x^4\right )} \, dx \]
Optimal antiderivative \[ {\mathrm e}^{\ln \left (5+\left (\frac {{\mathrm e}^{x}}{5+x}-4\right )^{2}-x -{\mathrm e}^{-2}\right )^{2}} \]
command
integrate(((4*x+16)*exp(2)*exp(x)^2+(-16*x^2-144*x-320)*exp(2)*exp(x)+(-2*x^3-30*x^2-150*x-250)*exp(2))*log((exp(2)*exp(x)^2+(-8*x-40)*exp(2)*exp(x)+(-x^3+11*x^2+185*x+525)*exp(2)-x^2-10*x-25)/(x^2+10*x+25)/exp(2))*exp(log((exp(2)*exp(x)^2+(-8*x-40)*exp(2)*exp(x)+(-x^3+11*x^2+185*x+525)*exp(2)-x^2-10*x-25)/(x^2+10*x+25)/exp(2))^2)/((5+x)*exp(2)*exp(x)^2+(-8*x^2-80*x-200)*exp(2)*exp(x)+(-x^4+6*x^3+240*x^2+1450*x+2625)*exp(2)-x^3-15*x^2-75*x-125),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {could not integrate} \]
Giac 1.7.0 via sagemath 9.3 output
\[ e^{\left (\log \left (-\frac {x^{3} e^{2}}{x^{2} e^{2} + 10 \, x e^{2} + 25 \, e^{2}} + \frac {11 \, x^{2} e^{2}}{x^{2} e^{2} + 10 \, x e^{2} + 25 \, e^{2}} - \frac {x^{2}}{x^{2} e^{2} + 10 \, x e^{2} + 25 \, e^{2}} + \frac {185 \, x e^{2}}{x^{2} e^{2} + 10 \, x e^{2} + 25 \, e^{2}} - \frac {8 \, x e^{\left (x + 2\right )}}{x^{2} e^{2} + 10 \, x e^{2} + 25 \, e^{2}} - \frac {10 \, x}{x^{2} e^{2} + 10 \, x e^{2} + 25 \, e^{2}} + \frac {525 \, e^{2}}{x^{2} e^{2} + 10 \, x e^{2} + 25 \, e^{2}} + \frac {e^{\left (2 \, x + 2\right )}}{x^{2} e^{2} + 10 \, x e^{2} + 25 \, e^{2}} - \frac {40 \, e^{\left (x + 2\right )}}{x^{2} e^{2} + 10 \, x e^{2} + 25 \, e^{2}} - \frac {25}{x^{2} e^{2} + 10 \, x e^{2} + 25 \, e^{2}}\right )^{2}\right )} \]