\[ \int \frac {4^{\frac {16 x}{4 e^x+20 x-4 x^2+e^4 x^2}} \left (e^x (64-64 x) \log (4)+\left (64 x^2-16 e^4 x^2\right ) \log (4)\right )}{16 e^{2 x}+400 x^2-160 x^3+16 x^4+e^8 x^4+e^x \left (160 x-32 x^2+8 e^4 x^2\right )+e^4 \left (40 x^3-8 x^4\right )} \, dx \]
Optimal antiderivative \[ {\mathrm e}^{\frac {8 \ln \left (2\right )}{\frac {x \,{\mathrm e}^{4}}{4}+\frac {x \left (5-x \right )+{\mathrm e}^{x}}{x}}} \]
command
integrate((2*(-64*x+64)*log(2)*exp(x)+2*(-16*x^2*exp(4)+64*x^2)*log(2))*exp(32*x*log(2)/(4*exp(x)+x^2*exp(4)-4*x^2+20*x))/(16*exp(x)^2+(8*x^2*exp(4)-32*x^2+160*x)*exp(x)+x^4*exp(4)^2+(-8*x^4+40*x^3)*exp(4)+16*x^4-160*x^3+400*x^2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ 2^{\frac {32 \, x}{x^{2} e^{4} - 4 \, x^{2} + 20 \, x + 4 \, e^{x}}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int -\frac {32 \, {\left (4 \, {\left (x - 1\right )} e^{x} \log \left (2\right ) + {\left (x^{2} e^{4} - 4 \, x^{2}\right )} \log \left (2\right )\right )} 2^{\frac {32 \, x}{x^{2} e^{4} - 4 \, x^{2} + 20 \, x + 4 \, e^{x}}}}{x^{4} e^{8} + 16 \, x^{4} - 160 \, x^{3} + 400 \, x^{2} - 8 \, {\left (x^{4} - 5 \, x^{3}\right )} e^{4} + 8 \, {\left (x^{2} e^{4} - 4 \, x^{2} + 20 \, x\right )} e^{x} + 16 \, e^{\left (2 \, x\right )}}\,{d x} \]________________________________________________________________________________________