\[ \int \frac {8 e^3 x}{64+192 x^2+144 x^4+e^2 \left (1+4 x^2+4 x^4\right )+e^{20} \left (1+4 x^2+4 x^4\right )+e \left (16+56 x^2+48 x^4\right )+e^{10} \left (-16-56 x^2-48 x^4+e \left (-2-8 x^2-8 x^4\right )\right )} \, dx \]
Optimal antiderivative \[ \frac {{\mathrm e}^{3}}{\frac {1}{\frac {1}{2}+x^{2}}+6-{\mathrm e}^{10}+{\mathrm e}} \]
command
integrate(8*x*exp(3)/((4*x^4+4*x^2+1)*exp(5)^4+((-8*x^4-8*x^2-2)*exp(1)-48*x^4-56*x^2-16)*exp(5)^2+(4*x^4+4*x^2+1)*exp(1)^2+(48*x^4+56*x^2+16)*exp(1)+144*x^4+192*x^2+64),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {2 \, e^{3}}{{\left (2 \, x^{2} e^{10} - 2 \, x^{2} e - 12 \, x^{2} + e^{10} - e - 8\right )} {\left (e^{10} - e - 6\right )}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________