\[ \int \frac {4 e^6 x+2 x^3-4 x^5+2 x^7+e^2 \left (1-3 x^2\right )+e^4 \left (-2 x^2-6 x^6\right )}{x^2-2 x^4+4 e^8 x^4+x^6+e^4 \left (4 x^3-4 x^5\right )} \, dx \]
Optimal antiderivative \[ \frac {1-\frac {{\mathrm e}^{2}}{x}-x^{4}}{2 x \,{\mathrm e}^{4}-x^{2}+1} \]
command
integrate((4*x*exp(2)^3+(-6*x^6-2*x^2)*exp(2)^2+(-3*x^2+1)*exp(2)+2*x^7-4*x^5+2*x^3)/(4*x^4*exp(2)^4+(-4*x^5+4*x^3)*exp(2)^2+x^6-2*x^4+x^2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ x^{2} + 2 \, x e^{4} + \frac {8 \, x^{2} e^{12} + 4 \, x^{2} e^{4} + 4 \, x e^{8} + e^{2}}{x^{3} - 2 \, x^{2} e^{4} - x} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________