\[ \int \frac {-1+e^4 \left (-5-2 x^2\right )+e^8 \left (10 x+5 x^2-x^4\right )+e^{2+x} \left (5 e^4+e^8 \left (-10 x+5 x^2\right )\right )}{5 e^4+10 e^8 x^2+5 e^{12} x^4} \, dx \]
Optimal antiderivative \[ \frac {{\mathrm e}^{2+x}-x -1}{1+x^{2} {\mathrm e}^{4}}-\frac {x \,{\mathrm e}^{-4}}{5} \]
command
integrate((((5*x^2-10*x)*exp(4)^2+5*exp(4))*exp(2+x)+(-x^4+5*x^2+10*x)*exp(4)^2+(-2*x^2-5)*exp(4)-1)/(5*x^4*exp(4)^3+10*x^2*exp(4)^2+5*exp(4)),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {x^{3} e^{4} + 10 \, x e^{4} + x + 10 \, e^{4} - 5 \, e^{\left (x + 6\right )}}{5 \, {\left (x^{2} e^{8} + e^{4}\right )}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________