\[ \int \frac {648+8 e^8-8 x^2}{6561+e^{16}+4 e^{12} x+162 x^2+x^4+e^8 \left (162+6 x^2\right )+e^4 \left (324 x+4 x^3\right )} \, dx \]
Optimal antiderivative \[ \frac {8 x}{81+\left (x +{\mathrm e}^{4}\right )^{2}} \]
command
integrate((8*exp(4)^2-8*x^2+648)/(exp(4)^4+4*x*exp(4)^3+(6*x^2+162)*exp(4)^2+(4*x^3+324*x)*exp(4)+x^4+162*x^2+6561),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {8 \, x}{x^{2} + 2 \, x e^{4} + e^{8} + 81} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int -\frac {8 \, {\left (x^{2} - e^{8} - 81\right )}}{x^{4} + 162 \, x^{2} + 4 \, x e^{12} + 6 \, {\left (x^{2} + 27\right )} e^{8} + 4 \, {\left (x^{3} + 81 \, x\right )} e^{4} + e^{16} + 6561}\,{d x} \]________________________________________________________________________________________