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