\[ \int \frac {e^4 x^2+e^4 \left (8 e x+5 e^2 x^2\right )+e^6 x^2 \log (4)-e^6 x^2 \log \left (3 e^2\right )}{e^2 x^2+e^2 \left (8 e x+10 e^2 x^2\right )+e^4 \left (16+40 e x+25 e^2 x^2\right )+\left (2 e^4 x^2+e^4 \left (8 e x+10 e^2 x^2\right )\right ) \log (4)+e^6 x^2 \log ^2(4)+\left (-2 e^4 x^2+e^4 \left (-8 e x-10 e^2 x^2\right )-2 e^6 x^2 \log (4)\right ) \log \left (3 e^2\right )+e^6 x^2 \log ^2\left (3 e^2\right )} \, dx \]
Optimal antiderivative \[ \frac {x}{2 \ln \left (2\right )+{\mathrm e}^{-2}+\frac {4 \,{\mathrm e}^{-1}}{x}+5-\ln \left (3 \,{\mathrm e}^{2}\right )} \]
command
integrate((-x^2*exp(1)^2*exp(2)^2*log(3*exp(2))+2*x^2*exp(1)^2*exp(2)^2*log(2)+(5*x^2*exp(1)^2+8*x*exp(1))*exp(2)^2+x^2*exp(1)^2*exp(2))/(x^2*exp(1)^2*exp(2)^2*log(3*exp(2))^2+(-4*x^2*exp(1)^2*exp(2)^2*log(2)+(-10*x^2*exp(1)^2-8*x*exp(1))*exp(2)^2-2*x^2*exp(1)^2*exp(2))*log(3*exp(2))+4*x^2*exp(1)^2*exp(2)^2*log(2)^2+2*((10*x^2*exp(1)^2+8*x*exp(1))*exp(2)^2+2*x^2*exp(1)^2*exp(2))*log(2)+(25*x^2*exp(1)^2+40*x*exp(1)+16)*exp(2)^2+(10*x^2*exp(1)^2+8*x*exp(1))*exp(2)+x^2*exp(1)^2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {2 \, x e^{4} \log \left (2\right ) - x e^{4} \log \left (3 \, e^{2}\right ) + 5 \, x e^{4} + x e^{2}}{4 \, e^{4} \log \left (2\right )^{2} - 4 \, e^{4} \log \left (2\right ) \log \left (3 \, e^{2}\right ) + e^{4} \log \left (3 \, e^{2}\right )^{2} + 20 \, e^{4} \log \left (2\right ) + 4 \, e^{2} \log \left (2\right ) - 10 \, e^{4} \log \left (3 \, e^{2}\right ) - 2 \, e^{2} \log \left (3 \, e^{2}\right ) + 25 \, e^{4} + 10 \, e^{2} + 1} + \frac {16 \, e^{4}}{{\left (2 \, x e^{2} \log \left (2\right ) - x e^{2} \log \left (3 \, e^{2}\right ) + 5 \, x e^{2} + x + 4 \, e\right )} {\left (4 \, e^{4} \log \left (2\right )^{2} - 4 \, e^{4} \log \left (2\right ) \log \left (3 \, e^{2}\right ) + e^{4} \log \left (3 \, e^{2}\right )^{2} + 20 \, e^{4} \log \left (2\right ) + 4 \, e^{2} \log \left (2\right ) - 10 \, e^{4} \log \left (3 \, e^{2}\right ) - 2 \, e^{2} \log \left (3 \, e^{2}\right ) + 25 \, e^{4} + 10 \, e^{2} + 1\right )}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________