\[ \int \frac {4^{\frac {-16-8 x-x^2}{-2-2 x+2 \log \left (\log \left (x^2\right )\right )}} \left (\left (32+48 x+18 x^2+2 x^3\right ) \log (4)+\left (2+8 x+10 x^2+4 x^3+\left (-8 x-6 x^2+3 x^3+x^4\right ) \log (4)\right ) \log \left (x^2\right )+\left (-4-12 x-8 x^2+\left (-8 x-10 x^2-2 x^3\right ) \log (4)\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+(2+4 x) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )\right )}{\left (2+4 x+2 x^2\right ) \log \left (x^2\right )+(-4-4 x) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+2 \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )} \, dx \]
Optimal antiderivative \[ \left (x^{2}+x \right ) {\mathrm e}^{\frac {\left (4+x \right )^{2} \ln \left (2\right )}{x +1-\ln \left (\ln \left (x^{2}\right )\right )}} \]
command
integrate(((4*x+2)*ln(x**2)*ln(ln(x**2))**2+(2*(-2*x**3-10*x**2-8*x)*ln(2)-8*x**2-12*x-4)*ln(x**2)*ln(ln(x**2))+(2*(x**4+3*x**3-6*x**2-8*x)*ln(2)+4*x**3+10*x**2+8*x+2)*ln(x**2)+2*(2*x**3+18*x**2+48*x+32)*ln(2))*exp(2*(-x**2-8*x-16)*ln(2)/(2*ln(ln(x**2))-2*x-2))/(2*ln(x**2)*ln(ln(x**2))**2+(-4*x-4)*ln(x**2)*ln(ln(x**2))+(2*x**2+4*x+2)*ln(x**2)),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Exception raised: TypeError} \]
Sympy 1.8 under Python 3.8.8 output
\[ \left (x^{2} + x\right ) e^{\frac {\left (- 2 x^{2} - 16 x - 32\right ) \log {\left (2 \right )}}{- 2 x + 2 \log {\left (\log {\left (x^{2} \right )} \right )} - 2}} \]