\[ \int \frac {16 \log ^2\left (x^2\right )+e^{2 e^{e^{\frac {x}{\log \left (x^2\right )}}}+4 x} \left ((1+4 x) \log ^2\left (x^2\right )+e^{e^{\frac {x}{\log \left (x^2\right )}}+\frac {x}{\log \left (x^2\right )}} \left (-4 x+2 x \log \left (x^2\right )\right )\right )+e^{e^{e^{\frac {x}{\log \left (x^2\right )}}}+2 x} \left ((8+16 x) \log ^2\left (x^2\right )+e^{e^{\frac {x}{\log \left (x^2\right )}}+\frac {x}{\log \left (x^2\right )}} \left (-16 x+8 x \log \left (x^2\right )\right )\right )}{\log ^2\left (x^2\right )} \, dx \]
Optimal antiderivative \[ x \left ({\mathrm e}^{{\mathrm e}^{{\mathrm e}^{\frac {x}{\ln \left (x^{2}\right )}}}+2 x}+4\right )^{2}-4 \]
command
integrate((((2*x*ln(x**2)-4*x)*exp(x/ln(x**2))*exp(exp(x/ln(x**2)))+(1+4*x)*ln(x**2)**2)*exp(exp(exp(x/ln(x**2)))+2*x)**2+((8*x*ln(x**2)-16*x)*exp(x/ln(x**2))*exp(exp(x/ln(x**2)))+(16*x+8)*ln(x**2)**2)*exp(exp(exp(x/ln(x**2)))+2*x)+16*ln(x**2)**2)/ln(x**2)**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
\[ 8 x e^{2 x + e^{e^{\frac {x}{\log {\left (x^{2} \right )}}}}} + x e^{4 x + 2 e^{e^{\frac {x}{\log {\left (x^{2} \right )}}}}} + 16 x \]