\[ \int \frac {e^x x^3+\left (324+324 x+e^{e^{4 x}} (324+324 x)\right ) \log \left (1+e^{e^{4 x}}\right )+2 e^x x^2 \log (x)+e^x x \log ^2(x)+e^{e^{4 x}} \left (-1296 e^{4 x} x^2+e^x x^3+\left (-1296 e^{4 x} x+2 e^x x^2\right ) \log (x)+e^x x \log ^2(x)\right )}{x^3+2 x^2 \log (x)+x \log ^2(x)+e^{e^{4 x}} \left (x^3+2 x^2 \log (x)+x \log ^2(x)\right )} \, dx \]
Optimal antiderivative \[ {\mathrm e}^{x}-\frac {324 \ln \! \left ({\mathrm e}^{{\mathrm e}^{4 x}}+1\right )}{x +\ln \! \left (x \right )} \]
command
integrate((((324*x+324)*exp(exp(4*x))+324*x+324)*ln(exp(exp(4*x))+1)+(x*exp(x)*ln(x)**2+(-1296*x*exp(4*x)+2*exp(x)*x**2)*ln(x)-1296*x**2*exp(4*x)+exp(x)*x**3)*exp(exp(4*x))+x*exp(x)*ln(x)**2+2*x**2*exp(x)*ln(x)+exp(x)*x**3)/((x*ln(x)**2+2*x**2*ln(x)+x**3)*exp(exp(4*x))+x*ln(x)**2+2*x**2*ln(x)+x**3),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Exception raised: TypeError} \]
Sympy 1.8 under Python 3.8.8 output
\[ e^{x} - \frac {324 \log {\left (e^{e^{4 x}} + 1 \right )}}{x + \log {\left (x \right )}} \]