\[ \int \frac {8 x-16 x^3-16 x^4+e^8 \left (-48 x^3-48 x^4\right )+e^{12} \left (16 x^3+16 x^4\right )+e^4 \left (-8 x+48 x^3+48 x^4\right )+\left (-4+24 x^2+16 x^3-8 x^4+e^8 \left (24 x^2-24 x^4\right )+e^{12} \left (8 x^3+8 x^4\right )+e^4 \left (-48 x^2-24 x^3+24 x^4\right )\right ) \log (x)+\left (-12 x+12 x^3+e^4 \left (12 x-12 x^2-24 x^3\right )+e^8 \left (12 x^2+12 x^3\right )\right ) \log ^2(x)+\left (2-4 x-6 x^2+e^4 \left (6 x+6 x^2\right )\right ) \log ^3(x)+(1+x) \log ^4(x)+\left (16-32 x+32 e^4 x+\left (8-16 x+16 e^4 x\right ) \log (x)\right ) \log (2+\log (x))}{-16 x^4+48 e^4 x^4-48 e^8 x^4+16 e^{12} x^4+\left (24 x^3-8 x^4+8 e^{12} x^4+e^8 \left (24 x^3-24 x^4\right )+e^4 \left (-48 x^3+24 x^4\right )\right ) \log (x)+\left (-12 x^2+12 x^3+12 e^8 x^3+e^4 \left (12 x^2-24 x^3\right )\right ) \log ^2(x)+\left (2 x-6 x^2+6 e^4 x^2\right ) \log ^3(x)+x \log ^4(x)} \, dx \]
Optimal antiderivative \[ x +\ln \! \left (x \right )-\frac {\ln \! \left (\ln \! \left (x \right )+2\right )}{\left (x \,{\mathrm e}^{4}-x +\frac {\ln \left (x \right )}{2}\right )^{2}} \]
command
integrate((((16*x*exp(4)-16*x+8)*ln(x)+32*x*exp(4)-32*x+16)*ln(ln(x)+2)+(1+x)*ln(x)**4+((6*x**2+6*x)*exp(4)-6*x**2-4*x+2)*ln(x)**3+((12*x**3+12*x**2)*exp(4)**2+(-24*x**3-12*x**2+12*x)*exp(4)+12*x**3-12*x)*ln(x)**2+((8*x**4+8*x**3)*exp(4)**3+(-24*x**4+24*x**2)*exp(4)**2+(24*x**4-24*x**3-48*x**2)*exp(4)-8*x**4+16*x**3+24*x**2-4)*ln(x)+(16*x**4+16*x**3)*exp(4)**3+(-48*x**4-48*x**3)*exp(4)**2+(48*x**4+48*x**3-8*x)*exp(4)-16*x**4-16*x**3+8*x)/(x*ln(x)**4+(6*x**2*exp(4)-6*x**2+2*x)*ln(x)**3+(12*x**3*exp(4)**2+(-24*x**3+12*x**2)*exp(4)+12*x**3-12*x**2)*ln(x)**2+(8*x**4*exp(4)**3+(-24*x**4+24*x**3)*exp(4)**2+(24*x**4-48*x**3)*exp(4)-8*x**4+24*x**3)*ln(x)+16*x**4*exp(4)**3-48*x**4*exp(4)**2+48*x**4*exp(4)-16*x**4),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Exception raised: TypeError} \]
Sympy 1.8 under Python 3.8.8 output
\[ x + \log {\left (x \right )} - \frac {4 \log {\left (\log {\left (x \right )} + 2 \right )}}{- 8 x^{2} e^{4} + 4 x^{2} + 4 x^{2} e^{8} - 4 x \log {\left (x \right )} + 4 x e^{4} \log {\left (x \right )} + \log {\left (x \right )}^{2}} \]