\[ \int \frac {-2 x-x^2+\left (-2-x-2 x^2-2 x^3-x^4\right ) \log (x)+\left (-2 x-4 x^2-2 x^3\right ) \log ^2(x)+\left (-2 x-x^2\right ) \log ^3(x)+\left (2+3 x+x^2\right ) \log (x) \log \left (\left (8+8 x+2 x^2\right ) \log (x)\right )}{\left (2 x^3+x^4\right ) \log (x)+\left (4 x^2+2 x^3\right ) \log ^2(x)+\left (2 x+x^2\right ) \log ^3(x)} \, dx \]
Optimal antiderivative \[ 13-x -\frac {\ln \! \left (2 \ln \! \left (x \right ) \left (2+x \right )^{2}\right )}{x +\ln \! \left (x \right )} \]
command
integrate(((x**2+3*x+2)*ln(x)*ln((2*x**2+8*x+8)*ln(x))+(-x**2-2*x)*ln(x)**3+(-2*x**3-4*x**2-2*x)*ln(x)**2+(-x**4-2*x**3-2*x**2-x-2)*ln(x)-x**2-2*x)/((x**2+2*x)*ln(x)**3+(2*x**3+4*x**2)*ln(x)**2+(x**4+2*x**3)*ln(x)),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Exception raised: TypeError} \]
Sympy 1.8 under Python 3.8.8 output
\[ - x - \frac {\log {\left (\left (2 x^{2} + 8 x + 8\right ) \log {\left (x \right )} \right )}}{x + \log {\left (x \right )}} \]