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