\[ \int \frac {\left (-208 x-60 x^2+108 x^3-36 x^4+4 x^5\right ) \log (2 x)+\left (-60+164 x-57 x^2-15 x^3+9 x^4-x^5\right ) \log \left (\frac {400-1920 x+2424 x^2-48 x^3-607 x^4+132 x^5+30 x^6-12 x^7+x^8}{81-108 x+54 x^2-12 x^3+x^4}\right )}{\left (60 x-164 x^2+57 x^3+15 x^4-9 x^5+x^6\right ) \log ^2(2 x)} \, dx \]
Optimal antiderivative \[ \frac {\ln \! \left (\left (\left (x +\frac {4}{3-x}\right )^{2}-4\right )^{2}\right )}{\ln \! \left (2 x \right )} \]
command
integrate(((-x**5+9*x**4-15*x**3-57*x**2+164*x-60)*ln((x**8-12*x**7+30*x**6+132*x**5-607*x**4-48*x**3+2424*x**2-1920*x+400)/(x**4-12*x**3+54*x**2-108*x+81))+(4*x**5-36*x**4+108*x**3-60*x**2-208*x)*ln(2*x))/(x**6-9*x**5+15*x**4+57*x**3-164*x**2+60*x)/ln(2*x)**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
\[ \frac {\log {\left (\frac {x^{8} - 12 x^{7} + 30 x^{6} + 132 x^{5} - 607 x^{4} - 48 x^{3} + 2424 x^{2} - 1920 x + 400}{x^{4} - 12 x^{3} + 54 x^{2} - 108 x + 81} \right )}}{\log {\left (2 x \right )}} \]