\[ \int \frac {3750 x+300 x^2+6 x^3+\left (4000 x^2+300 x^3+6 x^4\right ) \log (x)+\left (\left (4000 x^2+310 x^3+6 x^4\right ) \log (x)+\left (3750 x+300 x^2+6 x^3\right ) \log (x) \log (\log (x))\right ) \log \left (\frac {6400 x^2+480 x^3+9 x^4+\left (12000 x+930 x^2+18 x^3\right ) \log (\log (x))+\left (5625+450 x+9 x^2\right ) \log ^2(\log (x))}{5625+450 x+9 x^2}\right )}{\left (2000 x+155 x^2+3 x^3\right ) \log (x)+\left (1875+150 x+3 x^2\right ) \log (x) \log (\log (x))} \, dx \]
Optimal antiderivative \[ x^{2} \ln \! \left (\left (\frac {x}{\frac {3 x}{5}+15}+x +\ln \! \left (\ln \! \left (x \right )\right )\right )^{2}\right ) \]
command
integrate((((6*x**3+300*x**2+3750*x)*ln(x)*ln(ln(x))+(6*x**4+310*x**3+4000*x**2)*ln(x))*ln(((9*x**2+450*x+5625)*ln(ln(x))**2+(18*x**3+930*x**2+12000*x)*ln(ln(x))+9*x**4+480*x**3+6400*x**2)/(9*x**2+450*x+5625))+(6*x**4+300*x**3+4000*x**2)*ln(x)+6*x**3+300*x**2+3750*x)/((3*x**2+150*x+1875)*ln(x)*ln(ln(x))+(3*x**3+155*x**2+2000*x)*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
\[ \left (x^{2} - \frac {625}{6}\right ) \log {\left (\frac {9 x^{4} + 480 x^{3} + 6400 x^{2} + \left (9 x^{2} + 450 x + 5625\right ) \log {\left (\log {\left (x \right )} \right )}^{2} + \left (18 x^{3} + 930 x^{2} + 12000 x\right ) \log {\left (\log {\left (x \right )} \right )}}{9 x^{2} + 450 x + 5625} \right )} + \frac {625 \log {\left (\log {\left (\log {\left (x \right )} \right )} + \frac {3 x^{2} + 80 x}{3 x + 75} \right )}}{3} \]