\[ \int \frac {-562500-130000 \log (x)-10200 \log ^2(x)-336 \log ^3(x)-4 \log ^4(x)+\left (-337500-64500 \log (x)-3540 \log ^2(x)-60 \log ^3(x)\right ) \log (4+\log (x))+\left (-67500-10200 \log (x)-300 \log ^2(x)\right ) \log ^2(4+\log (x))+(-4500-500 \log (x)) \log ^3(4+\log (x))}{4 x+x \log (x)} \, dx \]
Optimal antiderivative \[ -\frac {1}{4}-\left (\ln \left (x \right )+25+5 \ln \left (\ln \left (x \right )+4\right )\right )^{4} \]
command
integrate(((-500*ln(x)-4500)*ln(ln(x)+4)**3+(-300*ln(x)**2-10200*ln(x)-67500)*ln(ln(x)+4)**2+(-60*ln(x)**3-3540*ln(x)**2-64500*ln(x)-337500)*ln(ln(x)+4)-4*ln(x)**4-336*ln(x)**3-10200*ln(x)**2-130000*ln(x)-562500)/(x*ln(x)+4*x),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \left (- 500 \log {\left (x \right )} - 12500\right ) \log {\left (\log {\left (x \right )} + 4 \right )}^{3} + \left (- 150 \log {\left (x \right )}^{2} - 7500 \log {\left (x \right )} - 93750\right ) \log {\left (\log {\left (x \right )} + 4 \right )}^{2} + \left (- 20 \log {\left (x \right )}^{3} - 1500 \log {\left (x \right )}^{2} - 37500 \log {\left (x \right )}\right ) \log {\left (\log {\left (x \right )} + 4 \right )} - \log {\left (x \right )}^{4} - 100 \log {\left (x \right )}^{3} - 3750 \log {\left (x \right )}^{2} - 62500 \log {\left (x \right )} - 625 \log {\left (\log {\left (x \right )} + 4 \right )}^{4} - 312500 \log {\left (\log {\left (x \right )} + 4 \right )} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________