\[ \int \frac {100+40 x+4 x^2+\left (-50-40 x-6 x^2\right ) \log (x)+\left (10 x+2 x^2\right ) \log ^2(x)-4 x \log ^3(x)+(-4+2 x) \log ^4(x)+2 \log ^5(x)+\left ((-100-20 x) \log (x)-20 \log ^3(x)\right ) \log (-2+\log (x))+\left (-200-40 x+(100+40 x) \log (x)-10 x \log ^2(x)\right ) \log ^2(-2+\log (x))+100 \log (x) \log ^3(-2+\log (x))+(100-50 \log (x)) \log ^4(-2+\log (x))}{-2 x \log ^3(x)+x \log ^4(x)} \, dx \]
Optimal antiderivative \[ \left (\ln \! \left (x \right )+\frac {x +\frac {5 x -5 x \ln \left (\ln \left (x \right )-2\right )^{2}}{x}}{\ln \! \left (x \right )}\right )^{2} \]
command
integrate(((-50*ln(x)+100)*ln(ln(x)-2)**4+100*ln(x)*ln(ln(x)-2)**3+(-10*x*ln(x)**2+(40*x+100)*ln(x)-40*x-200)*ln(ln(x)-2)**2+(-20*ln(x)**3+(-20*x-100)*ln(x))*ln(ln(x)-2)+2*ln(x)**5+(2*x-4)*ln(x)**4-4*x*ln(x)**3+(2*x**2+10*x)*ln(x)**2+(-6*x**2-40*x-50)*ln(x)+4*x**2+40*x+100)/(x*ln(x)**4-2*x*ln(x)**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
\[ 2 x + \frac {\left (- 10 x - 10 \log {\left (x \right )}^{2} - 50\right ) \log {\left (\log {\left (x \right )} - 2 \right )}^{2}}{\log {\left (x \right )}^{2}} + \frac {x^{2} + 10 x + 25}{\log {\left (x \right )}^{2}} + \log {\left (x \right )}^{2} + \frac {25 \log {\left (\log {\left (x \right )} - 2 \right )}^{4}}{\log {\left (x \right )}^{2}} \]