\[ \int \frac {16 x^2-112 x^3-128 x^2 \log (x)+\left (-64 x^2+32 x^3+\left (-64 x+32 x^2\right ) \log (x)\right ) \log (x+\log (x))+\left (-8 x+12 x^2+(-8+12 x) \log (x)\right ) \log ^2(x+\log (x))+\left (8 x^2-88 x^3-96 x^2 \log (x)+\left (-48 x^2+24 x^3+\left (-48 x+24 x^2\right ) \log (x)\right ) \log (x+\log (x))+\left (-6 x+8 x^2+(-6+8 x) \log (x)\right ) \log ^2(x+\log (x))\right ) \log \left (\frac {-4 x^2+\left (-x+x^2\right ) \log (x+\log (x))}{4 x+\log (x+\log (x))}\right )+\left (-16 x^3-16 x^2 \log (x)+\left (-8 x^2+4 x^3+\left (-8 x+4 x^2\right ) \log (x)\right ) \log (x+\log (x))+\left (-x+x^2+(-1+x) \log (x)\right ) \log ^2(x+\log (x))\right ) \log ^2\left (\frac {-4 x^2+\left (-x+x^2\right ) \log (x+\log (x))}{4 x+\log (x+\log (x))}\right )}{-16 x^3-16 x^2 \log (x)+\left (-8 x^2+4 x^3+\left (-8 x+4 x^2\right ) \log (x)\right ) \log (x+\log (x))+\left (-x+x^2+(-1+x) \log (x)\right ) \log ^2(x+\log (x))} \, dx \]
Optimal antiderivative \[ \left (-2-\ln \! \left (\frac {x}{\frac {4}{\ln \left (x +\ln \left (x \right )\right )}+\frac {1}{x}}-x \right )\right )^{2} x \]
command
integrate(((((-1+x)*ln(x)+x**2-x)*ln(x+ln(x))**2+((4*x**2-8*x)*ln(x)+4*x**3-8*x**2)*ln(x+ln(x))-16*x**2*ln(x)-16*x**3)*ln(((x**2-x)*ln(x+ln(x))-4*x**2)/(ln(x+ln(x))+4*x))**2+(((8*x-6)*ln(x)+8*x**2-6*x)*ln(x+ln(x))**2+((24*x**2-48*x)*ln(x)+24*x**3-48*x**2)*ln(x+ln(x))-96*x**2*ln(x)-88*x**3+8*x**2)*ln(((x**2-x)*ln(x+ln(x))-4*x**2)/(ln(x+ln(x))+4*x))+((12*x-8)*ln(x)+12*x**2-8*x)*ln(x+ln(x))**2+((32*x**2-64*x)*ln(x)+32*x**3-64*x**2)*ln(x+ln(x))-128*x**2*ln(x)-112*x**3+16*x**2)/(((-1+x)*ln(x)+x**2-x)*ln(x+ln(x))**2+((4*x**2-8*x)*ln(x)+4*x**3-8*x**2)*ln(x+ln(x))-16*x**2*ln(x)-16*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
\[ x \log {\left (\frac {- 4 x^{2} + \left (x^{2} - x\right ) \log {\left (x + \log {\left (x \right )} \right )}}{4 x + \log {\left (x + \log {\left (x \right )} \right )}} \right )}^{2} + 4 x \log {\left (\frac {- 4 x^{2} + \left (x^{2} - x\right ) \log {\left (x + \log {\left (x \right )} \right )}}{4 x + \log {\left (x + \log {\left (x \right )} \right )}} \right )} + 4 x \]