\int \frac{\ln  x}{x^2(\ln  x-1)} \, dx=| \ln  x=t,x=\left.e^t\right|=\int \frac{t-1+1}{e^t(t-1)} \, dt=\int \frac{1}{e^t} \, dt+\int \frac{e^{-t}}{(t-1)} \, dt=|t-1=s,t=s+1|=-e^{-t}+\int \frac{e^{-s-1}}{s} \, ds=-e^{-t}+\frac{1}{e} \mathrm{Ei}(-s)=-e^{-t}+\frac{1}{e} \mathrm{Ei}(-t+1)=-e^{-\ln  x}+\frac{1}{e}\mathrm{Ei}(-\ln  x +1)=-\frac{1}{x}+\frac{1}{e} \mathrm{Ei}(1-\ln  x)+c