Answer
$$\ln \left| {\ln \left( {\ln x} \right)} \right| + C$$
Work Step by Step
$$\eqalign{
& \int {\frac{{dx}}{{x\ln x\ln \left( {\ln x} \right)}}} \cr
& {\text{substitute }}u = \ln \left( {\ln x} \right),{\text{ }}du = \frac{{1/x}}{{\ln x}}dx \cr
& du = \frac{1}{{x\ln x}} \cr
& \int {\frac{{dx}}{{x\ln x\ln \left( {\ln x} \right)}}} = \int {\frac{1}{u}du} \cr
& = \int {\frac{1}{u}} du \cr
& {\text{find the antiderivative}} \cr
& = \ln \left| u \right| + C \cr
& {\text{replace }}u = \ln \left( {\ln x} \right) \cr
& = \ln \left| {\ln \left( {\ln x} \right)} \right| + C \cr} $$
