Answer
$$\ln \left| {\ln \left( {\ln x} \right)} \right| + C$$
Work Step by Step
$$\eqalign{
& \int {\frac{1}{{x \cdot \ln x \cdot \ln \left( {\ln x} \right)}}} dx \cr
& {\text{Write the integrand as}} \cr
& \int {\left( {\frac{1}{{\ln \left( {\ln x} \right)}}} \right)\left( {\frac{1}{{x\ln x}}} \right)} dx \cr
& {\text{Integrate by using the substitution method}}{\text{,}} \cr
& {\text{Let }}u = \ln \left( {\ln x} \right),\,\,\,\,\,\,\,\,\,du = \frac{{\frac{1}{x}}}{{\ln x}}dx,\,\,\,\,\,\,du = \frac{1}{{x\ln x}}dx \cr
& \cr
& {\text{Write the integrand in terms of }}u \cr
& \,\,\int {\left( {\frac{1}{{\ln \left( {\ln x} \right)}}} \right)\left( {\frac{1}{{x\ln x}}} \right)} dx = \int {\frac{1}{u}du} \cr
& {\text{Integrate}} \cr
& = \ln \left| u \right| + C \cr
& \cr
& {\text{Write in terms of }}x,{\text{ substitute }}\ln \left( {\ln x} \right){\text{ for }}u \cr
& = \ln \left| {\ln \left( {\ln x} \right)} \right| + C \cr} $$
