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(x*Log[Log[a*x]] - LogIntegral[a*x]/a)/2

Integrate[Log[Sqrt[Log[a*x]]], x] ==
(x*Log[Log[a*x]] - LogIntegral[a*x]/a)/2 

Integrate[Log[Sqrt[Log[a x]]], x] == 

                  LogIntegral[a x]
x Log[Log[a x]] - ----------------
                         a
----------------------------------
                2
Time to compute: < 0.01 second
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