Akra-Bazzi method integral diverges
Your recurrence isn't really defined for $n = 1$. This suggests using a different value for the lower bound in the integral. The formula given by Akra–Bazzi doesn't actually depend on the exact value of the lower bound that you choose – it can at most affect the big Theta constant or, in your case, some lower order term. If you choose any lower bound which is large enough, then the Akra–Bazzi formula gives you $\Theta(n\log\log n)$ as wanted.