$T(n)=2T(n/2)+O(n{\log }^{2}n\log \log n)$

This always comes up.

Without loss of generality, let

$n=2^k$ so

$k=\log n$.

Then:

$$\begin{array}{rl}T(2^k)& =2T(2^{k-1})+O(2^k{\log }^2(2^k)\log \log 2^k)\\ & =2T(2^{k-1})+O(2^k{k}^2\log k)\\ & =2(2T(2^{k-2})+O(2^{k-1}{(k-1)}^2\log (k-1)))+O(2^k{k}^2\log k)\\ & =\dots \\ & =2^{k}O(1)+\sum _{i=0}^{k}O(2^k{i}^2\log i)\\ & =O(2^k)+\sum _{i=1}^{k}O(2^k{i}^2\log i)\\ & \le O(2^k)+k\cdot O(2^k{k}^2\log k)\\ & \le O(2^k)+O(2^k{k}^3\log k)\\ & =O(n)+O(n{\log }^{3}n\log \log n)\end{array}$$