It depends on your definition of integrated analysis. The most simple (and also least powerful) way of integration is to do differential analysis on mRNA and miRNA between your timepoints seperately and afterwards compare the differential results. You can look if annotated mRNA-miRNA pairs according to the miranda database change over time in opposite directions. However, a much more powerful approach is to use regression of miRNA on mRNA expression estimates to look for significant negative interactions between any miRNA-mRNA pair. You don't need a special package, linear regression is in the base functionality of R. The advantage of the latter approach is explained by the following example. Supposed you have a set of 50 control and 60 tumor samples. In all samples, the miRNA expression and mRNA expression, which are known to be interacting, are similar except for 1 tumor. This tumor has low expression of the mRNA and high expression of the miRNA. So you see silencing of the gene that is most likely caused by the increased expression of the miRNA. With regression analysis you will find this very easily. However if you did differential expression analysis between tumor and control, both miRNA and mRNA would not have been significant because the one tumor sample does not make the means of the tumor and control cohorts different.
There seem to be quite a few tools out there capable of integrative miRNA-mRNA analysis. This one seems quite comprehensive: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4696828/