An Integrated culture is a culture that has the civic rights to promote to Citizens or Nobles, this makes that culture part of the elite in your country.
Each integrated culture will reduce the State Culture Happiness by 5% and exempt all Pops of that culture from cultural assimilation completely.
I really loved that DD and brings many great changes!
However, integrated cultures not assimilating makes no sense at all. If anything, they should assimilate faster in most cases. Romans arguably quickly "integrated" all the italic ethnic group, giving them several rights and after the social wars gave citizenship to everyone. They assimilated rather quickly into Rome and there were very few left of them in the end of the time frame. Furthermore one of the reasons the Etruscans disapeared so quickly (despite of how different they were) was the aggressive way Rome gave citizenship and "integrated" their former cities. Furthermore, after the Roman Gallic wars, the tribes that sided with Rome were almost immediately "integrated" becoming not only "citzens" but also "nobles" (as Caesar stuffed the senate with some of his gaullish allies), within even the same generation at least the elite of those people was thoroughly romanized (though the common people still probably retained their culture for much longer).
Also same culture groups, should assimilate MUCH faster than different culture groups.
I think that this system is a rather simplistic representation of integration and assimilation. I believe that each culture should have some kind of "prestige" (representing the traditions existing way before the start of the game), "civilized" cultures should have high prestige, while tribals should have low.
High prestige culture should be cheaper to integrate (representing the fact that your people are less reluctant to see them as equals, but they have a lot of literature and traditions so they don't assimilate and borrow as much from yours). Meanwhile, low prestige cultures should be easier to assimilate but harder to integrate (representing that they adopted many of their conqueror culture as they were "civilized" over time; but your main people would get upset if you stuff the senate with dirty barbarians).
With that system cultures like Greek, Egyptian, Persian and Phoenicians are better of "integrated"; while gaulish, iberians and Illyrians will eventually assimilate into your main culture overtime.
Moreover, I hope that the mallus to you primary culture is not permanent, but rather is a modifier that wears off over time. There are about 10 different Italic cultures and over 15 different greek cultures. If in order to accept each new culture you need to give up 5% of primary and assimilated cultures for the rest of the game, it could cost well over 70% happiness just to integrated the greeks and italics in the region as Rome (and Rome arguably, integrated all of them along with many others).
Finally, I hope that for each culture you integrate inside a certain group it become cheaper to integrate cultures inside that same culture group. For example, it doesn't make any sense that you integrate the Athenias, Peloponesans and Macedonians, but not the Boeotian because they are a smaller greek culture and not worth the political influence and happiness debuff to integrate them.