Int seems to affect linger of phrases. The tooltips are a bit weird though. Let's look at Kana's "At the Sight of their Comrades, their Hearts grew Bold" with 18 Int (+40% duration). This is what it looks like to me:
Duration: 4.0 sec
Linger: 2.8 sec
Friendly Aura: +10 Fortitude, +10 Will for 8.4 sec
Here the linger time has been modified by Int, it also looks like the total time in the end of the tooltip has been modified by Int ( (4+2)*1,4)=8.4 ). Now we have three possible outcomes that are easily distinguished, 6 second if Int has no effect on duration, 6,8 seconds if Int only modifies Linger duration, and 8.4 seconds if it modifies both Duration+Linger.
When I tried it the phrase is applied for 6.8 seconds so Int only affects Linger time. Whether this is intended behaviour or not I don't know (though I think this is how it's supposed to work) but the tooltip needs to be cleaned up regardless.