I have several Fujitsu ductless split ACs. I'm very happy with them. I don't know if they're any quieter than the competition, but they sure are quiet. They have four fan speeds: high, medium, low and "quiet"... in "quiet" mode, you can barely hear anything. But of course, in quiet mode, the fan is spinning VERY slowly, not moving much air.
For tracking anything somewhat "loud", as long as you are a decent distance from the AC unit, I think you could run the Fujitsu on quiet mode and you'd be totally fine.
However, as quiet as the Fujitsu is, I would say you could not record a very low level source within close proximity of the AC while it is on... you might be able to get away with it, but I would not recommend it... if you're recording a soft vocal and have to compress it hard later, ANY tiny degree of extra noise in the room could be an issue, at least if you're fussy. In this case, ANY AC unit will be a problem. Finally, there' no such thing as an AC unit / fan that can run inside a studio and be 100% perfectly quiet.
Also keep in mind, you can keep the AC running all day and just shut it off when you're about to record a low-level source. If your room is kept nice and cool and dehumidified all day long, it should be fine to shut the AC off for 20 minutes or whatever if you need to track something soft... then turn it back on when you're done. The key is to make sure the room is fully cool and dehumidified BEFORE you plan on shutting it off to track. Hopefully your room is tight and well insulated, if so, you should be fine... (and hopefully you don't have ten 21" CRT monitors in there).
In one old room I used to use, I had a pretty loud AC in there, I'd turn it on early in the morning, wait until the room was fully cool, then I'd shut it off, track for say 20 minutes, then turn it back on, take a 10 or 15 minute break, return some phone calls, whatever, then go back, shut it off, track for another 20 minutes, and so on. You'll often want to take breaks when tracking anyway.
In sum, in my opinion, you can't go wrong with the Fujitsu. I've heard good things about the Mitsubishi units too, but have not experienced one up close in recent times. But I'm really happy with my Fujitsus. If you're doing anything loud like tracking drums or re-amping guitars etc, you can certainly run a Fujitsu in quiet mode, you'll never hear it on tape, no way.