From what little I can find out there on the Internet regarding this combination of subjects, I see some holes. Filling in holes is a pretty good reason to use a blog, so here I go!
Firstly, for a background on Integral Theory, I highly recommend that you read Ken Wilber's work, and some of his compatriots like Robert Kegan. There are already whole blogs that do nothing but paraphrase his many volumes, and I am not going to do that here. I am going to assume that the reader has a basic knowledge of Integral Theory and Martial Arts as well. If you do not, probably the best place to start is with The Integral Vision: A Very Short Introduction to The Integral Approach to Life. It's a quick read with lots of pictures and not too many words. The "hard core" Ken Wilber stuff is exactly the opposite and needs to be distilled.
When I read current web pages on "Integral Martial Arts", they are usually written by or written from the perspective of fighters. These can be cage fighters, Mixed Martial Arts (MMA) people, or just folks that have had tough lives and learned to fight for survival. The sum total of that body of literature seems to be summed up in one sentence: If you fight long enough and enough times, you start to realize that there is a "higher state" to fighting and a higher mindset that you reach which makes you stronger and more able to win fights.
That's excellent, but it's not Integral in the sense that Integral Theory talks about. Winning more fights is a good goal for those fighters, I suppose, and tapping into whatever higher states can enhance fighting will definitely win them more trophies and save their lives, even. However, again, this is not Integral. For a practice to be Integral, it must include awareness of all States, Stages, Lines of development, and Quadrants. The abbreviation for this is "AQAL", which is a loose abbreviation, ok, but it roughly stands for All Quadrants All Lines, Levels (i.e. Stages), and States.
In light of AQAL, what these admirable martial artists are pursuing is becoming the perfect martial artists at the Orange Level or Stage. For some of them, it is actually a less broad Red Stage. The people that want to pursue these States and abilities for the purpose of beating other martial artists in contests are operating at Orange, and the ones that argue the only reason for martial arts is to preserve your life during a catastrophe are at the Red Stage. In fact, both of them are correct. They are also partial. My idea is to broaden the scope of what they are trying to say.
As the world becomes more complex, it calls for more complex thinking and more complex solutions. This is the basis for emergence of the broader Stages. From a fighting perspective, if you live in a sub-urban upper-middle-class neighborhood, chances are you are going to live much of your life, if not all of it, without ever needing to fight. If the toughest thing on your streets is a 12-year-old skater with a helicopter mom in tow, there must be some other reason at that point to practice martial arts. Of course, if you are growing up on a street corner in Mogadishu, Somalia, forget what I just said. You shouldn't be reading this blog. You should be learning to fight. Different parts of the world exist at different Stages, and different people do too. The Red and Orange Stage Martial Arts like MMA suit these people amazingly well, but what if that is not you? What if you see the Martial Arts differently, like I do?
In my view, the Martial Arts has a ton of other benefits besides learning to fight. In fact, fighting is probably one of the least useful benefits that Martial Arts teaches in today's world. Rather, Martial Arts can be a whole Life Practice, especially if done from an Integral perspective. From a Four Quadrant view, the Upper Left (UL) quadrant which deals with your internal self is served immensely by Martial Arts. You can learn to feel and manipulate your own subtle energies, including your thoughts. In the Upper Right (UR) quadrant you have all your physical benefits from exercise and physiological enhancements. In the Lower Left (LL), Martial Arts pursues understanding between individuals and among the members of groups. In other words, you learn how to get along with people, which increases your Emotional Intelligence (ala Daniel Goleman), which research shows increases your success at your job and career. In the Lower Right (LR), you can learn about systems for anything from running a business to the practice of healing arts. There is so much there!
Look at it this way.... If you are an average sub-urbanite, how many times in your life do you fight? On the other hand, how many times do you breathe in and out? If Martial Arts can be used to optimize your breathing and improve your health because of that, how useful is that to you? Like breathing, there are a lot of other things the Martial Arts can teach you to improve, and these are things you do every day, rather than something you do maybe once or twice in your life, i.e. combat.