Pretty sure and LLC is taxed the same as an individual. The profits flow through to your individual return. Corporations, however, may be different depending how they are set up.
But, horses are assets and therefore depreciate, even if in our minds they appreciate with training and a show record. So in some instances you MIGHT be better off paying a short term capital gain than the long term gain on a larger gap between the sale price and the depreciated value of the horse.
Generally speaking, capital gains rates apply if an asset is bought and sold for investment purposes outside the ordinary course of business. For example, if one purchased some shares of IBM as an investment and later sold them, the capital gains tax rates would apply. Conversely, if a horse trainer purchased and sold a horse, that profit would be taxed at "ordinary income" rates since buying, selling and training horses is part of a horse trainer's ordinary course of business.
It is probably worth noting that short term capital gains are taxed at the same rates as ordinary income (no tax advantage). Long term capital gains (asset held for more than a year) are taxed at a lower rate than ordinary income.
Limited Liabilty Corporations (LLC's) are pass through vehicles, meaning income is passed through to the owner(s) and the income is taxed at the respective owner's individual tax rate. Generally, folks adopt the LLC structure over a sole-proprietorship for the limited liability protection it affords.