"Inauthentic" is more commonly used than "unauthentic", at least these days, but they are both valid. Several sources (World Wide Words for example) suggest what prefix the negative form of an English adjective takes depends on its roots.
In general, words take un- when they are of English (Germanic) origin and in- if they come from Latin. (The forms im-, il-, and ir- are variations on in-.)
So logically the negative of "authentic" takes "in". "Inauthentic" it is! Problem solved. Or is it? A quick Google Ngram lookup shows that "unauthentic" was much more common before the 1950s and "inauthentic" had barely occurred by that point. Why does "unauthentic" seem to be the older, more traditional form of negation, making "inauthentic" seem inauthentic?
This popped up on the side bar after I posted my question. For reference: