Complete permutation polynomials from exceptional polynomials

Complete permutation monomials of degree $1+rac{q^n-1}{q-1}$ over the finite field with $q^n$ elements in odd characteristic, for $n+1$ a prime and (n+1)^4 less than q, are classified. As a corollary, a conjecture by Wu, Li, Helleseth, and Zhang is proven in odd characteristic. When n + 1 is a power of the characteristic we provide some new examples. Indecomposable exceptional polynomials of degree 8 and 9 are also classified.