A nonpolynomial one-dimensional quantum potential in the form of an isotonic oscillator (harmonic oscillator with a centripetal barrier) is studied. We provide the nonrelativistic bound state energy spectrum En and the wave functions ψn(x) in terms of the associated Laguerre polynomials in the framework of the Nikiforov-Uvarov method. Under the spin and pseudospin symmetric limits, the analytic eigenvalues and the corresponding two-component upper- and lower-spinors of the Dirac particle are obtained in closed form.