Dynamical Equations from a First-Order Perturbative Superspace   Formulation of 10D N=1 String-Corrected Supergravity (I)
  Utilizing a first-order perturbative superspace approach, we derive the bosonic equations of motion for the 10D, N = 1 supergravity fields. We give the Lagrangian corresponding to these equations derived from superspace geometry. Moreover, the equivalence of this Lagrangian to the first-order perturbative component level Lagrangian of anomaly-free supergravity is proven. Our treatment covers both the two-form and six-form formulations. 
S. J. Gates Jr, Annamaria Kiss, Willie Merrell