Computed tomography (CT) sequentially interrogates the object of interest from a complete set of view angles. Sequential scanning in CT introduces an acquisition delay and high radiation dose. This paper proposes a compressive sensing based “snapshot” coded X-ray CT (CXCT) method, where the object is simultaneously illuminated by multiple fan-beam X-ray sources equipped with coding masks in a fixed circular gantry. Low radiation dose is achieved by the use of incomplete projection measurements and encoded structured illuminations. Since all the measurement data are produced in one snapshot, the inspection time and motion artifacts are effectively reduced. Due to the overlap of X-rays in the measurements from several sources, a nonlinear reconstruction framework is established based on rank, intensity and sparsity priors. Then, a Newton split Bregman algorithm is exploited to reconstruct the object from a small set of nonlinear encoded measurements. Compared to the state-of-the-art CXCT approaches based on a linear model, the proposed method reduces the inspection time and motion artifacts significantly, achieving higher or comparable reconstruction accuracy.