We construct a new class of charged rotating black string solutions coupled to dilaton and exponential nonlinear electrodynamic fields with cylindrical or toroidal horizons in the presence of a Liouville-type potential for the dilaton field. Due to the presence of the dilaton field, the asymptotic behaviors of these solutions are neither flat nor (A)dS. We analyze the physical properties of the solutions in detail. We compute the conserved and thermodynamic quantities of the solutions and

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... the first law of thermodynamics on the black string horizon. When the nonlinear parameterβ2goes to infinity, our results reduce to those of black string solutions in Einstein-Maxwell-dilaton gravity.