A uniform asymptotic expansion of the time-harmonic antenna current with large parameter ln <2 kz/(ka)2> is given. As usual, the expansion is based on Hall¿n's filament integral and is restricted by the condition z≫a but is otherwise uniform in kz. The use of averaging leads to the vanishing of the second term of the asymptotic series. Two different averaging procedures are used to derive two accurate formulas: The first one, which is very simple, is useful for practical applications. The second formula is of theoretical interest. An improved asymptotic series for large kz is derived to relate the averaging results to existing asymptotic results. Comparison of these two formulas and the exact antenna integral demonstrates that these formulas are more accurate than any other existing formulas for infinite cylindrical antenna current. The Hall¿n filament integral is found to approximate closely the exact integral event when the condition z≫a is violated. Discrepancies in the literature are discussed.