Impedance of a lossless line
EDIT this is valid only for the original question. The edited version of the question has simpler calculation principle and the result is right.
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Your result is wrong. It resembles the right formula, but there's wrong signs. Your extra trip via the equation between impedance and reflection factor creates numerous possibilities to make errors.
You should calculate the impedance as Ut/It where Ut is the total voltage at the beginning of the line and It is the total current at the beginning of the line.
Assume there's at the beginning of the line an arbitary incident wave with voltage amplitude Uo and summed with it there's the reflected wave Uo*r where r is the reflection factor with the delay exponential term.
Calculate the incident and reflected current components with the wave impedance. Keep in mind that for the reflected current component you must take it's direction as reversed, so there's not Io+Ir, but Io-Ir as the total current. That can be shown with the energy flow direction rule of the general circuit theory.