OK, got the formula...


Of course, as a general rule starships want to slow down and stop at their destinations, not zip past them at 0.9999 of the speed of light. You need a standard torchship brachistochrone flight plan: accelerate to halfway, skew flip, then decelerate to the destination (which makes sense, since such starships will have to be torchships). To use the above equations, instead of using the full distance for d, divide the distance in half and use that instead. Run that through the equations, then take the resulting T or t and double it.

Example: The good scout starship Peek-A-Boo is doing a 1 g brachistochrone for Vega, which is 27 light-years away. Half of that is 13.5 light-years. How long will the journey be from the crew's standpoint (the proper time) ?

T = (c/a) * ArcCosh[a * d / (c2) + 1]
T = (1/1.03) * ArcCosh[1.03 * 13.5 / (12) + 1]
T = 0.971 * ArcCosh[13.9 / 1 + 1]
T = 0.971 * ArcCosh[13.9 + 1]
T = 0.971 * ArcCosh[14.9]
T = 0.971 * 3.39
T = 3.29 years
That's the crew time to the skew flip. The total time is twice this
T = 3.29 * 2
T = 6.58 years


But if you have more mathematical skills than I have, you could easily derive this short cut:

Tt = 1.94 * ArcCosh[dly/1.94 + 1]

where
Tt = Proper Time experienced during a brachistochrone flight (years)
dly = total distance to destination(light-years)

Remember this equation assumes a constant 1 g acceleration.