How to know history URLs

I have a program which creates links to FlightAware flight history pages from flight info that I have.

I’ve figured out that the URL format is like this:
https://flightaware.com/live/flight/{flightNumber}/history/{flightDate}/{utcTakeOffTime}/{departureAirport}/{arrivalAirport}

Example:
https://flightaware.com/live/flight/AAL2613/history/20180524/1953Z/KOMA/KDFW

My question is, is there a way to use scheduled departure time rather than take off time? I have the scheduled departure time, but no idea how I would obtain the take off time. When I try using scheduled departure, I always get a “Not Found” page directing me to the most recent flight.

The URL will accept actual departure time and filed departure time. The filed departure time is generally 10 minutes later than scheduled departure time (sometimes they are the same time).

So in your example link for AAL2613 with a 1943 scheduled time, adding 10 minutes gets you the 1953 in that URL.

Thank you for your reply. So the URL does not accept the scheduled departure time? I think it would be more useful if it also accepted this as it would be the time most people would actually know. I can’t work with a general rule because there will be too many exceptions to the rule.