Quite right. What was the Captain thinking .....?
In fact a solution does exist: There is a special function called the Fantastic F-function of the Second Kind that's exactly the answer to your problem. The properties of this function are left for you to explore as I don't think much literature on it exists at the moment.
(The Fantastic F-function of the First Kind was suggested to someone quite a few years ago now as the answer to an unrelated problem. No doubt a function of the Third Kind will be suggested in the future to someone else).
You are joking, right?! I am desperately looking for a closed form solution for that integral to save cpu time, and was glad to see the strange F-function, but a second thought realizes that it does not exist at the time being ....
Seems there are really very good mathematicians on this forum, , thanks for letting me know the Risch algorithm ...