I noticed a question on this forum, whether functions are known, for which it is known that a closed-form antiderivate exists, but the antiderivate is unknown.
Since this question would interest me as well, I wonder whether this question was deleted. I could not find it anymore.
To repeat the question
Are there functions with unknown antiderivates known to have a closed-form antiderivate ?
The answer should be no, since the Risch-algorithm finds the antiderivate, if it exists, but I am not sure.