Is the following function real analytic in $t>0$:
where $-a$ and $b$ are positive, and $c\not=a$?
I have consulted a large table of integrals looking for a closed form (for $t=1$), but without success.
This question arised during my efforts to show non-degeneracy of certain integrable systems. Real analyticity would make showing the non-degeneracy quite easy.