Is the following function real analytic in $t>0$:

$$F(t)=\int_0^1\frac{\exp(ctx)}{\sqrt{(\exp(bt)-1)(1-\exp(atx))-(1-\exp(at))(\exp(btx)-1)}} dx,$$

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.

**Motivation:**

This question arised during my efforts to show non-degeneracy of certain integrable systems. Real analyticity would make showing the non-degeneracy quite easy.