Let $\alpha,\beta\in \mathbb{R}$ such that $\alpha<\beta$ and let consider the following integral : $$\displaystyle \int\limits_{[\alpha,\beta]} \dfrac{1}{\sqrt{(\beta-t)(t-\alpha)}}\mathrm{d}\lambda(t) \ $$(where $\lambda$ is the Lebesgue's measure).
I was wondering if we could link this integrals with elliptic integral and to go further, with Jacobi's functions ?
A first type of elliptic integral can be written as follows : $$f(x,k)= \displaystyle \int \limits_{[0,x]} \dfrac{1}{\sqrt{(1-s^2)(1-k^2s^2)}}\mathrm{d}\lambda(s) \ $$ with $0\le k\le 1.$
We can notice that $(\beta-t)(t-\alpha)\in\mathbb{R}_2[t]$ with simple roots. Hence it leads to the problem that for an elliptic integral, it should be at least of degree $3$ or $4$ with simple roots. Does this fact block the problem ?
Notice also that using $t=\alpha+u(\beta-\alpha)$ we obtain : $$\displaystyle \int\limits_{[\alpha,\beta]} \dfrac{1}{\sqrt{(\beta-t)(t-\alpha)}}\mathrm{d}\lambda(t) =\displaystyle \int\limits_{[0,1]} \dfrac{1}{\sqrt{u(1-u)}}\mathrm{d}\lambda(u).$$
Thanks in advance !