I would like to evaluate an integral: $$\int_t^1\frac{1}{B(1+s\phi,1+\phi(1-s))}p^{s\phi}(1-p)^{\phi(1-s)}ds,$$ where $B(a,b)$ is a beta function and $p\in(0,1)$ and $\phi>0$ are some parameters. Notice that the integration is with respect to $s$, which appears both in the exponent and in the beta function. This is integrating over a family of pdfs of beta distributions, indexed by the parameters in beta.

As an analogy with this question, I would expect the integral to be connected to $B(t\phi,1+\phi(1-t))$ distribution, presumably, equal to the incomplete beta function evaluated at $p$: $$B(p;t\phi,1+\phi(1-t)).$$

However, I have no idea whether this intuition is correct and how to prove it.

Edit: apparently, this integral has sth do to with $\nu$ function. However, I have not found any formula to help me with the problem.

Edit2: since I got no hints, I am thinking of a workaround. In principle, I do not need to caculate the integral, I just need to determine for which $t$ we have: $$\int_t^1\frac{1}{B(1+s\phi,1+\phi(1-s))}p^{s\phi}(1-p)^{\phi(1-s)}ds>c,$$ for some predefined $c\in[0,1]$. Maybe there is a way to derive it without explicit formula for the integral.


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