Suppose that we have a CDF $F(x)$ and its associated pdf $f(x)$, $x\in[0,1]$.
What I want to evaluate is the following definite integral:
$$\int^1_0I_x(a,a)f(x)dx.$$
Here, $I_x(a,a)$ is the regularized incomplete beta function with parameters $(a,a)$ which is defined as $$I_x(a,a)=\frac{B_x(a,a)}{B(a,a)}=\frac{\int^x_0t^{a-1}(1-t)^{a-1}dt}{B(a,a)},$$ for some $a>0$. For simplicity, we may assume $a$ is an integer.
My question is: Can this integral be further simplified? I've tried to solve it using the integration by parts. However, the value after the integration becomes $$1-\frac{1}{B(a,a)}\int^1_0F(t)t^{a-1}(1-t)^{a-1}dt$$ and I don't know how to further simplify this expression..