Finding an elementary evaluation of $B_{1/2}(a,1-a)$

Question

I'm trying to prove $$B_{1/2}(a,1-a):=\int_0^{1/2}x^{a-1}(1-x)^{-a}dx=\int_0^1\frac{x^{a-1}-x^a}{1-x^2}dx$$ $(a>0)$ (where $B$ denotes the incomplete beta function) with elementary method.
I have already found a way (with hypergeometric function) to prove this identity. $$\begin{aligned}LHS&=2^{-a}\int_0^1x^{a-1}(1-x/2)^{-a}dx\\&=\frac{2^{-a}}{a}{}_2F_1(a,a;1+a;1/2)\\&=\frac1a{}_2F_1(1,a;1+a;-1)\\&=\frac12\left(\psi\left(\frac{1+a}2\right)-\psi\left(\frac a2\right)\right)\\&=RHS\end{aligned}$$ I wonder if there is a solution without using hypergeometric functions.