I recently came across the following integral identity
$\int_0^1 \,\frac{x^2}{(1-x^2)^2}[P_{-1/2}(x)-x\,P_{1/2}(x)]^2\,\mathrm{d}x=\frac{3}{2}-\frac{4}{\pi}\,,$
where $P_\alpha(x)$ is a Legendre function of order $\alpha$.
I managed to write the integral above in terms of a particular ${}_2 F_1(a,b;c,z)$ hypergeometric function. In particular, I use the relation
$P_{-1/2}(x)-x\,P_{1/2}(x)=\frac{3}{2}(1-x)\;{}_2F_1\left(-\frac{1}{2},\frac{3}{2};2,\frac{1-x}{2}\right)$
which brings the integral identity to
$\frac{9}{4}\int_0^{1}\frac{x^2}{(1+x)^2}\;{}_2F_1\left(-\frac{1}{2},\frac{3}{2};2,\frac{1-x}{2}\right)^2=\frac{3}{2}-\frac{4}{\pi}$
But I am still stuck in proving the above identity! Any thoughts?
Thanks for your help!