Let $p$ and $q$ be probability densities on $\mathbb R$, with respect to the Lebesgue measure $dx$. The corresponding Hellinger integral is 
$H(p,q):=\int_{\mathbb R}\sqrt{pq}\,dx$. 

Let now $p$ be the density of Student's distribution with $d$ degrees of freedom, so that 
$$p(x)=C_d\,(1+x^2/d)^{-(d+1)/2}$$
for real $x$, where $d\in(0,\infty)$ and $C_d$ does not depend on $x$. 

At least for $d=1$, is there a closed form expression for $H(p,p_t)$ for real $t\ne0$, where $p_t(x):=p(x-t)$ for real $x,t$? 

An obviously equivalent form of this question: Is there a closed form expression for the integral 
$$\int_0^\infty\frac{dx}{(1+x^2)^a\,(1+(x-t)^2)^a}
$$
for all real $t\ne0$ and all real $a>1/4$ (or at least for $a=1/2$)? Of course, this is not a problem for natural $a$.