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 $x,t$?