Denote $t=2\tau$. $$
\int_{\mathbb R}\frac{dx}{(1+x^2)^a\,(1+(x-2\tau)^2)^a}=\int_{\mathbb R}\frac{dx}{(1+(x+\tau)^2)^a\,(1+(x-\tau)^2)^a}.
$$
We have 
$$
\left(1+(x+\tau)^2\right)\,\left(1+(x-\tau)^2\right)=\left(x^2+\tau^2+1\right)^2-4\tau^2x^2=x^4+2x^2(1-\tau^2)+(\tau^2+1)^2=:x^4+Ax^2+B,
$$
so our problem for reduces to evaluation of integral 
$$
\int_{\mathbb{R}} \frac {dx}{(x^4+Ax^2+B)^a}=2\int_0^{\infty} \frac {dx}{(x^4+Ax^2+B)^a}.
$$ 
Denote $x^2=y$, we get 
$$
\int_0^{\infty} \frac {y^{-1/2}dy}{(y^2+Ay+B)^a}.
$$
Well, this is number 2.2.9.7 in Prudnikov-Brychkov-Marichev. In my home edition this specific integral is corrected by my father (what a coincidence!). I asked him, hopefully it was just a misprint in the exponent of $a$, and the result is
$$
\int_0^{\infty} \frac {x^{\alpha-1}dx}{(ax^2+2bx+c)^\rho}=
a^{-\alpha/2} c^{\alpha/2-\rho} B(\alpha,2\rho-\alpha)\,_2F_1(\frac{\alpha}2,\rho-\frac{\alpha}2;\rho+\frac12,1-\frac{b^2}{ac})
$$
for natural assumptions $a>0$, $b^2<ac$, $0<\Re \alpha<2\Re \rho$.