Let $x\in \mathbb{R}^{d}$, $d\geq 1$ and $p>\frac{2}{d}$. For any $R>0$ and any $0<s<\frac{dp}{2}-1$
\begin{align} &\int_{\mathbb{R}}\int_{|x|>R}\left(\frac{1}{1+t^{2}}\right)^{\frac{d(p+2)}{4}}e^{-\frac{|x|^{2}(p+2)}{4(1+t^{2})}}dxdt\\ &\lesssim R^{-s}\int\int |x|^{s}\left(\frac{1}{1+t^{2}}\right)^{\frac{d(p+2)}{4}}e^{-\frac{|x|^{2}(p+2)}{4(1+t^{2})}}dxdt\\ &\lesssim R^{-s}\int (1+t^{2})^{-\frac{dp}{4}+\frac{s}{2}}dt\\ &\lesssim_{s} R^{-s} \end{align}
Trying to understand the sequence of inequalities above, the first one simply applies Chebyshev's inequality. However, from the second to the third one, I don't understand the upper bound on the exponential function that is used.