The inequality $\int^\infty_0 \frac{\sin(rt)}{rt}\frac{r^4}{\sinh^2(r)} e^{-ar\coth(r)}dr\leq c \big(e^{-At}\big)$ Let $a>0$. How to prove the following inequality $$\exists c>0,\exists A>0,\forall t>0:\quad\int^\infty_0 \frac{\sin(rt)}{rt}\frac{r^4}{\sinh^2(r)} e^{-ar\coth(r)}dr\leq c \big(e^{-At}\big)$$
for some constants $A>0,c>0$
 A: $\newcommand{\R}{\mathbb R}\renewcommand{\S}{\mathcal S}$This follows from Theorem IX.14, which states the following:

Let $T\in\S'(\R^n)$. Suppose that the Fourier transform $\hat T$ of $T$ can be continued analytically to the set $\{z\colon|\Im z|<A\}$ for some $A>0$. Suppose also that for each $B<A$ we have $\sup\{\|\hat T(\cdot+iy)\|_1\colon y\in(-B,B)\}<\infty$. Then $T$ is a bounded continuous function and for each $B<A$ there is a real number $C_B$ such that
\begin{equation*}
    |T(x)|\le C_B e^{-B|x|}
\end{equation*}
for $x\in\R^n$.

Indeed, for the integral in question,
\begin{equation*}
    I(t):=\int^\infty_0 \frac{\sin rt}{rt}\frac{r^4}{\sinh^2r} e^{-ar\coth(r)}\,dr,  
\end{equation*}
and
\begin{equation*}
    T(t):=t I(t), \tag{1}\label{1}
\end{equation*}
we have $T=\frac{\sqrt{2\pi}}{2i}\,\check f$, where
\begin{equation*}
    f(r):=\frac{r^3}{\sinh^2r} e^{-ar\coth(r)}
\end{equation*}
and $\check f$ is the inverse Fourier transform of $f$ (defined by the formula $\check f(t):=\frac1{\sqrt{2\pi}}\,\int_\R e^{irt}f(r)\,dr$), so that
\begin{equation*}
    \hat T=\frac{\sqrt{2\pi}}{2i}\,f. 
\end{equation*}
It is not hard to see that the function $T$ defined by \eqref{1} satisfies all the conditions of Theorem IX.14, cited above, with $A=\pi$. So, for each $B\in(0,\pi)$ there is a real number $C_B$ such that $|T(t)|\le C_B e^{-Bt}$ for all $t\ge0$ and hence
\begin{equation*}
    |I(t)|\le C_B e^{-Bt} 
\end{equation*}
for $t\ge1$. On the other hand, for all $t\in[0,1]$,
\begin{equation*}
    |I(t)|\le c_a:=\int^\infty_0\frac{r^4}{\sinh^2r} e^{-ar\coth(r)}\,dr
    \le c_{a,B}\, e^{-Bt},
\end{equation*}
where $c_{a,B}:=c_a e^B\in(0,\infty)$.
Thus, for each $B\in(0,\pi)$ and all $t\ge0$
\begin{equation*}
    |I(t)|\le\max(C_B,c_{a,B}) e^{-Bt},
\end{equation*}
as desired.
