Let $s>0$ be fixed, and consider for $p>0$, $\alpha>0$, the integral $$I_s(p,\alpha)= \int_0^1 t^p e^{-s\left(1-t^2\right)^{-\alpha}} dt $$
For fixed $\alpha$, one has $I_s(p,\alpha)\to0$ as $p\to\infty$, and for fixed $p$, one has $I_s(p,\alpha)\to1/(e(p+1))$ for $\alpha\to0$. I wonder if there holds a bound of the form
$$I_s(p,\alpha)\leq\frac{c_s}{\alpha^{q_s}}\,e^{-a_s\,p^{\beta_s}}$$
for all $p,\alpha$, with suitable parameters $c_s,a_s,q_s,\beta_s>0$, i.e. (stretched) exponential decay in $p$ times polynomial behaviour in $\alpha$.
Of course, it would be best if the integral could be calculated, or related to known functions, so that this behaviour could be checked.