Any idea how to attack this integral $\int_0^{\infty} p^2 e^{-\beta\sqrt{p^2+1}} \, dp$

Question

I need to do this integral:

$\int_0^{\infty} p^2 e^{-\beta\sqrt{p^2+1}} \, dp$

($\beta > 0$)

Any technique how to do this integral? (apparently not possible in closed form) or at least produce the result as a rapidly converging series.

A closely related formula (just a Gaussian integral):

$\int_0^{\infty} p^2 e^{-\beta p^2 } \, dp = \frac{\sqrt{\pi }}{4 \beta ^{3/2}}$

and on the other hand:

$\int_0^{\infty} p^2 e^{-\beta p } \, dp = \frac{2}{\beta^3}$

Progress 1: So, intuition tells that for $\beta\to 0$ the integral behaves like $\beta^{-3}$ and for $\beta\to\infty$ as $\beta^{-3/2}$.

Progress 2: From related texts in the field I am getting the hint that it maybe expressable in terms of a modified Bessel function $K_2$.

Progress 3 (solution): From experimenting with Bessel functions I get that

$\int_0^{\infty} p^2 e^{-\beta\sqrt{p^2+1}} \, dp = K_2(\beta)/\beta$

I'll still accept an answer with the derivation

Answer 1

Substitute $p=\sinh{t}$ and get the integral is equal to

$$\int_0^{\infty} dt \, \left ( \cosh^3{t}-\cosh{t}\right) e^{-\beta \cosh{t}}$$

Use the fact that

$$K_1(\beta) = \int_0^{\infty} dt \, \cosh{t}\, e^{-\beta \cosh{t}}$$

and the recurrence relations for modified Bessels; I get as the value of the integral

$$K_1''(\beta)-K_1(\beta) = \frac14 [K_3(\beta)- K_1(\beta)]$$