$$\int_{t_2}^{t_1} \exp(-\sqrt{a^2+z^2}) dz$$
where $a>0$
One related question is here in which the integration range is $(-\infty, +\infty)$.
In Gradshteyn and Ryzhik's book 3.461, I found the following:
$$\int_{0}^{\infty} \exp(-\beta\sqrt{a^2+z^2}) dz=aK_1(a\beta)$$
Let's simplify it to solve
$$\int_{0}^{t} \exp(-\sqrt{a^2+z^2}) dz$$
Do you get to something interesting if you try thr variable change z=acos(x)? Then maybe some complex analysis