Finding success probability for wireless channels
As I have determined the answer, I am just posting it so that it is helpful for others.

First, let $$X = |h_{su_i}|^2, Y = |h_{se_i}|^2 \text{ where } X,Y\sim \exp(1) \ $$

This detail is missing in the research paper. Then:

$$P_{secure} = \text{Pr}\{X&gt; \frac{d_{se_i}^{-n}Y}{\epsilon d_{su_i}^{-n}}\}$$

$$P_{secure} = \text{Pr}\{X&gt; sY\}$$
Following Complimentary CDF of exponential random variable:
$$P_{secure} = \exp(-sY)$$

Since Y is a exponentially distribued random variable with mean 1, we will take its Expected value:

$$P_{secure} = \mathbb{E}[\exp(-sY)]$$

Following the Laplace Transform property for exponentially distributed Random variable, we can write as:

$$P_{secure} = \frac{1}{1+s}$$

which can be simplified as
$$P_{secure} = \frac{\epsilon d_{se_i}^n}{d_{su_i}^n+\epsilon d_{se_i}^n}$$