Assume $X, Y$ to be random variables with Marshall-Olkin bivariate exponential distribution with parameters $\lambda_1, \lambda_2$ and $\lambda_{12} > 0$.
It is known that the following distributions hold: $$ \mathbb{P}(X > x, Y > y) = e^{-\lambda_1 x - \lambda_2 y - \lambda_{12} \max(x,y)}$$
$$ \mathbb{P}(X > x) = e^{-(\lambda_1 + \lambda_{12})x}$$
$$ \mathbb{P}(Y > y) = e^{-(\lambda_2 + \lambda_{12})y}$$
I'm interested in finding the $\text{Cov}(X,Y)$, for which I'm following the Hoeffding's Identity:
$$ Cov(X,Y) = \int\int_{\mathbb{R}^2} \left[ \mathbb{P}(X \leq x, Y \leq y) - \mathbb{P}(Y \leq y)\mathbb{P}(X \leq x) \right]dydx$$
It's clear that the identity holds for the right tail:
$$ Cov(X,Y) = \int\int_{\mathbb{R}^2} \left[ \mathbb{P}(X > x, Y > y) - \mathbb{P}(Y > y)\mathbb{P}(X > x) \right]dydx$$
By just plugging in the known facts we get that
$$ Cov(X,Y) = \int\int_{\mathbb{R}^2} \left[ e^{-\lambda_1 x - \lambda_2 y - \lambda_{12} \max(x,y)} - e^{-(\lambda_1 + \lambda_{12})x} e^{-(\lambda_2 + \lambda_{12})y} \right]dydx$$
It is known that $Cor(X,Y) = \frac{\lambda_{12}}{\lambda_1 + \lambda_2 + \lambda_{12}}$. However, this seem to diverge, according to WolframAlpha.
What am I missing or where am I wrong?