I need to solve a double integral where the integrand converges to infinity in the middle of the integral's range. Following is the integral in question:
\begin{equation} \iint \limits_{-\infty}^{+\infty} \Phi^{-1}(1 - 2(1 - \Phi(|t_i|))) \times \Phi^{-1}(1 - 2(1 - \Phi(|t_j|))) \times f_{T_i, T_j}(t_i, t_j) \mathrm{d}t_i \mathrm{d}t_j \end{equation}
where $t_i$ and $t_j$ follow $\textrm{N}(0, 1)$, $f_{T_i, T_j}(t_i, t_j)$ is the probability distribution function of $\textrm{MVN}\left(\left(\begin{array}{c}0 \\ 0\end{array}\right), \left(\begin{array}{cc}1 & \rho \\ \rho & 1\end{array}\right)\right)$ with a known covariance, $\rho$, and $\Phi(.)$ is the density function of a standard normal distribution, whereas $\Phi^{-1}(.)$ is its inverse.
The problem arises due to that when either $t_i$ or $t_j$ converges to 0, the integrand converges to $-\infty$. If I replace $t_i$ with 0, for instance, the first term becomes $-\infty$ as shown step-by-step below:
$$ \Phi(|0|) = 0.5 $$ $$ 2(1 - \Phi(|0|)) = 1 $$ $$ 1 - 2(1 - \Phi(|0|)) = 0 $$ $$ \Phi^{-1}(1 - 2(1 - \Phi(|t_i|))) = -\infty $$
Note that $f_{T_i, T_j}(t_i, t_j)$ still has a density when either $t_i$ or $t_j$ is 0, so the $-\infty$ is multiplied with a non-zero value. The problem can also be shown on a univariate case (in order to simplify the problem a little) where the integral can be defined as
\begin{equation} \int \limits_{-\infty}^{+\infty} \Phi^{-1}(1 - 2(1 - \Phi(|t_i|))) \times f_{T_i}(t_i) \mathrm{d}t_i \end{equation}
where $t_i \sim \textrm{N}(0, 1)$ and $f_{T_i}(t_i)$ is its pdf. The univariate case also has a similar problem, i.e., the integrand converges to $-\infty$ as $t_i$ converges to 0 (please see the image below that plots the integrand, denoted by X(.), over a reasonably wide interval, (-5, 5), for the standard normal distribution).
I tried to find a closed form of both the functions by using substitutions, but unfortunately I wasn't successful. I tried to solve them numerically with some software. R and Mathematica gave results with numerical approaches, but I am not sure if these numerical results are mathematically meaningful in such a case.
So, this might be a very easy question (I admit that I have been too idle since the last time I took a calculus course); however, is the first integral (bivariate case) solvable, and if so how can I solve it? Or, can I trust the numerical results, do they make sense? I would appreciate any help or explanation on this matter.
Thanks in advance.