Since the convergence is different, it is necessary to use different equations for positive and negative λ.
Since p(λ) has periodicity due to sin, this is used.
If evaluated here for each cycle of sin, the integration point can always be positive.
Evaluate the peak point of the integrated function. It can be seen that t < 1 always.
Evaluate the upper bound.
Similarly, a lower bound is required.
For negative λ, do a variable transformation and use an equation with small oscillations that decay rapidly.
When λ → -∞, the integral J converges to 1.
Evaluate the upper bound.
Also, variable transformation is performed so that the integral interval is finite.
W.Börsch-Supan, "On the Evaluation of the Function Φ(λ) for Real Values of λ" (1961)