How to calculate the integral of some form of error funciton?

Question

Let $\text{erf}(x)$ be the error function given by $$\text{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}dt.$$ I want to calculate the following integral $$\int_{-1}^{+\infty} e^{-(t+1)^2} \text{erf}(t)dt.$$ To solve this, I have checked some reference, but it only tells the result when then integral domain is $[-\infty,+\infty]$. How can I handle this integral with the required domain? If we can not get the solution, how can we obtain an approximation?