I would like to know whether this integral could be solved analytically

$\int_{-\infty }^{\infty } e^{-x^2} \text{erf}(k-x)^2 \, \text{d}x$

where $k$ is a real constant.

P.S. I know that $\int e^{-x^2} \text{erf}(x)^2 \, \text{d}x = \frac{1}{6} \sqrt{\pi } \text{erf}(x)^3$ .


For what it's worth, I can give you the approximate solution

$$ \int_{-\infty }^{\infty } e^{-x^2} \text{erf}(k-x)^2 \, \text{d}x\approx \sqrt{\pi }-\sqrt{\frac{8 \pi }{8+\pi ^2}}\text{exp}\left(-\frac{k^2 \pi ^2}{8+\pi ^2}\right) $$


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