$$0.5 \int_{-\infty }^{\infty } \frac{e^{-\frac{x^2}{4 (c+k L)^2}} -\frac{-2 c e^{-\frac{x^2}{4 c^2}} +\frac{\sqrt{\pi } x (\text{erf} x)}{2 (c+k L)} -\frac{\sqrt{\pi } x (\text{erf} x)}{2 c} +2 c e^{-\frac{x^2}{4 (c+k L)^2}} +2 k L e^{-\frac{x^2}{4 (c+k L)^2}}} {2 k L} } {\frac{-2 c e^{-\frac{x^2}{4 c^2}}+\frac{\sqrt{\pi } x (\text{erf} x)}{2 (c+k L)}-\frac{\sqrt{\pi } x (\text{erf} x)}{2 c}+2 c e^{-\frac{x^2}{4 (c+k L)^2}}+2 k L e^{-\frac{x^2}{4 (c+k L)^2}}}{2 k L}+e^{-\frac{x^2}{4 (c+k L)^2}}} \, dx $$
Will settle for:
$$ 0.5 \int_{-\infty }^{\infty } \frac{e^{-\frac{x^2}{4 (c+k L)^2}}-\frac{-2 c e^{-\frac{x^2}{4 c^2}}+\sqrt{\pi } x \text{erf}\left(\frac{x}{2 (c+k L)}\right)-\sqrt{\pi } x \text{erf}\left(\frac{x}{2 c}\right)+2 c e^{-\frac{x^2}{4 (c+k L)^2}}+2 k L e^{-\frac{x^2}{4 (c+k L)^2}}}{2 k L}}{\frac{-2 c e^{-\frac{x^2}{4 c^2}}+\sqrt{\pi } x \text{erf}\left(\frac{x}{2 (c+k L)}\right)-\sqrt{\pi } x \text{erf}\left(\frac{x}{2 c}\right)+2 c e^{-\frac{x^2}{4 (c+k L)^2}}+2 k L e^{-\frac{x^2}{4 (c+k L)^2}}}{2 k L}} \, dx $$