Integral $\int_0^\infty F(x)\,F\left(x\,\sqrt2\right)\frac{e^{-x^2}}{x^2} \, dx$ involving Dawson's function

Notice that for $a>0$, we have $$F(ax) = e^{-a^{2}x^{2}}\int_{0}^{ax} e^{y^{2}} \mathrm dy = e^{-a^{2}x^{2}} \int_{0}^{a} u e^{x^{2}u^{2}} \, \mathrm du . \tag{1}$$

Then using $(1)$, we get

$$ \begin{align} I &= \int_{0}^{\infty} F(x) F(x \sqrt{2}) \, \frac{e^{-x^{2}}}{x^{2}} \, \mathrm dx \\&= \int_{0}^{\infty} \int_{0}^{\sqrt{2}} \int_{0}^{1} x e^{-x^{2}} e^{x^{2} y^{2}} x e^{-2x^{2}} e^{x^{2}z^{2}} \, \frac{e^{-x^{2}}}{x^{2}} \, \mathrm dy \, \mathrm dz \, \mathrm dx \\ &= \int_{0}^{\sqrt{2}} \int_{0}^{1} \int_{0}^{\infty} e^{-(4-y^{2}-z^{2})x^{2}} \, \mathrm dx \, \mathrm dy \, \mathrm dz \\ &= \frac{\sqrt{\pi}}{2} \int_{0}^{\sqrt{2}} \int_{0}^{1} \frac{1}{\sqrt{4-y^{2}-z^{2}}} \, \mathrm dy \, \mathrm dz \\&= \frac{\sqrt{\pi}}{2} \int_{0}^{\sqrt{2}} \int_{0}^{\arcsin ( \frac{1}{\sqrt{4-z^{2}}})} \, \mathrm d \theta \, \mathrm d z \tag{2} \\ &= \frac{\sqrt{\pi}}{2} \int_{0}^{\sqrt{2}} \arcsin \left( \frac{1}{\sqrt{4-z^{2}}} \right) \, \mathrm dz \\ &= \frac{\sqrt{\pi}}{2} \left( \frac{ \sqrt{2} \pi}{4} - \int_{0}^{\sqrt{2}} \frac{z^{2}}{\sqrt{3-z^{2}} (4-z^{2})} \, \mathrm dz \right) \tag{3} \\ &= \frac{\sqrt{\pi}}{2} \left( \frac{\sqrt{2} \pi}{4} - \int_{1 / \sqrt{2}}^{\infty} \frac{1}{\sqrt{3u^{2}-1} (4u^{2}-1)} \frac{\mathrm du}{u}\right) \tag{4} \\ &= \frac{\sqrt{\pi}}{2} \left( \frac{\sqrt{2} \pi}{4} - 3 \int_{1 /\sqrt{2}}^{\infty} \frac{1}{ (4w^{2}+1)(w^{2}+1)} \, \mathrm dw\right) \tag{5}\\ &=\frac{\sqrt{\pi}}{2} \left( \frac{\sqrt{2} \pi}{2} - 4 \int_{1/ \sqrt{2}}^{\infty} \frac{1}{4w^{2}+1} + \int_{1/ \sqrt{2}}^{\infty} \frac{1}{w^{2}+1} \, \mathrm dw\right) \\ &= \frac{\sqrt{\pi}}{2} \left[ \frac{\sqrt{2} \pi}{4} - \pi +2 \arctan \left( \sqrt{2} \right) +\frac{\pi}{2} - \arctan \left( \frac{1}{\sqrt{2}} \right) \right] \\ &= \frac{\sqrt{\pi}}{2} \left( \frac{\sqrt{2} \pi}{4} - \pi + 3 \arctan{\sqrt{2}} \right). \end{align}$$

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$(2)$ Let $y=\sqrt{4-z^{2}}\sin \theta$.

$(3)$ Integrate by parts.

$(4)$ Let $z = \frac{1}{u}$.

$(5)$ Let $w^{2}=3u^2-1$.

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EDIT:

Using the same approach, I get

$$ \int_{0}^{\infty} F(ax) F(bx) \, \frac{e^{-p^{2}x^{2}}}{x^{2}} \, \mathrm dx $$

$$ = \frac{\sqrt{\pi}}{2} \left[b \arcsin \left( \frac{a}{\sqrt{a^{2}+p^{2}}} \right) - \sqrt{a^{2}+b^{2}+p^{2}} \arctan \left(\frac{ab}{p \sqrt{a^{2}+b^{2}+p^{2}}} \right) + a \arctan \left( \frac{b}{p} \right)\right]$$

where $a, b,$ and $p$ are all positive parameters.

$$I=\frac{\pi^{3/2}}8\left(\sqrt2-4\right)+\frac{3\,\pi^{1/2}}2\arctan\sqrt2$$