In my research about distribution theory in the topic of probability and statistic, I came across the following integral: $$\int_0^\infty \frac{\operatorname{erf}(1/x)\operatorname{erfc}(1/x)}{x}dx$$ I run it using WolframAlpha and as shown here I have got $\dfrac{2C}{\pi}$, where $C$ is Catalan's constant. The latter let me to believe that $$\int_0^t \frac{\operatorname{erf}(1/x)\operatorname{erfc}(1/x)}{x}dx$$ have a nice closed form which i didn't Get it using integration by part and series asymptotic of both error function and complementary error function, Any way to get that closed form?
Note: $\mathrm{erfc}$ is the complementary error function.