Closed form of integral of $\operatorname{erfc} \log t$

Question

Is there any closed form expression for the following integral?

$$ \int\limits_t^\infty \left(1- \operatorname{erf}(\log x) \right )dx $$

or equivalently:

$$ \int\limits_t^\infty \operatorname{erfc}(\log x ) dx $$

I just wish to know if there is any way I can do better than sampling the values at particular points and calculating the area under the curve numerically. Any help is appreciated. Thanks.

Answer 1

Not terribly hard; one only needs simple substitutions and an application of integration by parts here. To wit,

$$\begin{align*} \int_t^\infty \mathrm{erfc}(\log u)\,\mathrm du&=\int_{\log\,t}^\infty \exp\,v\;\mathrm{erfc}\,v\,\mathrm dv\\ &=\left.\exp\,v\;\mathrm{erfc}\,v\right|_{\log\,t}^\infty+\frac2{\sqrt\pi}\int_{\log\,t}^\infty \exp\,(v-v^2)\,\mathrm dv\\ &=-t\;\mathrm{erfc}(\log\,t)+\frac{2\sqrt[4]{e}}{\sqrt\pi}\int_{\log\,t-\frac12}^\infty \exp\,(-w^2)\,\mathrm dw\\ &=\sqrt[4]{e}\,\mathrm{erfc}\left(\log\,t-\frac12\right)-t\;\mathrm{erfc}(\log\,t)\\ \end{align*}$$