# Integral of $\exp[\text{erfc}[C x]]$

I have the complementary CDF of a continuous random variable $$X$$ that looks like...

$$1-F_{X}(x)=\frac{\exp\left(\frac{\pi}{2\sqrt{e}}\text{ erfc}\left(\sqrt{2}x\right)\right)-1}{\exp\left(\frac{\pi}{2\sqrt{e}}\right)-1}$$

and I'm trying to use $$\int_{0}^{\infty}\left(1-F_{X}\left(x\right)\right) \mathrm{d}x=\mathbb{E}[X]$$, relating the CDF to the expectation.

To calculate this means evaluating the definite integral above. I think it can be done, leaving the answer in terms of the complementary error function. Is this true?