An integral of a complementary error function I really appreciate it if someone help me solving this integral:
$$ \int \frac 1x \cdot \operatorname{Erfc}^n x\, dx,$$
where $\operatorname{Erfc}$ is the complementary error function, defined as $\operatorname{Erfc}=\frac 2{\sqrt \pi}\int_x^{+\infty}e^{-t^2}dt$.
thank you

A Taylor series at $x=0$ may be found here:
$$
\int \frac{\text{Erfc}^n(x)}{x}dx=\log(x)-\frac{2nx}{\sqrt{\pi}}+\frac{(n-1)nx^2}{\pi}+\cdots
$$
There is also a result for $n=1$ given: $\log(x)-\frac{2x}{\sqrt{\pi}}{ _2F_2}\left(1/2,1/2;3/2,3/2;-x^2\right)$.
EDIT:
You get a series expansion for $\text{Erfc}^n(x)$ at $x=\infty$ here:
$$
\text{Erfc}(x)^n=\left(1-2 \sum _{k=0}^{\infty } \frac{(-1)^k x^{2 k+1}}{\sqrt{\pi }(2 k+1) k!}\right){}^n
$$
