Indefinite integral that cannot be expressed in terms of elementary functions

Question

I have the following indefinite integral which I cannot evaluate by integration by part, change of variable etc.

$$\int\frac{1}{y}e^{-(y-A)^2}dy$$

The output of Wolfram Alpha is in the link. Note that A is a constant parameter.

Answer 1

$\int\dfrac{e^{-(y-A)^2}}{y}dy$

$=\int\sum\limits_{n=0}^\infty\dfrac{(-1)^n(y-A)^{2n}}{n!y}dy$

$=\int\sum\limits_{n=0}^\infty\sum\limits_{k=0}^{2n}\dfrac{(-1)^nC_k^{2n}(-1)^{2n-k}A^{2n-k}y^k}{n!y}dy$

$=\int\sum\limits_{n=0}^\infty\sum\limits_{k=0}^{2n}\dfrac{(-1)^{n+k}(2n)!A^{2n-k}y^{k-1}}{n!k!(2n-k)!}dy$

$=\int\left(\dfrac{1}{y}+\sum\limits_{n=1}^\infty\sum\limits_{k=0}^{2n}\dfrac{(-1)^{n+k}(2n)!A^{2n-k}y^{k-1}}{n!k!(2n-k)!}\right)dy$

$=\int\left(\dfrac{1}{y}+\sum\limits_{n=1}^\infty\dfrac{(-1)^nA^{2n}}{n!y}+\sum\limits_{n=1}^\infty\sum\limits_{k=1}^{2n}\dfrac{(-1)^{n+k}(2n)!A^{2n-k}y^{k-1}}{n!k!(2n-k)!}\right)dy$

$=\int\left(\sum\limits_{n=0}^\infty\dfrac{(-1)^nA^{2n}}{n!y}+\sum\limits_{n=1}^\infty\sum\limits_{k=1}^{2n}\dfrac{(-1)^{n+k}(2n)!A^{2n-k}y^{k-1}}{n!k!(2n-k)!}\right)dy$

$=\int\left(\dfrac{e^{-A^2}}{y}+\sum\limits_{n=1}^\infty\sum\limits_{k=1}^{2n}\dfrac{(-1)^{n+k}(2n)!A^{2n-k}y^{k-1}}{n!k!(2n-k)!}\right)dy$

$=e^{-A^2}\ln y+\sum\limits_{n=1}^\infty\sum\limits_{k=1}^{2n}\dfrac{(-1)^{n+k}(2n)!A^{2n-k}y^k}{n!k!k(2n-k)!}+C$