I am looking for a solution to the following integral and finding it quite hard to find one ($|x| < a$):
$$ \int e^{\frac{1}{x^2 - a^2}} dx $$
I've tried to solve it with several substitutions, such as $u := x^2 - a^2$, but it yielded no other integral that seemed easier to solve. I've also tried to separate the exponent into something like $\frac{A}{x-a} + \frac{B}{x+a}$ and then integrating by parts, but also this approach didn't work out.
The reason for this question is the following: I need to modify this function, such that the integral from $-a$ to $a$ equals 1. That's why I came up with this integral, but if you see a way to find the required modification without having to integrate, that'll be fine as well. Thanks!
Well, if you just want $-a$ to $a$, use change of variable $u=-1/(x^2-a^2 )$ as follows: $$ \int_{-a}^a \exp\frac{1}{x^2-a^2}\;dx = 2 \int_{0}^a \exp\frac{1}{x^2-a^2}\;dx = \int_{1/a^2}^\infty \frac{e^{-u}}{u^{3/2}\sqrt{a^2u-1}}\;du = \frac{e^{-1/(2a^2)}}{a}\left(K_1\left(\frac{1}{2a^2}\right)-K_0\left(\frac{1}{2a^2}\right)\right) $$