Integral of $\frac{e^{-x^2}}{\sqrt{1-x^2}}$ I am stuck at an integral $$\int_0^{\frac{1}{3}}\frac{e^{-x^2}}{\sqrt{1-x^2}}dx$$
My attempt is substitute the $x=\sin t$, however there may be no primitive function of $e^{-\sin^2 t}$.
So does this integral has a definitive value? If does, how can we solve it? Thank you!

We have:
$$ I = \int_{0}^{\arcsin\frac{1}{3}}\exp\left(-\sin^2\theta\right)\,d\theta \tag{1}$$
but since:
$$ \exp(-\sin^2\theta) = \frac{1}{\sqrt{e}}\left(I_0\left(\frac{1}{2}\right)+2\sum_{n\geq 1}I_n\left(\frac{1}{2}\right)\cos(2n\theta)\right)\tag{2} $$
we have:
$$ I = e^{-1/2}\left(\arcsin\frac{1}{3}\right) I_0\left(\frac{1}{2}\right)+e^{-1/2}\sum_{n\geq 1}\frac{1}{3n}\, I_n\left(\frac{1}{2}\right)U_{2n-1}\left(\sqrt{\frac{8}{9}}\right)\tag{3}$$
where $I_m$ is a modified Bessel function and $U_k$ is a Chebyshev polynomial of the second kind.
