Definite integral involving the exponential of an algebraic function We define the following integral for $\omega>0$ and $\lambda\neq0$ such that $\omega^2>\lambda$:
$$\begin{align}
f(\omega,\lambda)
&:= \int_\omega^\infty e^{-x\sqrt{x^2-\lambda}} \; \varphi(x) \, \text{d}x
\\[6pt]
\tag{1}
&=\frac{1}{\sqrt{2\pi}}\int_\omega^\infty e^{-\sqrt{x^4-\lambda x^2}-\frac{x^2} 2} \;\text{d}x
\end{align}$$
where $\varphi$ is the probability density function of a Gaussian random variable. I am wondering if it is possible to derive a closed-form expression, where by "closed-form" I do include expressions involving special functions such as the gamma function or elliptic integrals.
I am not sure how to proceed from expression $(1)$. I have scanned the book by Gradshteyn & Ryzhik to search for similarly-looking expressions, but have been unlucky $-$ or at least I haven't been able to find anything of the same vein. While it seems possible to solve this type of integral for exponents of the form $\sqrt{ax^2+bx+c}$, in my particular case the issues seem to be:

*

*The polynomial is of degree greater than $2$;

*The additional square term coming from the Gaussian pdf.

Any hints would be much appreciated. If a closed-form exists for a different domain of integration, please also do let me know.

This is not a closed form solution.
As you already found it, I did not find anything for such integrands.
A possible solution would be a Taylor expansion around $\lambda=0$. This would give for the integrand
$$e^{-\sqrt{x^4-\lambda x^2}-\frac{x^2} 2}=e^{-\frac{3 }{2}x^2}\Bigg[1+\frac 12 \lambda+\sum_{n=2}^\infty\frac {P_{n}(x) }{2^n \,n!\, x^{2n-2}}\lambda^n\Bigg]$$ where the first polynomials are
$$\left(
\begin{array}{cc}
 2 & x^2+1 \\
 3 & x^4+3 x^2+3 \\
 4 & x^6+6 x^4+15 x^2+15 \\
 5 & x^8+10 x^6+45 x^4+105 x^2+105 \\
 6 & x^{10}+15 x^8+105 x^6+420 x^4+945 x^2+945 \\
 7 & x^{12}+21 x^{10}+210 x^8+1260 x^6+4725 x^4+10395 x^2+10395 \\
8 & x^{14}+28 x^{12}+378 x^{10}+3150 x^8+17325 x^6+62370 x^4+135135 x^2+135135
\end{array}
\right)$$ where some interesting patterns appear (o be explored with $OEIS$.
This means that we should face linear combinations of integrals $(k \geq 0)$
$$I_k=\int x^{-2k}\,e^{-\frac{3 }{2}x^2}\,dx=-\frac 12 \left(\frac{3}{2}\right)^{k-\frac{1}{2}}\,\Gamma \left(\frac{1}{2}-k,\frac{3 x^2}{2}\right)$$
$$J_k=\int_\omega ^\infty x^{-2k}\,e^{-\frac{3 }{2}x^2}\,dx=\frac 12 \left(\frac{3}{2}\right)^{k-\frac{1}{2}}\,\Gamma \left(\frac{1}{2}-k,\frac{3 \omega ^2}{2}\right)$$
Using $\lambda=\omega=\pi$ and only the terms given in the table, I obtained a value of $2.11383 \times 10^{-7}$ while numerical integration gives $2.11601 \times 10^{-7}$
