Can this difficult Gaussian integral be done analytically?

[Ben D.]

Here is a tough integral that I'm not quite sure how to do. It's the Gaussian average:

$$
I = \int_{-\infty}^{\infty}dx\, \frac{e^{-\frac{x^2}{2}}}{\sqrt{2\pi}}\sqrt{1+a^2 \sinh^2(b x)}
$$

for $0 < a < 1$ and $b > 0$. Obviously the integral can be done for $a = 0$ (or $b=0$) and for $a=1$. But otherwise, I'm stomped? Expanding the root in powers of $a$, we can do all the integrals in the series and get a power series. But the sequences I get don't seem easy to work with.

I'm curious if there is an elegant way to do this? Is it even doable? To clarify, I'm looking for a closed form analytical solution (if it exists).

Ben

P.S. Solutions in term of known special functions are acceptable.

 

[Haborix]

Integrals like that make me think of the saddle-point method. However, it is a method for approximation and not exact aside from trivial cases.

 

[Ben D.]

Integrals like that make me think of the saddle-point method. However, it is a method for approximation and not exact aside from trivial cases.

Yes, the saddle point approximation is useful in a couple of limits, but these limits are basically just the trivial limits because the integrand is almost exactly Gaussian (or a pair of Gaussians) in these limits.

 

[Keith_McClary]

https://www.integral-calculator.com/
could not solve it with this input:
exp(-x^2/2)sqrt(1+a^2 sinh^2(bx))

 

[BvU]

I'm looking for a closed form analytical solution (if it exists).

Out of curiosity: why, exactly ?

 

[BvU]

https://www.integral-calculator.com/
could not solve it with this input:
exp(-x^2/2)sqrt(1+a^2 sinh^2(bx))

Maybe because it's an integrator, not an equation solver ?
Filling in numbers works with wolframalpha. However, especially for $a$, the results don't give a clue.