Steepest descent, non-analytic roots

[CompuChip]

Hi,

For a physics class, I am supposed to evaluate the following integral
$$I_n(a) = \int_{-\infty}^\infty \mathrm dx \, e^{-n x^2/2 + n a x} \cosh^n(x)$$
as a function of the real non-zero parameter a, in the limit as $n \to \infty$ using the method of steepest descent. The question adds: "Express the ersult in parametric form as a function of the saddle point position."

The problem I ran into, is that the extremum position $x_0$ satisfies the equation
$$x_0 - a - \tanh(x_0) = 0$$
which cannot be solved analytically. So I'm getting a bit confused, whether I should just leave the result
$$I_n(a) \simeq e^{- n x_0^2 + n a x_0} \sqrt{\frac{2\pi}{f''(x_0)}}$$
where f''(x_0) = tanh(x_0) = x_0 - a, or whether I am missing something here.

I'd appreciate your input.

 

[CompuChip]

I've tried some more approaches, but in all of them, the problem is in the log(cosh(x)) that's in the exponent. There is no way it can be rewritten without having $\log\cosh(x_0))$, with x₀ the position of the minimum of the exponent, in the final expression.

I will try to see the TA tomorrow, but your suggestions are still welcome.
(The exercise was given Thursday afternoon, due on Tuesday).