Suppose that $\alpha$ is a very large number ($\alpha \gg 1$), and $b\in(-1,1)$, then let us define an integral as
$$ I(\alpha,b) = \int_0^{\pi/2} dx e^{- \alpha \sin x \cdot(1+b\cos x) }. $$
When $b = 0$, it relates to the Bessel function. Generally, I found that its leading term is roughly $\frac{1}{\alpha(1+b)}$.
My question is that: (1) Could we reexpress $I(\alpha,b)$ by some special functions? (2) Could we get the closed form, or could we get the asymptotic behavior when $\alpha \to \infty$? Any hint or comment are welcome.
Note that $f(x)=\sin x(1 + b\cos x)$ has a sole minimum at $x=0$ on the interval $0\leq x \leq \frac{\pi}{2}$ and $f'(x) \neq 0$ when $-1<b<1$. Thus, by Laplace's method (cf. http://dlmf.nist.gov/2.3.iii), $$ I(\alpha ,b) \sim \sum\limits_{n = 0}^\infty {\frac{{c_n(b) }}{{\alpha ^{n + 1} }}} $$ as $\alpha \to +\infty$ with $$ c_n (b) = \left[ {\frac{{d^n }}{{dt^n }}\left( {\frac{t}{{\sin t(1 + b\cos t)}}} \right)^{n + 1} } \right]_{t = 0} . $$ In particular, $c_0(b) =\frac{1}{1+b}$, $c_1(b)=0$ and $c_2 (b)= \frac{{1+4b}}{{(1 + b)^4 }}$. Observe that the coefficients blow up at $b=-1$, thus the asymptotics becomes worse as $b$ approaches $-1$. This is because when $b=-1$, $x=0$ is a saddle point of the phase function $f(x)$.
To obtain a uniform approximation near $b=-1$, we can proceed as follows. Assume that $-1<b<0$, say. We have $$ f(x) \sim \frac{{x^3 }}{3}\left( {\frac{1}{2} - 2b} \right) + x(1 + b)$$ near $x=0$. Consequently, \begin{align*} I(\alpha ,b) & \sim \int_0^{\pi /2} {\exp \left( { - \alpha \left( {\frac{{x^3 }}{3}\left( {\frac{1}{2} - 2b} \right) + x(1 + b)} \right)} \right)dx} \\ & \sim \frac{{2^{1/3} \pi }}{{(1 - 4b)^{1/3} }}\frac{1}{{\alpha ^{1/3} }}\operatorname{Hi}\left( { - \frac{{2^{1/3} (1 + b)}}{{(1 - 4b)^{1/3} }}\alpha ^{2/3} } \right) \end{align*} as $\alpha \to +\infty$, where $\operatorname{Hi}$ is the Scorer function. It is possible to obtain a complete uniform asymptotic expansion in terms of $\operatorname{Hi}$ and $\operatorname{Hi}'$ using the method of Chester, Friedman and Ursell (see, Chapter 9, $\S$11 of F. W. J. Olver's book Asymptotics and Special Functions).