I am trying to solve this: $$\int_0^{\infty} e^{-ab\cosh x}\cos\left(ac\sinh x+\frac{ix}{2}\right)\,dx$$
I don't have much ideas about the problem. I thought of writing $\cos x=\dfrac{e^{ix}+e^{-ix}}{2}$ but couldn't proceed after that.
$$\Large \frac{1}{2}\int_0^{\infty} \left(e^{ -(ab\cosh x-iac \sinh x)-\frac{x}{2}}+e^{ -(ab\cosh x+iac \sinh x)+\frac{x}{2}}\right)\,dx$$
$\displaystyle \begin{aligned} ab\cosh x-iac\sinh x &=a\sqrt{b^2+c^2}\left(\frac{b}{\sqrt{b^2+c^2}}\cos(ix)-\frac{c}{\sqrt{b^2+c^2}}\sin(ix)\right)\\ &=a\sqrt{b^2+c^2}\sin\left(\alpha-ix\right)\\ \end{aligned}$
Similarly,
$\displaystyle \begin{aligned} ab\cosh x+iac\sinh x &=a\sqrt{b^2+c^2}\left(\frac{b}{\sqrt{b^2+c^2}}\cos(ix)+\frac{c}{\sqrt{b^2+c^2}}\sin(ix)\right)\\ &=a\sqrt{b^2+c^2}\sin\left(\alpha+ix\right)\\ \end{aligned}$
where $\alpha=\arcsin\left(\dfrac{b}{\sqrt{b^2+c^2}}\right)$
So the integral can be simplified to:
$$\Large \frac{1}{2}\int_0^{\infty} \left(e^{-a\sqrt{b^2+c^2}\sin\left(\alpha-ix\right)-\frac{x}{2}}+e^{-a\sqrt{b^2+c^2}\sin\left(\alpha+ix\right)+\frac{x}{2}}\right)\,dx$$
A solution without using contour integration is appreciated. Thanks!