I want to solve following integral $$\int_0^{\infty}\exp{(-i\omega x-a\sqrt{ib\omega}\tanh^{-1}(c/\sqrt{ib\omega}))}d\omega$$ where $i$ means $\sqrt{-1}$ and $a,b,c,x$ are all real positive values. I will be very thankful for your help in this regard.
Edit:
In my strategy I used $b\omega=u$ and a formula for inverse of hyperbolic tangent then the above integral has following form $$\frac{1}{b}\int_0^{\infty}\cos(\frac{ux}{b})\left[\frac{\sqrt{iu}+c}{\sqrt{iu}-c}\right]^{-\frac{a\sqrt{iu}}{2}}du-i\frac{1}{b}\int_0^{\infty}\sin(\frac{ux}{b})\left[\frac{\sqrt{iu}+c}{\sqrt{iu}-c}\right]^{-\frac{a\sqrt{iu}}{2}}du$$
Is there any formula that can help in solving the above integral further. Thanks in advance.
Edit 2:
In my other attempt (in which I divided the above expression by the derivative of the power of exponential) I have obtained following form $$\frac{1}{\left[\frac{a\sqrt{ib\omega}bc}{2\sqrt{ibx}(bx+ic^2)}+\frac{a\sqrt{ib}}{2\sqrt{\omega}\tanh^{-1}(\frac{c}{\sqrt{ib\omega}})}-ix\right]}\exp{(-i\omega x-a\sqrt{ib\omega}\tanh^{-1}(c/\sqrt{ib\omega}))}\Bigg{|}_0^{\infty}$$ Now in this case the limits needs to be solved. Please help. Thank you.