Is it possible to compute the following Fourier transform analytically?
$$\psi(x) = \frac{1}{\sqrt{4 \pi}}\int \Gamma \left (\frac{1}{2}-i \frac{p}{2 \pi} \right) \frac{e^{i p x}}{\sqrt{ \cosh(p/2)}} dp $$

I am aware that 
$$ \Gamma \left (\frac{1}{2}-i \frac{p}{2 \pi} \right) \Gamma \left (\frac{1}{2}+i \frac{p}{2 \pi} \right)= \frac{\pi}{\cosh(p/2)}$$
and both sides of the identity have poles only at $p=(\text{odd integer}) \times 2 \pi i$, but can not figure out how this can help with the integral.

Mathematica v10 returns the integral unevaluated, both with Integrate[] and InverseFourierTransform[].

The question comes from a research problem in quantum physics.

**Edit-3:** (makes Edit-1 obsolete).
The function is real-valued and positive, see continuous line on the numerically produced graph. Dashed lines are $\sqrt{2} \, e^{-\pi x}/\sqrt{x}$ for $x \to +\infty$ and $20 \, e^{+\pi x} \sqrt{-x}$ for $x \to -\infty$, confirming the asymptotics derived by @tired.

[![graph of the function + asymptotics](https://i.stack.imgur.com/CA4n4.png)](https://i.stack.imgur.com/CA4n4.png)

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**Edit-2:** Here is what is relevant for the publication on the physics side. We are are proposing a method to generate quantum particles described by the wave-function $\psi(x)$. The unprecedented feature of the method (in the particular physics context) is that the Heisenberg product of coordinate-momentum uncertainty $\Delta x \, \Delta p \approx 0.537541$ is close to the so-called [Kennard bound](https://en.wikipedia.org/wiki/Uncertainty_principle) of $1/2$ (known to be satisfied by Gaussian wave-packet only).

For the momentum the result is neat:
$$(\Delta p)^2 = \int \frac{p^2}{4 \cosh^2(p/2)} dp= \frac{\pi^2}{3}$$

For the coordinate I need to compute $\Delta x \equiv \langle x^2 \rangle-\langle x \rangle^2$ where 

$$ \langle x^n \rangle =\frac{1}{2\pi} \int x^n \, |\psi(x)|^2 \, dx$$

Numerically I have $ \langle x \rangle \approx -0.251022$ and $\langle x^2 \rangle \approx 0.150842$ which gives the above quoted uncertainty product.

So it would be very nice to find an analytic form for the first two moments of the probability distribution corresponding to $| \psi(x)|^2$ and also to prove analytically the exponential asymptotics (which distinguishes these wave-packets from other types known in the literature in my field of physics).