# Numerically Inverting Characteristic Function with Inverse Fourier Transform

I'm trying to numerically invert this characteristic function:

cf = (((-I)*t + n*(λ + μ) -
n*Sqrt[-4*λ*μ + (I*t - n*(λ + μ))^2/
n^2])/(n*Sqrt[λ*μ*ρ]))^n/2^n


via a numerical Inverse Fourier Transform that looks like this...

Re[(1/(2*Pi))*
NIntegrate[
cf /. {λ -> 0.5, μ -> 1, ρ -> 0.5,
n -> 5}, {t, -Infinity, 0, Infinity},
Method -> DoubleExponential]]


but I keep on getting errors that the integral cannot be evaluated at the boundaries.

Does anyone have recommendations on how to compute this thing so that I get numerical values?

The Integrate works for this function:
cf = (((-I)*t + n*(λ + μ) -