Calculate difficult Fourier Transform

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I have to calculate a quite difficult Fourier Transform for my class $$\int_{-\infty}^\infty dw\frac{(\varGamma-iw)w^3}{(\varGamma-iw)^2+1}\frac{J_{1}(|wr|)}{|wr|}e^{-iwt}$$ $J_1$ is the normal Bessel function of first order. Please help me.