Question Number 222389 by MrGaster last updated on 25/Jun/25
$$\mathrm{Prove}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{cos}\left({nx}\right)\mathrm{cos}\left({p}\:\mathrm{arctan}\:{x}\right)}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{\frac{{p}}{\mathrm{2}}} }=\frac{\pi}{\mathrm{2}}\:\frac{{n}^{{p}−\mathrm{1}} {e}^{−{n}} }{\Gamma\left({p}\right)}\:\left({p}>\mathrm{0}\right) \\ $$