I am pretty close to finding the right-hand side of the desired proof (see below) but not able to finish it.
I have done this far but, it seems too long to continue.
2026-04-06 06:15:05.1775456105
Using a contour integral of $f(z)=x^{4n+3}e^{-z}$ to show $\int_0^\infty x^{4n+3}e^{-x}\cos x\;dx=(-1)^{n+1}(4n+3)!/2^{2n+2}$
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