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While reading a chapter on approximation of prime counting function $\pi(x)$ from the book Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work by G. H. Hardy I came across an identity concerning integral of $a^{x + 1}/\Gamma(x + 1)$ namely $$\int_{0}^{\infty}\frac{a^{x + 1}}{\Gamma(x + 1)}\,dx = e^{a} - \int_{0}^{\infty}\frac{e^{-ax}}{x\{\pi^{2} + (\log x)^{2}\}}\,dx\tag{1}$$ I don't see any easy route to proving the above (probably because I am not so much used to evaluation of integrals via residue theorem).

Any hints or a proof of $(1)$ (preferably via elementary techniques) will be greatly appreciated.

Paramanand Singh
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    http://math.stackexchange.com/questions/1802176/how-was-the-difference-of-the-frans%C3%A9n-robinson-constant-and-eulers-number-found/1802219?sfb=2#1802219 it is essentially the same. – Marco Cantarini May 29 '16 at 20:46
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    @MarcoCantarini: Got it!! How easy things are once we look in the right direction. Thanks a lot – Paramanand Singh May 30 '16 at 04:21
  • @ParamanandSingh do you still have your notes of the full calculations somewhere? If so, could you send them to me? I've been stuck with this problem for quite a while now! – Flammable Maths Jul 08 '19 at 16:36

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