Help in proving, that $\int_{0}^{\infty} \frac{1}{\Gamma(x)} d x=e+\int_{0}^{\infty} \frac{e^{-x}}{\pi^{2}+(\ln x)^{2}} d x$ using real methods only

Question

I hope this does not seem like a too easy question for Overflow. I would like to find an easier method than mine to prove the above statement for the Fransén-Robinson Constant. My first method was to use the reflection formula, and then a laplace transform for the sin($\pi$x). Due to convergence problems with the integral definition of the gamma function, I was always missing the factor of $+e$. After days of work, I was actually able to prove and derive a way stronger argument, where the above equality is just a special case (see pictures). I would appreciate some help in simplifying the process of proving it!

My ideas and calculations

Answer 1

This is Exercise 100 of Chapter 9 of my GTM 240 (Exercise 99 gives the complex analytic method): set $$I(a,t)=\int_{-t}^\infty\dfrac{a^x}{\Gamma(x+1)}\,dx+\int_0^\infty\dfrac{e^{-ax}x^{t-1}}{\log^2(x)+\pi^2}\Bigl(\cos(\pi t)-\dfrac{\sin(\pi t)}{\pi}\log(x)\Bigl)\,dx$$ 1) After proving absolute convergence, prove that its derivative with respect to $t$ vanishes, so that $I(a,t)=I(a)$.

2) Prove that $I'(a)=I(a)$.

3) Make $a\to0^+$, set $x=\exp(-t)$, and deduce the value of $I(a)$.

4) Deduce for instance the value of $$\int_0^\infty \dfrac{x(x-1)\cdots(x-k+1)a^x}{\Gamma(x+1)}\,dx$$ I must confess that I forgot where I found this proof (perhaps in one of the Borwein's books ?)