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I would like to learn more about the behavior of the factorial function or Gamma function, so I decided to compute the following integral $$ \int_1^\infty\dfrac{1}{\sqrt{\Gamma(x)}}\,\mathrm dx. $$

According to Wolfram alpha, its value is approximately $3$:

My question is whether the exact value is rational or not.

Edit: The Motivation of this question is to know more about transcendence degree of the field generated by $\int_1^\infty\frac{1}{\sqrt{\Gamma(x)}}\mathrm dx$