Two integral representations for the logarithm of the Glaisher-Kinkelin constant

We present two integral representations of the logarithm of the Glaisher-Kinkelin constant. Both are based on a definite integral of $\ln [\mathrm{\Gamma }(x+1)]$, $\mathrm{\Gamma }$ being the usual Gamma function. The first one relies on an integral representation of $\ln [\mathrm{\Gamma }(x+1)]$ due to Binet, and the second one results from the so-called Malmstén formula. The numerical evaluation is easier with the latter expression than with the former.