Camelia Integral on the fractional part function

Let be a positive integer and let denote the fractional part function, then calculate in closed-form the following Triple Integral : ∫01∫01∫01lnk(xyz)\bigg{\bigg(xy\bigg)k\bigg}\bigg{\bigg(yz\bigg)k\bigg}\bigg{\bigg(zx\bigg)k\bigg}dx dy dz\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}\ln^k(xyz)\bigg\{\bigg(\frac{x}{y}\bigg)^k\bigg\}\bigg\{\bigg(\frac{y}{z}\bigg)^k\bigg\}\bigg\{\bigg(\frac{z}{x}\bigg)^k\bigg\}\mathrm dx\ \mathrm dy\ \mathrm dz