\[\int_{0}^{K\left(k\right)}\ln\left(\operatorname{cn}\left(t,k\right)\right)\,%
\mathrm{d}t=-\tfrac{\pi}{4}{K^{\prime}}\left(k\right)+\tfrac{1}{2}K\left(k%
\right)\ln\left(k^{\prime}/k\right),\]