\[\Pi\left(\phi,\alpha^{2},k\right)=\int_{0}^{\phi}\frac{\,\mathrm{d}\theta}{%
\sqrt{1-k^{2}{\sin}^{2}\theta}(1-\alpha^{2}{\sin}^{2}\theta)}=\int_{0}^{\sin%
\phi}\frac{\,\mathrm{d}t}{\sqrt{1-t^{2}}\sqrt{1-k^{2}t^{2}}(1-\alpha^{2}t^{2})}.\]