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Incomplete elliptic integral of the 3rd kind Π(φ,n,k)

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Calculate the 3rd-kind incomplete elliptic integral Π(φ,n,k).
\(\Pi(\phi,n,k)={\large\int_{\small 0}^{\phi}\frac{d\theta}{(1-n \sin^2\theta)\sqrt{1-k^2 \sin^2\theta}}}\)
φ 
n 
k 
-1≦k≦1
\(\normalsize Incomplete\ elliptic\ integral\\ \hspace{100px} of\ the\ 3rd\ kind\ \Pi(\phi,n,k)\\ (1)\ \Pi(\phi,n,k)={\large\int_{\small 0}^{\phi}\frac{d\theta}{(1-n \sin^2\theta)\sqrt{1-k^2 \sin^2\theta}}}\\ \hspace{80px}={\large\int_{\small 0}^{\small x}\frac{dt}{(1-nt^2)\sqrt{(1-t^2)(1-k^2t^2)}}}\\ \hspace{230px}x= \sin\phi\\\)


・If x = sin(φ), the related library function Π(x, n, k) can be used.

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