DLMF:19.2.E16 (Q6112)

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DLMF:19.2.E16
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    test
    el3 ⁡ ( x , k c , p ) = ∫ 0 arctan ⁡ x d θ ( cos 2 ⁡ θ + p ⁢ sin 2 ⁡ θ ) ⁢ cos 2 ⁡ θ + k c 2 ⁢ sin 2 ⁡ θ = Π ⁡ ( arctan ⁡ x , 1 - p , k ) , Bulirsch-integral-el-3 𝑥 subscript 𝑘 𝑐 𝑝 superscript subscript 0 𝑥 𝜃 2 𝜃 𝑝 2 𝜃 2 𝜃 superscript subscript 𝑘 𝑐 2 2 𝜃 elliptic-integral-third-kind-Pi 𝑥 1 𝑝 𝑘 {\displaystyle{\displaystyle\mathrm{el3}\left(x,k_{c},p\right)=\int_{0}^{% \operatorname{arctan}x}\frac{\mathrm{d}\theta}{({\cos^{2}}\theta+p{\sin^{2}}% \theta)\sqrt{{\cos^{2}}\theta+k_{c}^{2}{\sin^{2}}\theta}}=\Pi\left(% \operatorname{arctan}x,1-p,k\right),}}
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    DLMF id
    DLMF:19.2.E16
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