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Calculus Level 5

\[\large\sum _{ n=0 }^{ \infty }{ \dfrac { { L }_{ 2n+1 } }{ (2n+1)^{ 2 } \binom{2n}{n}} } \]

Let \(L_n\) denote the \(n^\text{th} \) Lucas Number, where \(L_1 = 1, L_2 = 3\) and \(L_n = L_{n-1} + L_{n-2} \) for \(n=3,4,\ldots \).

If the series above can be expressed as \[ -\dfrac\pi A \ln\left( B - C\sqrt D+ E\sqrt{F - G\sqrt H} \right) + \dfrac IJ K , \]

where \(A,B,C,D,E,F,H,I\) and |(J) are positive integers, with \(I,J\) coprime, \(D,E,H\) square-free and \(K\) denotes the Catalan's constant, find \( A+B+C+D+E+F+G+H+I+J \).

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