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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 \(n\)th Lucas number, where \(L_1 = 1\), \(L_2 = 3\) and \(L_n = L_{n-1} + L_{n-2} \) for \(n \ge 3\).

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

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