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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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