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Calculus Level 5

n=0L2n+1(2n+1)2(2nn)\large\sum _{ n=0 }^{ \infty }{ \dfrac { { L }_{ 2n+1 } }{ (2n+1)^{ 2 } \binom{2n}{n}} }

Let LnL_n denote the nnth Lucas number, where L1=1L_1 = 1, L2=3L_2 = 3 and Ln=Ln1+Ln2L_n = L_{n-1} + L_{n-2} for n3n \ge 3.

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

where AA, BB, CC, DD, EE, FF, HH, II and JJ are positive integers, with II and JJ being coprime integers; DD, EE and HH square-free; and KK denotes the Catalan's constant, find A+B+C+D+E+F+G+H+I+J A+B+C+D+E+F+G+H+I+J .

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