\[\large{\sum _{ k=0 }^{ \infty }{ { \left( -1 \right) }^{ k }\frac { \prod _{ r=0 }^{ k }{ { \left( 1+{ 4r }^{ 2 } \right) } } }{ { 9 }^{ k+1 }\left( 2k+2 \right) ! } } }\]

If the above summation can be expressed as

\[{A \sin^{B} \bigg(\frac{\ln \big(\frac{C+\sqrt{D}}{E} \big)}{2}\bigg)}\]

where \(A,B,C,D,E\) are positive integers (need not be distinct) and \(D\) being square free.

Find \(A+B+C+D+E\)

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