Time for easier problems

Calculus Level 4

$\large \displaystyle\sum _{ k=1 }^{ \infty }{ \frac { \sin { k } }{ { k }^{ 2 } } =\frac { A }{ B } i({ \text{Li} }_{ 2 }({ e }^{ -Ci } }) -{\text{Li} }_{ 2 }({ e }^{ Di }))$

The equation above holds true for positive integers $$A,B,C$$ and $$D$$, with $$A,B$$ coprime. Find $$A+B+C+D$$.

Notation: $${ \text{Li} }_{ n }(a)$$ denotes the polylogarithm function, $${ \text{Li} }_{ n }(a)=\displaystyle\sum _{ k=1 }^{ \infty }{ \frac { { a }^{ k } }{ { k }^{ n } } }.$$

Clarification:$$i=\sqrt{-1}$$

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