Not the ordinary kind

Calculus Level 5

\[\sum _{ n=1 }^{ \infty }{ \frac { 1 }{ n^{ 6 }+n^{ 4 } } } =\frac { 1 }{ 90 } \left( A{ \pi }^{ 4 }-B{ \pi }^{ C }+D{ \pi }-E+\frac { F\pi }{ { e }^{ G\pi }-1 } \right) \]

If \(A,B,C,D,E,F,G\) are integers. Find \(A+B+C+D+E+F+G\)

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