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Calculus Level 5

\[\large \int _{ 0 }^{ \infty }{ \dfrac { x\ln { x } }{ 1+{ e }^{ x } } \, dx } =\dfrac { 1 }{ B } \pi ^{ C }-{ \pi }^{ D }\ln { A } +\dfrac { { \pi }^{ E }\ln { F } }{ G } +\dfrac { { \pi }^{ H }\ln { \pi } }{ I } \]

The equation above holds true for positive integers \(B,C,D,F,G,H\) and \(I \) such that \(F\) is minimized.

Find \(B+C+D+E+F+G+H+I\).

Notation: \(A\) denotes the Glaisher–Kinkelin constant, \(A \approx 1.2824\).

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