\[\int _{ 0 }^{ 1 }{ x{ { \text{ Li} } }_{ 4 }^{ 2 }\left( x \right) \, dx } =\frac { A }{ B } -\frac { C{ \pi }^{ D } }{ E } -\frac { F }{ G } \zeta \left( H \right) +\frac { I{ \pi }^{ J } }{ K } -\frac { { \pi }^{ L } }{ M } \zeta \left( N \right) -\frac { O }{ P } \zeta \left( Q \right) +\frac { R{ \pi }^{ S } }{ T } +\frac { { \zeta }^{ U }\left( V \right) }{ W } -\frac { { \pi }^{ X } }{ Y } \zeta \left( Z \right) +\frac { \alpha { \pi }^{ \beta } }{ \gamma } \]

The equation is true for positive integers \(A,B,C,\ldots,Z,\alpha, \beta, \gamma\). Find the minimum value of the sum of those integers.

**Notations**:

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

\(\zeta(\cdot) \) denotes the Riemann zeta function.

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