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Calculus Level 5

\[\large \int _{ 0 }^{ 2\ln { \phi } }{ \ln { \left( 2\sinh { \left( \dfrac { x }{ 2 } \right) } \right) dx } } =-\dfrac { { \pi }^{ a } }{ b } \]

Let \(\phi\) denotes the Golden ratio, \(\phi = \dfrac{1+\sqrt5}2 \approx 1.618\).

If the equation above holds true for integers \(a\) and \(b\), find \(a\times b\).

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