Integrate Box Function?

Calculus Level 3

\[ \int_1^3 \lfloor x \rfloor \cos \left( \dfrac\pi2 \big( x - \lfloor x \rfloor \big) \right) \, dx \]

The integral above is equal to \( \frac a{b\pi}\), where \(a\) and \(b\) are coprime positive integers.

Find \(a+b\).

\(\)
Notation: \( \lfloor \cdot \rfloor \) denotes the floor function.

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