An integral with the floor function

Calculus Level 4

Let \( \lfloor \cdot \rfloor \) denotes the floor function. The integral \[\large \int_1^\infty\dfrac{\lfloor x\rfloor}{x^3}\, dx\] can be written in the form \[\dfrac{3}{a}+\dfrac{\pi b}{12}+\dfrac{\pi^2 c}{24}-1,\] where \(a,b\) and \(c\) are integers. Compute the value of \(c^{a+b}\).