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 $$c^{a+b}$$.

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