How to proceed with a definite integral of this form?

Question

How do I find the definite integral of this sort of function?

$$ \int_m^n \frac{\sin(a \sin(\sqrt{(b \lfloor x\rfloor^2 + b d + 1)}))}{a \sin(\sqrt{(b \lfloor x\rfloor^2 + b d + 1)})} dx $$

This function is actually a form of the $ \operatorname{sinc} $ function except it has a floor variable nested inside it, and also takes the sine of another sine function.

Note: a & b = 10^k (a & b are higher order powers of 10)

Answer 1

\begin{eqnarray} \int_m^n \frac{\sin(a \sin(\sqrt{(b \lfloor x\rfloor^2 + b d + 1)}))}{a \sin(\sqrt{(b \lfloor x\rfloor^2 + b d + 1)})} dx&=&\sum_{k=m}^n\int_k^{k+1} \frac{\sin(a \sin(\sqrt{(b k^2 + b d + 1)}))}{a \sin(\sqrt{(b k^2 + b d + 1)})} dx\\ &=&\sum_{k=m}^n \frac{\sin(a \sin(\sqrt{(b k^2 + b d + 1)}))}{a \sin(\sqrt{(b k^2 + b d + 1)})} \end{eqnarray}