The given mathematical expression is:

$\left( \frac{xy^{-1} + 1}{xy^{-1} - x^{-1}y} \cdot \frac{x^3 - y^{-3} - 1}{x^2y^{-2} + xy + 1} \right) \div \frac{x^3y^{-3} + 1}{xy^{-1} + x^{-1}y^{-1}}$

This is a complex expression involving variables $x$ and $y$, with several exponents and terms. Here's how we could approach simplifying it step-by-step:

1. Simplify individual fractions: Start by handling each fraction separately. Simplify the terms involving negative exponents.

2. Multiply and divide fractions: Use the rule $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$ to combine and reduce the fractions.

Would you like me to proceed with the detailed step-by-step simplification?