Type-reduction is a kernel block in type-2 fuzzy logic systems. Nie-Tan algorithms that do not involve iterative process can obtain the systems output directly, and have the advantage of saving computational cost. Moreover, the continuous Nie-Tan (CNT) algorithms have been proved to be the accurate centroid type-reduction algorithms in most recent studies. By analyzing the sum operation in discrete NT algorithms and the integral operation version in continuous NT (CNT) algorithms, the paper extends NT algorithms to three different forms of Weighted Nie-Tan (WNT) algorithms according to the formulas of Newton-Cotes quadrature of numerical integration techniques. In case of choosing the same sampling rate of primary variables, three computer simulation examples show that WNT algorithms have smaller absolute errors than NT algorithms and the computation speed is almost the same, so they are potentially feasible and effective on the real-time applications of interval type-2 fuzzy logic systems.