We introduce a finite class of weighted quadrature rules with the weight function |x|−2𝑎exp(−1/𝑥2) on (−∞,∞) as ∫∞−∞|𝑥|−2𝑎exp(−1/𝑥2∑)𝑓(𝑥)𝑑𝑥=𝑛𝑖=1𝑤𝑖𝑓(𝑥𝑖)+𝑅𝑛[𝑓], where 𝑥𝑖 are the zeros of polynomials orthogonal with respect to the introduced weight function, 𝑤𝑖 are the corresponding coefficients, and 𝑅𝑛[𝑓] is the error value. We show that the above formula is valid only for the finite values of 𝑛. In other words, the condition 𝑎≥{max𝑛}+1/2 must always be satisfied in order that one can apply the above quadrature rule. In this sense, some numerical and analytic examples are also given and compared.