In this paper we consider polynomials orthogonal with respect to the linear functional $\mathcal{L}:\mathcal{P}\to \mathbb{C}$, defined on the space of all algebraic polynomials $\mathcal{P}$ by$$\mathcal{L}[p] =\int_{-1}^1 p(x) (1-x)^{\alpha-1/2} (1+x)^{\beta-1/2}\exp(i\zeta x)dx,$$ where $\alpha,\beta >-1/2$ are real numbers such that $\ell=|\beta-\alpha|$ is a positive integer, and $\zeta\in\mathbb{R}\backslash\{0\}$. We prove the existence of such orthogonal polynomials for some pairs of $\alpha$ and $\zeta$ and for all nonnegative integers $\ell$. For such orthogonal polynomials we derive three-term recurrence relations and also some differential-difference relations. For such orthogonal polynomials the corresponding quadrature rules of Gaussian type are considered. Also, some numerical examples are included.