This paper discussed the numerical evaluation of highly oscillatory integrals involving logarithmic and algebraic singularities. For an analytic function in a sufficiently large region containing $[a,b]$, the integral was transformed into the sum of two line integrals where the integrands did not oscillate and decay exponentially. Thus, to approximate the line integrals, generalized Gauss-Laguerre quadrature and logarithmic Gauss-Laguerre quadrature were applied. The error bound and numerical results demonstrated that the proposed method efficiently obtained high-precision results even for high oscillations.