Generalized B-splines in non-polynomial space has been employed as analysis tools for IGA. However, the previous models used in IGA are not the unified mathematical representation of conics and polynomial parametric curves/surfaces. In this paper, we propose a new isogeometric analysis framework called NUAHT-IGA by using non-uniform algebraic hyperbolic trigonometric B-splines(NUAHT B-splines for short) in the space spanned by $\{\sin t,\cos t,\sinh t,\cosh t,1,t,\cdots,t^{n-5}\}$, which can be considered as an alternative to NURBS model for unified representation of conics and polynomial parametric curves/surfaces. The input CAD boundary with conics and polynomial representation can be converted into NUAHT B-spline form by matrix computation, and the solution space of NUAHT-IGA is spanned by the quasi-B-spline basis defined in the algebraic hyperbolic trigonometric space. Compared with the NURBS-IGA method, the NUAHT-IGA method has several advantages such as high accuracy, easy-to-compute derivatives and integrals due to the non-rational form of NUAHT B-splines, and some remarkable transcendental curves/surfaces such as the helix/helicoid, the cycloid and the catenary can be involved in the unified framework. Several 2D and 3D examples based on the heat conduction problem are presented to illustrate the efficiency of the proposed method.