The present paper proposes a new Strain Energy Function (SEF) for incompressible transversely isotropic hyperelastic materials, i.e. materials with a single fiber family. This SEF combines polyconvex invariants forming an integrity basis (Ta et al., 2014) in a polynomial and exponential form. Compared to a previous attempt for building a SEF based on the same invariants (Cai et al., 2016), we have reduced the number of material parameters from 23 to 10, without losing any accuracy on the numerical results. The 10 material parameters are identified by comparing the closed form solutions deriving from our model with experimental and numerical data extracted from the literature. These data concern uniaxial tension and shear tests, both parallel and transverse to the fiber direction (Ciarletta et al., 2011; Davis and De Vita, 2014) [3, 4], as well as shear calculations with 9 different fiber angles (Horgan and Murphy, 2017) [5]. Due to the variety of the considered situations, we have developed specific identification strategies based on: 1) the linear or nonlinear nature of the material parameters of the model; 2) the modeling of the free boundary conditions by a spectral approach.