  In this work we solve the Schr\"odinger equation for Bohr Hamiltonian with
Coulomb and Hulth\'en potentials within the formalism of minimal length in
order to obtain analytical expressions for the energy eigenvalues and
eigenfunctions by means of asymptotic iteration method. The obtained formulas
of the energy spectrum and wave functions, are used to calculate excitation
energies and transition rates of $\gamma$-rigid nuclei and compared with the
experimental data at the shape phase critical point X(3) in nuclei.
