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

Comparaison between Coulomb and Hulth\`en potentials within Bohr Hamiltonian for $\gamma$-rigid nuclei in the presence of minimal length