  Analytic and approximate solutions for the energy eigenvalues generated by
the hyperbolic potentials
$V_m(x)=-U_0\sinh^{2m}(x/d)/\cosh^{2m+2}(x/d),\,m=0,1,2,\dots$ are constructed.
A byproduct of this work is the construction of polynomial solutions for the
confluent Heun equation along with necessary and sufficient conditions for the
existence of such solutions based on the evaluation of a three-term recurrence
relation. Very accurate approximate solutions for the general problem with
arbitrary potential parameters are found by use of the {\it asymptotic
iteration method}.
