In a publication I found an equation (eq. 11of the linked publication) of the form
$$c_p = B + C \left( \frac{D/T}{\sinh(D/T)} \right)^2 + E \left( \frac{F/T}{\cosh(F/T)} \right)^2$$
$T$ is the temperature, so it is always larger than zero, $c_p$ is the heat capacity, it is also larger than zero and it is monotonically increasing iwth $T$. The parameters $B \dots F$ have a physical meaning, but here they are just fitted to measured data. The publication includes a table with recommended constant values for the parameters.
The equation can be integrated to yield enthalpy $H$ (eq. 12 of the linked publication)
$$H = BT + CT \left( \frac{D}{T} \right) \coth\left( \frac{D}{T} \right) - ET \left( \frac{F}{T} \right) \tanh\left( \frac{F}{T} \right) + A$$
$T$ cancels out, so that the equation becomes
$$H = BT + CD \coth\left( \frac{D}{T} \right) - EF \tanh\left( \frac{F}{T} \right) + A$$
Using $x=1/T$ then gives
$$H = B/x + CD \coth\left( Dx \right) - EF \tanh\left( Fx \right) + A$$
For my application, I know the value of $H$ and have to iteratively find the corresponding value of $x$. If possible, it should be nicer to invert the function symbolically once and have an expression $x=x \left( H \right)$ that can be evaluated directly. Does the shown equation have an inverse function? As a unexperienced user I was not able to find an inverse function using Sympy or Mathematica.
If an exact inverse does not exist, I would as well be interested in a new parametric equation that can be fitted to measured/calculated data.