Is it possible to find solutions to $C=e^{a*x}(A*\cosh(b*x)+B*\sinh(b*x))$

Question

I need to solve a problem in which I find an equation like:

$$C=e^{a*x}(A*\cosh(b*x)+B*\sinh(b*x))$$

I would like to express $x$ in function of $C,a,b,A,B$.

However I am starting to wonder if it is simply possible to find analytic solution to this... Is it a kind of non solvable analytically transcendent equation ?

Answer 1

Hint:

put $$A= M\sinh(b\,c)\quad B=M\cosh(b\,c)$$ which invert to $$B^2-A^2=M^2 \quad \tanh(bc)=a/B$$

Then use the fact that $\sinh(x+y)=\sinh(x)\cos(y)+\sinh(y) \cosh(x)$