Does the function $$y(x) = cx + a \sinh (bx) $$ have an inverse function $x=x(y)$ and if yes, what is it?
The part $$y(x) = a \sinh(bx)$$ can easily be inverted to $$ x(y) = asinh(y/a)/b$$
For $c>0$ the function is strictly monotonic, so by just looking at the graph I would assume an inverse should exist.
Rule of thumb: if the unknown is both outside and inside trig/log/exp functions, there is no closed form inverse (unless in terms of purpose-defined special functions).
So no... just looking at your expression, seeing $x$ in a linear term and inside $\sinh$, the only way to do this is solving it numerically.