I'm having trouble finding a reasonable way to integrate the following by parts:
$$\int_a^b dx \ e^{-c\sinh ^2 x}$$
The idea is to get an expansion in powers of $c$ (I have to later show that this expansion doesn't work in fact), but I have no clue how to do an integration by parts with this integrand. I tried adding a $\frac{d}{dx}(x)$ but this leads to a huge mess. I don't know what else to do, that is how to write
$$e^{-c\sinh ^2 x} = f(x)g'(x)$$
such that there is some sort of recurring structure to the successive integrations.
The constants $a$ and $b$ are arbitrary, $c$ is very large.
edit: I'm not looking for an antiderivative, but for an expansion in $c$