I'm trying to numerically evaluate $$ S_{\nu}(z) = \int_0^\infty \exp\bigl(-\nu t - z \sinh(t)\bigr) \mathrm{d}t $$ where $\nu, z \in \mathbb{C}$ with $\Re(z) > 0$. This integral is a part of the Schläfli representation of the Bessel function $J_\nu(z)$.
So I separate the real part and the imaginary part and I do the change of variables $t = \dfrac{u}{1-u}$. The problem is that after trying with two numerical integration calculators, the result is NaN for both the real part and the imaginary part.
For the real part of the integrand, using a computer algebra system, I get, letting $z = x+iy$ and $\nu = a+ib$,: $$ \exp\bigl(-x \sinh(t)-at\bigr) \cos\bigl(-y\sinh(t)-bt\bigr). $$ Then I replace $t$ with $\dfrac{u}{1-u}$ and I multiply by $\dfrac{1}{{(1-u)}^2}$ and I evaluate the integral from $0$ to $1$.
Why does it fail? Am I doing something bad or wrong ?
Here is the integrand in $u$: $$ \exp\left(-x\sinh\bigl(u/(1-u)\bigr)-au/(1-u)\right) \cos\left(-y\sinh\bigl(u/(1-u)\bigr)-bu/(1-u)\right) / {(1-u)}^2 $$