Why can't Wolfram calculate the Laplace transform of $\sinh(t)\sin(t)$ correctly?

Question

Question

Show that the Laplace transform of $\sinh(t)\sin(t) = \frac{2s}{s^4+4}$.

Wolfram can't calculate this as is, so I tried to simplify it a bit. I defined $\sinh(t)$ as $e^t-e^{-t}$ and split up the operation. This is what Wolfram outputs (after I pair them up myself):

$$ \frac{1}{(s-1)^2+1}-\frac{1}{(s+1)^2+1} $$

However, when you attempt to structure this to show that it answers the question, you get

$$ \frac{4s}{s^4+4} $$

Why is it not calculating this correctly? How do you solve this problem then?

Answer 1

Note how you have defined $\sinh(t)$. A factor of $\frac{1}{2}$ is missing, which when you correct for should give you the answer you're looking for.