### integral of exp(-x/a)*sinh(sqrt(b*x)) from 0 to infinity vanished?

Hi there.

I've tried to calculate integral, $$\int_{0}^{\infty} \exp{(-x/a)}\:\sinh{(\sqrt{bx})}\:dx,$$

```
a, b = var("a b")
Watt = exp(-x/a)*sinh(sqrt(b*x))
defInt = integrate(Watt,x,0,infinity)
```

For a>0 and b>0, the result seems to be
$$\frac{\sqrt{\pi}\:a^{3/2}\:\sqrt{b}\:e^{ab/4}}{2},$$ and actually

```
defInt_a1_b1 = integrate(Watt(a=1,b=1),x,0,infinity)
defInt_a1_b1.full_simplify()
```

yields $$\frac{1}{2}\sqrt{\pi}\:e^{\frac{1}{4}}.$$

But

```
defInt = integrate(Watt,x,0,infinity)
defInt
```

vanishes... $$0$$

Where did I make mistakes?

Thanks in advance,

Kazuyoshi