I have looked to show that this integral:

$$\int_{-\infty}^{+\infty} x^n 2\cosh( x)e^{-x^2}=0$$

for $n$ is an odd positive integer , but i don't succeed to show that using standard method for getting closed form , Wolfram alpha show that is $0$ for some odd positive integer as shown here for $n=3$ , then my question here is :

**Question :** How do I show that integral is $0$ for odd positive integer $n$ if it is true ?