$\int_{-\infty}^{\infty} (x+1)(x-1)e^{-(x+1)^2(x-1)^2}$

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I am tring to solve the following integral: $$I=\int_{-\infty}^\infty(x+1)(x-1)e^{-(x+1)^2(x-1)^2}dx$$ So far I cannot think of a method for doing this other than numerically, And I have no proof that it has an elementary solution. Any thoughts?