Definite integration of an algebraic expression 
Evaluate $$\int_{0}^{1}\frac{1-x}{1+x}\frac{dx}{\sqrt{x+x^2+x^3}}$$

I think none of the properties of definite integral will be useful here so I think I will have to integrate. But I am unable to do so. Some hints on how to integrate. Thanks.

Let $u = \sqrt{x+1+x^{-1}}$, we have $2 u du = (1 - x^{-2})dx$ and
$$
\int_0^1 \frac{1-x}{1+x}\frac{dx}{\sqrt{x+x^2+x^3}}
= \int_\infty^\sqrt{3} \frac{1-x}{1+x}\frac{2u du}{(x-x^{-1})u}
= 2 \int_\sqrt{3}^\infty \frac{du}{x+2+x^{-1}}\\
= 2 \int_\sqrt{3}^\infty \frac{du}{u^2+1}
= 2 \left[\tan^{-1}u\right]_{\sqrt{3}}^\infty
= 2 \left(\frac{\pi}{2} - \frac{\pi}{3}\right) = \frac{\pi}{3}
$$
