Use Complex Integrals/ Residue to evaluate $\int_0^\infty \frac{dx}{(x+1)^3 + 1}$ 
Use Complex Integrals/ Residue to evaluate $\int_0^\infty \frac{dx}{(x+1)^3 + 1}$

I'm not sure how to do this integration. It looks like partial fractions but I'm unsure. 

We have
$$I = \int_0^{\infty} \dfrac{dx}{(1+x)^3+1} = \int_1^{\infty} \dfrac{dx}{x^3+1} = \int_1^0 \dfrac{-dx/x^2}{1/x^3+1} = \int_0^1 \dfrac{xdx}{1+x^3}$$
Hence,
$$I = \int_0^1\dfrac{x+1}{3(x^2-x+1)}dx - \int_0^1\dfrac{dx}{3(x+1)} = \dfrac{\pi}{3\sqrt{3}}-\dfrac{\log2}3$$
