Double Integrals · Superchallenge problem

Evaluate $\displaystyle\int_0^{1} \dfrac{\arctan x}{x\sqrt{1 - x^2}}\,dx$. Give an exact answer. (Hint: $\dfrac{\arctan x}{x} = \displaystyle\int_0^1 \dfrac{dt}{1 + t^2x^2}$.)