TITLE: A closed form of $\int_0^\infty\frac{\sqrt[\phi]{x}\ \arctan x}{\left(x^\phi+1\right)^2}dx$
QUESTION [40 upvotes]: Is it possible to evaluate the following integral in a closed form?
$$\int_0^\infty\frac{\sqrt[\phi]{x}\ \arctan x}{\left(x^\phi+1\right)^2}dx,$$
where $\phi$ is the golden ratio:
$$\phi=\frac{1+\sqrt{5}}{2}.$$

REPLY [22 votes]: robjohn's result can be generalized to all real $a\ne0$:
$$\int_0^\infty\frac{x^{a-1}\arctan x}{(x^a+1)^2}dx=\frac\pi{4\,|a|}.$$