Prove inequality for $a, b , c, d$ for $a,b, c, d\ge 0$ How does one prove the following inequality:
$$ \frac{a}{a^2+1}+\frac{b}{b^2+1}+\frac{c}{c^2+1}+\frac{d}{d^2+1}\le\frac{a}{b^2+1}+\frac{b}{c^2+1}+\frac{c}{d^2+1}+\frac{d}{a^2+1}$$
without much computation? Is there a trick? 

Yes, there is.
You can consider the numbers $a,b,c,d$ on one side and the numbers $\frac1{a^2+1}$, $\frac1{b^2+1}$, $\frac1{c^2+1}$, $\frac1{d^2+1}$ on the other, and apply the Rearrangement Inequality.
