$P=ab+bc+ca+\frac{1}{a^{2}+1}+\frac{1}{b^{2}+1}+\frac{1}{c^{2}+1}\geq \frac{9}{2}$cm:
$P=ab+bc+ca+3-(\frac{a^{2}}{a^{2}+1}+\frac{b^{2}}{b^{2}+1}+\frac{c^{2}}{c^{2}+1})\geq ab+bc+ca+3-\frac{a+b+c}{2}$
$P\geq a(b-1)+b(c-1)+c(a-1)+\frac{a+b+c}{2}+3\geq \frac{9}{2}$