\(\frac{a\left(b^2+1\right)-ab^2}{b^2+1}+\frac{b\left(a^2+1\right)-a^2b}{a^2+1}\)
\(=a-\frac{ab^2}{b^2+1}+b-\frac{a^2b}{a^2+1}=\left(a+b\right)-\left(\frac{ab^2}{b^2+1}+\frac{a^2b}{a^2+1}\right)\)
\(\ge\left(a+b\right)-\left(\frac{ab^2}{2b}+\frac{a^2b}{2a}\right)=\left(a+b\right)-\left(\frac{ab}{2}+\frac{ab}{2}\right)=\left(a+b\right)-ab\)
\(a^2+b^2\ge2ab\Leftrightarrow\left(a+b\right)^2\ge4ab\Rightarrow ab\le\frac{\left(a+b\right)^2}{4}=1\)
\(\Rightarrow\left(a+b\right)-ab\ge2-1=1\)
\("="\Leftrightarrow a=b=1\)
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