\(\frac{a}{b^3-1}+\frac{b}{a^3-1}=\frac{a}{\left(b-1\right)\left(b^2+b+1\right)}+\frac{b}{\left(a-1\right)\left(a^2+a+1\right)}\)
\(=\frac{a}{-a\left(b^2+b+1\right)}+\frac{b}{-b\left(a^2+a+1\right)}=-\frac{1}{b^2+b+1}-\frac{1}{a^2+a+1}\)
\(=-\frac{a^2+a+1+b^2+b+1}{\left(b^2+b+1\right)\left(a^2+a+1\right)}=-\frac{a^2+b^2+3}{a^2b^2+b^2a+b^2+ba^2+ab+b+a^2+a+1}\)
\(=-\frac{\left(a+b\right)^2-2ab+3}{a^2b^2+ab\left(a+b\right)+a^2+b^2+ab+\left(a+b\right)+1}\)
\(=\frac{2ab-4}{a^2b^2+2ab+\left(a+b\right)^2-2ab+2}=\frac{2\left(ab-2\right)}{a^2b^2+3}\)
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