Let $a>2$ be a real variable. My objective is to find an approximation of the integral defined as \begin{equation} \int_b^{ + \infty } {\frac{x}{{1 + {x^a}}}}. \end{equation} Here $b$ is a positive real number.

Write $\int_b^\infty = \int_0^\infty - \int_0^b$. The first integral gives a Beta function, which evaluated yields $\frac{\pi}{a\sin(2\pi/a)}$. If $b^a\ll 1$, you can get a good approximation to the second integral by converting the integral to a weighted geometric series and integrating term by term, obtaining $$b^2\sum_{k=0}^\infty (-1)^k \frac{b^{ak}}{ak+2}.$$ In this mentioned regime, your integral is $$\frac{\pi}{a\sin(2\pi/a)}+b^2/2-\frac{b^{a+2}}{a+2}+\mathcal{O}(b^{2a+2}).$$

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