I am studying how to evaluate the integral $\int_{0}^{\pi/4}{d\theta \over \epsilon^2+\sin^2\theta}$ as $\epsilon \rightarrow 0$ with asymptotic methods.
The book perturbation methods by Hinch suggests to split the range of integration into two parts, which makes sense since the local behaviour at $\epsilon \rightarrow 0$ is os different from the global. The local contribution is easy to evaluate by rescaling with a parameter $\theta = \epsilon \, u$. For the global contribution the book suggests to use the expansion of sin for small angles: ${1 \over \epsilon^2 +\sin^2\theta}= {1 \over \epsilon^2 + \epsilon^2 u^2 -\frac 1 3 \epsilon^4 u^4 + \cdots}={1 \over \epsilon^2} \left( {1 \over 1+u^2}+{\epsilon^2 u^4 \over 3(1+u^2)^2}+\cdots\right)$
and then integrate out.
I understand that the second term if the Taylor expansion of $\sin^2\theta$ but could anyone tell me how was the third term obtained?