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Sum to N
Sum to infinity
Hyperbolic sum
\displastyle[br]\begin{equation*} I_n = \int^{\infty}_{-\infty}\frac{1}{(x^2 + 2ax + b)^n} \mathrm{d}x\end{equation*}
\displaystyle [br]\begin{align*}I_n = \int^{\infty}_{-\infty}\frac{1}{((x+a)^2 + b - a^2)^n} \mathrm{d}x & = \int^{\pi/2}_{-\pi/2}\frac{\sqrt{b-a^2}\sec ^2 u}{((b-a^2)\tan ^2 u + (b-a^2))^n } \mathrm{d}u \\ & = \int^{\pi/2}_{-\pi/2}\frac{\sqrt{b-a^2}\sec ^2 u}{((b-a^2)\sec ^2 u)^n } du \\ & = \int^{\pi/2}_{-\pi/2}\frac{1}{(b-a^2)^{n-\frac{1}{2}} \sec ^{2n-2} u } du \end{align*}
\displaystyle [br]\begin{align*}I_1 = \int^{\pi/2}_{-\pi/2}\frac{1}{(b-a^2)^{\frac{1}{2}}} \mathrm{d}u & = \left[ \frac{x}{(b-a^2)^\frac{1}{2}} \right]_{-\pi/2}^{\pi/2} \\& = \frac{\pi}{\sqrt{b-a^2}} \end{align*}
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