<s>[INST] How can we show that $\int_{-\infty}^{+\infty}{b\over 1+2ax+(cx)^2}\cdot{\mathrm dx\over 1-2ax+(cx)^2}={\pi\over 2}?$ Consider this integral

$$\int_{-\infty}^{+\infty}{b\over 1+2ax+(cx)^2}\cdot{\mathrm dx\over 1-2ax+(cx)^2}={\pi\over 2}\tag1$$
  Where $a^2+b^2=c^2$; Pythagoras  theorem.

Making an attempt:
Complete the square of $(cx)^2+2ax+1=c^2\left(x+{a\over c^2}\right)^2+{c^2-a^2\over c^2}$, I can't see how this would be of any useful.
$1+2x^2(c^2-2a^2)+c^4x^4=(1+2ax+c^2x^2)(1-2ax+c^2x^2)$
I am even sure how to go about making an attempt to find the equivalent of $(1)$.
How may we go about and prove $(1)?$
 [/INST] I will assume $b > 0$ (in which case $a^2$ can never equal $c^2$).
Employing a partial fraction decomposition we have
$$\frac{1}{1 + 2ax + c^2 x^2} \cdot \frac{1}{1 - 2ax + c^2 x^2} = \frac{1}{4a} \cdot \frac{2a + c^2 x}{1 + 2ax + c^2 x^2} + \frac{1}{4a} \cdot \frac{2a - c^2 x}{1 - 2ax + c^2 x^2}.$$
Now noting that
$$\int \frac{2a + c^2 x}{1 + 2ax + c^2 x^2} \, dx = \frac{a}{\sqrt{c^2 - a^2}} \tan^{-1} \left (\frac{c^2 x + a}{\sqrt{c^2 - a^2}} \right ) + \frac{1}{2} \ln |1 + 2ax + c^2 x^2| + \cal{C}_1,$$
and
$$\int \frac{2a - c^2 x}{1 - 2ax + c^2 x^2} \, dx = \frac{a}{\sqrt{c^2 - a^2}} \tan^{-1} \left (\frac{c^2 x - a}{\sqrt{c^2 - a^2}} \right ) - \frac{1}{2} \ln |1 - 2ax + c^2 x^2| + \cal{C}_2.$$
(note that each of these integrals can be solved by elementary means). Thus
\begin{align*}
\int_{-\infty}^{\infty}\frac{b}{1+2ax+c^2x^2}\cdot \frac{dx}{1-2ax+c^2 x^2} &= \left [\frac{b}{4 \sqrt{c^2 - a^2}} \left \{\tan^{-1} \left (\frac{c^2 x + a}{\sqrt{c^2 - a^2}} \right ) \right. \right.\\
& \left. \left. + \tan^{-1} \left (\frac{c^2 x - a}{\sqrt{c^2 - a^2}} \right ) \right \}+ \frac{b}{8a} \ln \left |\frac{1 + 2ax + c^2 x^2}{1 - 2ax + c^2 x^2} \right | \right ]^\infty_{-\infty}\\
&= \frac{b}{4 \sqrt{c^2 - a^2}} \left [\frac{\pi}{2} + \frac{\pi}{2} - \left (-\frac{\pi}{2} - \frac{\pi}{2} \right ) \right ]\\
&= \frac{\pi b}{2 \sqrt{c^2 - a^2}}.
\end{align*}
But given $a^2 + b^2 = c^2$, then $\sqrt{c^2 -a^2} = \sqrt{b^2} = |b| = b$, assuming $b > 0$ giving
$$\int_{-\infty}^\infty \frac{b}{1+2ax+c^2x^2} \cdot \frac{dx}{1-2ax+c^2 x^2} = \frac{\pi}{2},$$
as required to show
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