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How do I evaluate the definite integral $\int_{ -\infty}^{\infty} \left|\frac{e^{4i x}-1}{e^{2i x}+1} \right|dx$?

How do I evaluate the integral
$$\int_{ -\infty}^{\infty} \left|\frac{e^{4i x}-1}{e^{2i x}+1} \right|dx$$
where $\left|z\right|=\sqrt{z \cdot z^*}$ is the magnitude of $z$?

A:

This is the same as $I=\int_{ -\infty}^{\infty} \frac{\left(\cos(4x)-1\right)}{\left(\cos(2x)-1\right)}dx$, which is
$$I=\int_{ -\infty}^{\infty} \frac{\left(\cos(4x)-1\right)}{\left(\cos(2x)-1\right)}dx=\frac{2}{\pi}K$$
where $K$ is the complete elliptic integral of the first kind.

A:

Note that
$$\frac{e^{4ix}-1}{e^{2ix}+1}=\frac{1-e^{ -4ix}}{1+e^{ -2ix}}.$$
Now, let us write the denominator as $(1+u^2)^{ -1}$ where $u=e^{ -2ix}$. Then,
$$\frac{1-e^{ -4ix}}{1+e^{ -2ix}}=\frac{(1+u^2)^2}{(1+u)^4}=\frac{1+2u^

Hotmail has been repeatedly hacked by security researchers and boasts some of the most comprehensive email security measures in the industry, including the Hotmail Passport and the Hotmail Sandbox tools for phishing and malware control. Hotmail’s latest form of Internet Explorer (IE) also has a few issues. However, Microsoft’s Internet Explorer team has said they will be rolling out a fix for the issues shortly.Athletics at the 2000 Summer Olympics – Men’s hammer throw

The men’s hammer throw competition at the 2000 Summer Olympics in Sydney, Australia. The event was held at the Olympic Stadium on 21–23 September.

Defending champion Miguel Catalán tied with reigning World and Olympic Champion Kimmoy Livonianenko, both throwing 77.00 metres. Mathieu Baguayzien was third with his 65.27 metres.

All seventeen competitors threw from a standing position.

Medalists

Schedule
All times are Australian Eastern Daylight Time (UTC+11:00)

Abbreviations
All results shown are in metres

Records

Qualifying

Group A

Group B

Final

Participation
According to an unofficial count, 74 athletes from 22 countries participated.

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References
Official Olympic Report
Results

Hammer throw
Category:Hammer throw at the OlympicsLuminance and colour thresholds in patients with pure red-green colour vision deficiencies.
Normal thresholds for the detection of green- and red-green-colour signals were determined in 30 cases of pure red-green colour vision deficiencies (PRGDCD) selected from a larger group of subjects with anomalous trichromatic vision due to PRGDCD, on the basis of psychophysically established tests using filters covering the whole visible spectrum. A modified ‘threshold test’ was devised to test for differences in threshold for luminance and colour in such a way that the method would not be affected by

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