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Have this small, but beautiful integral:

$$\int \frac{dx}{x\ln x\ln (\ln x)}$$

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    Do you know the last words of a number theorist before drowning? They are $\log\log\log$. – 2017-02-23
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    Very happy that you possess such an integral. Then what else ? ... You should at least say what you have tried... – 2017-02-23
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    @JackD'Aurizio: ;-D ;-D – 2017-02-23

2 Answers 2

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Hint

You should notice that :

$$\frac{d}{dx}\left(\ln(\ln(x))\right)=\frac{\frac 1x}{\ln(x)}=\frac{1}{x\ln(x)}$$

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    Nice observation! – 2017-02-23
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Well, you can do repeated substitution of $\ln$. Substituting $u=\ln(x)$ gives: $$\int \frac{1}{u\cdot \ln{u}}~du$$ You can substitute $v=\ln{u}$, and from here on, it should be extremely easy.