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I want to integrate the flowing $$\int\sec^3x\ln (\arctan e^x)dx$$ i tried u substitution using $u=e^x$ and $ \arctan e^x=u$ and also $\sec^3x=u$, but i found nothing. What would you suggest?

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  • $\begingroup$ Have you tried integration by parts? $\endgroup$ – Maryam Dec 26 '15 at 17:56
  • $\begingroup$ @Maryam yes i did $\endgroup$ – zakaria hmimid Dec 26 '15 at 17:56
  • $\begingroup$ I don't think this has an elementary antiderivative. $\endgroup$ – Chappers Dec 26 '15 at 17:58
  • $\begingroup$ @Chappers .. but the function is continous so it has one .. $\endgroup$ – zakaria hmimid Dec 27 '15 at 12:39
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    $\begingroup$ It has an antiderivative. But not an antiderivative composed of a finite combination of elementary functions, which is what an elementary antiderivative is. See the links in @Lucian's answer. $\endgroup$ – Chappers Dec 27 '15 at 16:20
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What would you suggest ?

I would suggest acquainting yourself with Liouville's theorem and the Risch algorithm.

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