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?

  • $\begingroup$ 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
  • 1
    $\begingroup$ 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

What would you suggest ?

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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.