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I'm interested to know if there is a general formual for $$ \int\left[\arctan\left(x\right) \over x^{2} + 1\right]^{1/k}\mathrm{d}x $$ with $k$ is positive integer may present integral of fraction derivative in other context for the form : $u'\times u$ , For the power is integer there is for each $k$ an antiderivative , But for fraction power i didn't find that in web , then is there any reference or any paper where the closed form of the titled integral is defined and exists ?.

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  • $\begingroup$ I seriously doubt a closed form exists. Do you have any reason to think it does? Also: what have you tried? $\endgroup$ – clathratus Apr 1 at 1:20
  • $\begingroup$ $x=\tan u$ gives $$\int\left[\frac{u}{\sec(u)^2}\right]^{1/k}\sec(u)^2du$$ which is $$\int u^{1/k}\cos(u)^{2/k-2}du$$ I am sure there is no closed form $\endgroup$ – clathratus Apr 1 at 1:24

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