I am trying to simplify an integration expression


I know that for $a=2$, I can have it as $\arctan(x)$. But what about the general case?

Thanks for your help!

  • 2
    $\begingroup$ What about PFD , Partial Fraction Decomposition ? $\endgroup$ – Cardinal Dec 4 '15 at 14:12
  • $\begingroup$ How can I decompose it? $\endgroup$ – KennyYang Dec 4 '15 at 14:14
  • 2
    $\begingroup$ If you want the most general answer for any real $a$, it is given in terms of Hyper Geometric function $I = \int {{1 \over {{x^a} + 1}} = } x\,\,{}_2^{}F_1^{}\left( {1,{1 \over a};1 + {1 \over a}; - {x^a}} \right)$ $\endgroup$ – H. R. Dec 4 '15 at 16:08
  • 1
    $\begingroup$ For positive integer values of $a,$ see Solving this integral?. $\endgroup$ – Dave L. Renfro Dec 4 '15 at 16:30
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    $\begingroup$ @Cardinal Your suggestion of partial fractions is not useful. Prove me wrong with a full answer for general $a$ or even just arbitrary positive integer $a$ if you think otherwise. $\endgroup$ – Simon S Dec 4 '15 at 18:13

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