# Evaluating $\int{ \frac{\arctan\sqrt{n^{2}-1}}{\sqrt{n^{2}+n}}} dn$

How to integrate? $$\int{ \frac{\arctan\sqrt{n^{2}-1}}{\sqrt{n^{2}+n}}} dn$$

I have no idea how to do it. Tried to get some information from wiki, but its too hard :|

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Please try to use a more specific title, this one is very generic and will not assist users searching for similar queries in future. –  Simon Hayward Dec 1 '12 at 21:10
It's easier for people if you use conventional letters for variables. Since only one variable appears in your question, it would be more friendly to call it $x$. The letter $n$ normally denotes an integer. –  John Bentin Dec 1 '12 at 21:24
The substitution $n=\sec\theta$ looks messy. I've also gotten it into the form $\int\frac{\cos^{-1}xdx}{x\sqrt{x+1}}$ and tried integration by parts, but that doesn't seem to lead anywhere as eliminating the inverse trig function still leaves natural log terms... –  Mike Dec 2 '12 at 18:16
@user51402 looks like Wolfram Mathematica refuses symbolic integration. –  Caran-d'Ache May 18 '13 at 8:19
Hope you are not randomly writing integrands and asking us. And now I realize I just brought up an old question... –  Awesome Mar 26 at 14:20
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## 1 Answer

Mathematica returns a closed form solution for $\int\frac{\cos^{-1}xdx}{x\sqrt{x+1}}$, but it is several dozen lines long. I am not sure how to clean it up yet.

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The solution you speak of can be reduced to the size of a single screen (six or seven lines) if FullSimplify is employed. –  Lucian Oct 31 '13 at 2:07
Please show the amazing result. –  doraemonpaul Feb 9 at 0:02
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