Stumped by an Integral: Can it be Solved with Elementary Functions?

[silver-rose]

Homework Statement

$$\int\frac{\arctan{x}dx}{x(x^2+1)}$$

I've been thinking over this for the past few days...I'm still stuck though
Can this integral even be expressed with elementary functions?

Homework Equations

N/A

The Attempt at a Solution

use the substitution $$u=tan{x}$$, and then, use integration by parts.
However I end up with $$\int\left|ln(cosx)\right|$$, as a term, which I cannot manage to integrate.

 

[arunbg]

Hint: Use u=arctan(x). What is du?

 

[coomast]

I think this integral does not have a "classical" primitive. After the substitution and partial integration you end up with:

$$I=\frac{1}{2}\frac{arctan^2(x)}{x^2}+\frac{1}{2}\int \frac{u^2}{sin^2(u)}du$$

The remaining integral is not an elementary function, according to "the integrator" of mathematica.

@silver-rose: What is expected as a result? A classical function or an advanced one?

 

[Defennder]

I agree. I can't find an elementary function despite a few pages of calculations and random substitutions. Maybe it's because I'm dumb or something. Anyone else had better luck here?

Tried it out at http://integrals.wolfram.com/

The answer was given in some weird notation involving something called a polylogarithm. What's that?

 

[arunbg]

Sorry my bad, I thought the OP had got it wrong in substituting u=tan(x) instead of arctan(x), and didn't check further. Since the mathematica integrator doesn't report a solution in terms of elementary functions, it is highly unlikely that there actually exists one.

 

[silver-rose]

yea that's what i think so too..

I've spent days on this integral, basically trying tons and tons of substitutions.

ti-89 can't do it, and mathematica gives a non-elementary answer.

Thanks anyways guys.