Integration of this trigonometry function

[songoku]

Homework Statement

Find
$$\int x \sqrt{1+4 \cos^2 (x)} dx$$

Relevant Equations

High School Integration:
integration by substitution
integration by part
integration of trigonometry function

Is it possible to do the integration? That is the full question

I don't know where to start, try to use $u=\cos x$ and also $\cos^2 (x) = \frac{1}{2} + \frac{1}{2} \cos (2x)$ but failed.

Thanks

 

[fresh_42]

WolframAlpha and I assume that there is no closed solution. The fact that $x$ occurs as polynomial and as trigonometric function (whose derivative is not the polynomial in $x$) makes it impossible to use things like e.g. the Weierstraß substitution.

 

[songoku]

Thank you very much fresh_42

 

[Vanadium 50]

Are you 100% sure you wrote the problem down right? If instead of x in the cosine you have an x² you could make more progress. (How much progress would depend on what the new integral is)

 

[songoku]

Are you 100% sure you wrote the problem down right? If instead of x in the cosine you have an x² you could make more progress. (How much progress would depend on what the new integral is)

Yes 100% sure. I have re-checked several times when doing the question and before posting it here