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I'm having trouble solving:

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

Wolfram Alpha gives an answer with imaginary numbers. I am able to get somewhat close to the answer by doing a Weierstrass Substitution, but the partial fraction decomposition because very ugly. I wanted to know if there is an easier way of solving the integral? Or if someone could solve it with the partial fractions.

Thanks!

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  • $\begingroup$ Tangent half-angle substitution works fine :/ $\endgroup$
    – Célestin
    Jan 5, 2018 at 2:00
  • $\begingroup$ It does. I'm currently working it out again, but it's just really ugly. (I could very easily be doing something wrong though, which is why I'm doing it again) $\endgroup$
    – Tom Himler
    Jan 5, 2018 at 2:01
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    $\begingroup$ @TomHimler: the final outcome is really ugly, I agree, but the technique is fine, it converts such trigonometric integral into $\int\frac{1+t^2}{3+t^4}\,dt$. $\endgroup$ Jan 5, 2018 at 2:02
  • $\begingroup$ Yep, that's exactly what I got. I'm just trying to see if there's a simpler way of doing it. Thanks for the response though! $\endgroup$
    – Tom Himler
    Jan 5, 2018 at 2:03
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    $\begingroup$ Well, a more general problem: $$\mathscr{I}_{\space\text{n}}:=\int\frac{1}{1+\cos\left(x\right)+\dots+\cos^\text{n}\left(x\right)}\space\text{d}x=\int\frac{1}{1+\displaystyle\sum_{\text{k}=1}^\text{n}\cos^\text{k}\left(x\right)}\space\text{d}x=$$ $$\int\frac{1}{\left(\frac{\cos^{1+\text{n}}\left(x\right)-1}{\cos\left(x\right)-1}\right)}\space\text{d}x=\int\frac{\cos\left(x\right)-1}{\cos^{1+\text{n}}\left(x\right)-1}\space\text{d}x\tag1$$ $\endgroup$ Jan 5, 2018 at 9:05

1 Answer 1

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We have $\cos^2(x)+\cos(x)+1\geq \frac{3}{4}$ so the integral is well-defined over any bounded interval of the real line. By letting $x=2\arctan t$ the integral is converted into $$ \int \frac{2(1+t^2)}{3+t^4}\,dt $$ which only depends on $\arctan\left(1\pm \frac{\sqrt{2}\,t}{\sqrt[4]{3}}\right)$ and $\log\left(\sqrt{3} \pm \sqrt{2}\sqrt[4]{3}\,t + t^2\right)$ by partial fraction decomposition.

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