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I want to obtain the result of this integral where $\mathbb{R}\ni a,x>0$

$$\int{\sqrt{a^2 \left(\sin{a x}-2 \sin{2 a x}\right)^2+1} \, \mathrm{d}x}$$

but Mathematica does not give an answer. Of course, NIntegrate works for a given parameter $a$, but I am interested in a general result in terms of $a$.

Integrate[Sqrt[1 + a^2 (Sin[a x] - 2 Sin[2 a x])^2], x]

Any comments are appreciated.

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  • $\begingroup$ Any restriction on x and a? real? complex? range of values? $\endgroup$ – SHuisman Feb 9 at 18:02
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    $\begingroup$ it does not look like there is analytical solution to this. tried Rubi and FriCAS and both can't do it. $\endgroup$ – Nasser Feb 9 at 18:07
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    $\begingroup$ Is the indefinite integral required? If not, could you add the supposed range of integration to your post? Your NIntegrate results would be helpful too. $\endgroup$ – Gravifer Feb 9 at 18:15
  • $\begingroup$ @Gravifer Since I need to obtain the results on the domain in terms of $\pi$, I prefered to obtain the analytical result instead of numerical ones. The range of $x$ for example $ \frac{2\pi}{a} <x< \frac{3\pi}{a} $ $\endgroup$ – sara96 Feb 9 at 18:20
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    $\begingroup$ @sara69 Sometimes explicit formula exists for specific definite integrals but non for indefinite ones. $\endgroup$ – Gravifer Feb 9 at 18:28

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