# How to get a result for $\int{\sqrt{a^2 \left(\sin{a x}-2 \sin{2 a x}\right)^2+1} \, \mathrm{d}x}$?

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]


• 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. – Gravifer Feb 9 at 18:15
• @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 $\frac{2\pi}{a} <x< \frac{3\pi}{a}$ – sara96 Feb 9 at 18:20