Despite my best attempts, I have been unable to evaluate the following integral: $$ \int_s^t\sqrt{9+(2+\cos3u)^2}\,du. $$ This integral showed up during an investigation of torus knots. It represents the length of a subarc of the (1,3)-torus knot, which I've parametrized as $$ \gamma_{1,3}(t)=\begin{bmatrix}(2+\cos 3t)\cos t\\(2+\cos3t)\sin t\\-\sin3t\end{bmatrix}. $$ Is there any closed form for this integral? I suspect that the answer is no, and a previous question lends evidence to this suspicion, but I couldn't find a statement anywhere to confirm this.
2026-04-06 06:31:34.1775457094
Integral of the square root of a trigonmetric function
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