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$$\int \frac{1}{x\sin x}dx$$

Is this integral computable? I tried with Wolfram Alpha; it says

Standard computation time exceeded.

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$x=1/y \implies \int \frac{y}{\sin \frac{1}{y}}dy$ but still seems problematic – Jonas12 Aug 16 '13 at 17:53
See answer..... – MathApprentice Aug 16 '13 at 17:54
You forgot to multiply by $\frac{-1}{y^2}$ in your substitution. Making that substitution would actually yield $\int \frac{-1}{y \sin (1/y)} dy$. (But this doesn't really help!) – Clive Newstead Aug 16 '13 at 18:00

As another answer pointed out, this can't be computed in terms of elementary functions. In general, you can use the Risch algorithm to determine if a function has an anti-derivative expressible in terms of elementary functions (see e.g. ) and the Risch algorithm correctly identifies this example as not being integrable in terms of elementary functions.

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