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Solve the following integral : $$\int_{0}^{e}\frac{\sin(\operatorname{W(x)})-1}{\sin(\operatorname{W(x)})+1}\frac{\sin(\operatorname{2W(x)})-1}{\sin(\operatorname{2W(x)})+1}dx=?$$


My attempt:

We are lucky because there exists an antiderivative in terms of Hypergeometric function and generalized Hypergeometric function

I have tried the obvious substitution $\operatorname{W(x)}=t$ and integration by parts, but it generates integrals where I can't find the issue.

Moreover, I have tried to reveal some Gamma function but I can't.

Finally, I think (it's just an intuition) that we can use complex integral and residue theorem.

If you could help me, I would be thankful.

Thank you in advance!

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1 Answer 1

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You seem to like Lambert function as much as I do !

Concerning the integral, make $$W(x)=t \implies x=t\, e^t\implies dx=(1+t)\, e^t$$ and then we face the problem of $$\int_0^1 \frac{(\sin (t)-1)\, (\sin (2 t)-1)}{(\sin (t)+1)\, (\sin (2 t)+1)} \,(t+1)\, e^t\,dt$$ As you wrote, there is a messy antiderivative and the definite integral express in terms of a bunch of hypergeometric functions and Hurwitz-Lerch functions.

Evaluated, the result is $$0.21053628117921200861$$ which is not recognized by inverse symbolic calculators.

If you want a meaningless approximation $$\frac{541+596 \pi -212 \pi ^2}{361+449 \pi -25 \pi ^2}=0.21053628117921200854$$

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