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$$\int \tanh \left ( \cos x \right )\mathrm{d}x$$

Is this analytically Solvable?

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This is not wolframalpha :) – Dominic Michaelis Jul 15 '13 at 8:03
And Wolframalpha says no – Dominic Michaelis Jul 15 '13 at 8:07
How did you land on this integral? Also, I've found to be fairly useful. – dfeuer Jul 15 '13 at 8:39

Since the radius of convergence of maclaurin series of $\tanh x$ is $\dfrac{\pi}{2}$ , the following procedure holds for at least all real numbers of $x$ :

$\int\tanh(\cos x)~dx=\int\sum\limits_{n=1}^\infty\dfrac{B_{2n}4^n(4^n-1)\cos^{2n-1}x}{(2n)!}dx$

For $\int\cos^{2n-1}x~dx$ , where $n$ is any natural number,


$=\int\cos^{2n-2}x~d(\sin x)$

$=\int(1-\sin^2x)^{n-1}~d(\sin x)$

$=\int\sum\limits_{k=0}^{n-1}C_k^{n-1}(-1)^k\sin^{2k}x~d(\sin x)$



Or you can directly use the formula

$\int\tanh(\cos x)~dx=\int_0^x\tanh(\cos x)~dx+C=x\sum\limits_{n=1}^\infty\sum\limits_{k=1}^{2^n-1}\dfrac{(-1)^{k+1}}{2^n}\tanh\left(\cos\dfrac{kx}{2^n}\right)+C$

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