Integral of $\sqrt{\cosh(x)}$ with respect to x

Question

I am trying to obtain a solution for the integral \begin{equation} \int^{x}_{0} \sqrt{\cosh(x)} dx. \end{equation}

A CAS system yields an answer depending on an elliptic integral of the second kind without giving me the reference to find the elliptic integral in question.

If I instead write $\cosh(x)$ above as \begin{equation} \cosh(x) = \frac{e^{x}+e^{-x}}{2} = \frac{e^{x}}{2}(1+e^{-2x}), \end{equation} I end up obtaining an answer in a CAS as dependent on a hypergeometric function, but again without a reference to find exactly where to look up what that function would be.

Would anyone have any suggestions on how to solve that integral? If one could further provide some references for my further education on these areas, that would be great.

Answer 1

WolframAlpha provides links to EllipticE and Hypergeometric2F1. Not sure how helpful they are for your purpose.

I don't think there's any “nice” way to express this integral. I'd recommend evaluating it numerically.

Here's a Taylor series for $x$ near 0:

$$x + \frac{x^3}{12} - \frac{x^5}{480} + \frac{19 x^7}{40320} - \frac{559 x^9}{5806080} + \frac{2651 x^{11}}{116121600} - \frac{2368081x^{13}}{398529331200} + O(x^{14})$$

For larger $x$, the $e^x$ term will dominate the $e^{-x}$ term, and you can just approximate the integral as $\sqrt{2}e^{x/2}$.

Answer 2

In the same spirit as @Dan, at least for small $x$, use $$\sqrt{\cosh (x)}=1+\sum_{n=1}^\infty \frac {(-1)^{n+1}}{2^n \,(2n)!}\,a_n\,x^{2n}$$ the first $a_n$ being $$\{1,1,19,559,29161,2368081,276580459,43947282079,\cdots\}$$ which correspond to sequence $A008990$ in $OEIS$ (use their absolute values).

Answer 3

$$I=\int \sqrt{\cosh(x)}~ dx=\int \sqrt{2\cosh^2\left(\frac x2\right)-1} ~dx=\int \sqrt{2\sinh^2\left(\frac x2\right)+1} ~dx$$ Putting, $u=\dfrac{ix}{2}\implies du=\dfrac{i}{2}~dx$, $$I=-2i\int \sqrt{1-2\sin^2 u}~du=-2i~E(u|2)+c=-2i E\left(\frac {ix}{2}\bigg| 2\right)+c$$ where $~c~$ in integrating constant.

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Note: Here $E(\phi|k^2)$ is called the incomplete elliptic integral of the second kind.

In trigonometric form, $$E(\phi|k^2)=\int \sqrt{1-k^2\sin^2 \phi}~~d\phi $$

For more details about elliptic integral, you may visit the following links:

Elliptic integral
Elliptic functions: Introduction course
Elliptic Functions and Elliptic Integrals by Viktor Vasilʹevich Prasolov
Handbook of Elliptic Integrals for Engineers and Physicists by Paul F. Byrd & Morris D. Friedman
Elliptic Functions: An Elementary Text-Book for Students of Mathematics by Arthur L. Baker