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How to calculate

$L= \int\limits_0^1\sqrt{1+\cosh^2(x)}dx$?

I tried substituting $t=e^x$, but it did not help.

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    $\begingroup$ I think the indefinite integral cannot be expressed in terms of elementary functions. See WA for example. $\endgroup$
    – user0102
    Jul 30 '21 at 23:45
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    $\begingroup$ Mathematica: $$-i \sqrt{2} E\left(i\left|\frac{1}{2}\right.\right) ,$$ where $E()$ is the elliptic integral. $\endgroup$ Jul 31 '21 at 0:15
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The problem would be the same with $\sin(x)$, the integration of it leading to elliptic integrals.

If you want to avoid it, the only solution left (beside numerical integration) would be a series expansion of the integrand

By Taylor $$\sqrt{1+\cosh^2(x)}=\sqrt 2 \,\Bigg[1+\sum_{n=1}^\infty a_n\,x^{2n}\Bigg]$$ where the first coefficients are $$\left\{\frac{1}{4},\frac{5}{96},-\frac{11}{5760},-\frac{11}{129024},\frac{18121}{11 6121600},-\frac{216599}{6131220480},\cdots\right\}$$ Integrated between $0$ and $1$, this truncated series would give $$\int_0^1\sqrt{1+\cosh^2(x)}\,dx\sim \frac{435783525797}{199264665600 \sqrt{2}} =1.546413085\cdots$$ while the exact solution given by @David G. Stork $$\int_0^1\sqrt{1+\cosh^2(x)}\,dx=-i \sqrt{2} E\left(i \left|\frac{1}{2}\right.\right)=1.546413264\cdots$$

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