I am using $\operatorname{arcosh} x = \operatorname{arcsinh}(\sqrt { x^2-1 } ) $ with bad results. The standard power series $$ \operatorname{arcosh} x = \ln(2x) - \left( \left( \frac {1} {2} \right) \frac {x^{-2}} {2} + \left( \frac {1 \cdot 3} {2 \cdot 4} \right) \frac {x^{-4}} {4} + \left( \frac {1 \cdot 3 \cdot 5} {2 \cdot 4 \cdot 6} \right) \frac {x^{-6}} {6} +\cdots \right) \\ = \ln(2x) - \sum_{n=1}^\infty \left( \frac {(2n)!} {2^{2n}(n!)^2} \right) \frac {x^{-2n}} {2n} , \qquad \left| x \right| > 1 $$ is also messy for me since it wants to use $ \ln(2x) $ and it has the slow calculation of $ x^{-6} / 6 $ and others like it. I have a $ \ln() $ power series but it also takes a long time to calculate. Also, this one is not good $$ \operatorname {arcosh} (x) = \ln \left(x + \sqrt{x^{2} - 1} \right) \qquad x \geqslant 1 \\ $$ It uses an $ \ln() $ and square root function. Calculating a square root is as bad as a $\ln()$. Please, no Bernoulli or Euler numbers also. They are messy in a computer system. Least way, my system. Sorry for being picky.
Is there another series to be used? Thank you.
$Blue$, 1st Thank you for all of your help before. Yes, I followed your process, and it made sense, but I can not see how to remove $ln(2x)$ from the series.
$ Jean-Claude Arbaut $, The problem, for me only, is that 1. Within the C code, they do use the $log$ function built within the computer system. 2. They are also using fixed-length (20 decimal places) constants, example::$ln2 = 6.93147180559945286227e-01; $ I want and can go longer than this, if I need to.
$Yves Daoust $, Bernoulli or Euler are "messy" from the view point that I would have to carry them in a table and they are only given a certain fixed-length. Also, I have been trying to find "how do you create Bernoulli or Euler numbers". I have found nothing simple to use.
For "small" values of $x$ (say $x<3$), you could use Puiseux-Laurent series since $$\frac{\cosh^{-1}(x)}{\sqrt{2(x-1)} }=1-\frac{x-1}{12}+\frac{3(x-1)^2}{160} -\frac{5(x-1)^3}{896} +\frac{35 (x-1)^4}{18432}-\frac{63 (x-1)^5}{90112}+\frac{231 (x-1)^6}{851968}-\frac{143 (x-1)^7}{1310720}+\frac{6435 (x-1)^8}{142606336}-\frac{12155 (x-1)^9}{637534208}+\frac{46189 (x-1)^{10}}{5637144576}-\frac{88179 (x-1)^{11}}{24696061952}+\frac{676039 (x-1)^{12}}{429496729600}-\frac{1300075 (x-1)^{13}}{1855425871872}+\frac{5014575 (x-1)^{14}}{15942918602752}+O\left((x-1)^{15}\right) $$
Have a look at $OEIS$ sequences $A055786$ and $A091019$ if you want more terms.
Otherwise, using $$\cosh^{-1}(x)= \log(2x)- \sum_{n=1}^\infty \left( \frac {(2n)!} {2^{2n}(n!)^2} \right) \frac {x^{-2n}} {2n} $$ let $$a_n=\left( \frac {(2n)!} {2^{2n}(n!)^2} \right) \frac {x^{-2n}} {2n}\implies a_{n+1}=\frac{n (2 n+1)}{2 (n+1)^2 }\,a_n\,x^{-2}$$ which is quite good for programming.