\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]

↓

\[\left(\left(\log 2 + \log x\right) - \frac{\frac{1}{4}}{x \cdot x}\right) - \frac{\frac{3}{32}}{{x}^{4}}\]

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Derivation

Initial program 30.5
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
Taylor expanded around inf 0.4
\[\leadsto \color{blue}{\log 2 - \left(\log \left(\frac{1}{x}\right) + \left(\frac{3}{32} \cdot \frac{1}{{x}^{4}} + \frac{1}{4} \cdot \frac{1}{{x}^{2}}\right)\right)}\]
Simplified0.4
\[\leadsto \color{blue}{\left(\left(\log 2 + \log x\right) - \frac{\frac{1}{4}}{x \cdot x}\right) - \frac{\frac{3}{32}}{{x}^{4}}}\]
Final simplification0.4
\[\leadsto \left(\left(\log 2 + \log x\right) - \frac{\frac{1}{4}}{x \cdot x}\right) - \frac{\frac{3}{32}}{{x}^{4}}\]

herbie shell --seed 2018216 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))