\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;n \le -32555399.34096443:\\ \;\;\;\;\left(\sqrt[3]{1 + \frac{\frac{-1}{2}}{x}} \cdot \sqrt[3]{1 + \frac{\frac{-1}{2}}{x}}\right) \cdot \left(\frac{\frac{1}{n}}{x} \cdot \sqrt[3]{1 + \frac{\frac{-1}{2}}{x}}\right) + \frac{\log x}{\left(x \cdot n\right) \cdot n}\\ \mathbf{elif}\;n \le 4231354.817802664:\\ \;\;\;\;\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{1 + \frac{\frac{-1}{2}}{x}} \cdot \sqrt[3]{1 + \frac{\frac{-1}{2}}{x}}\right) \cdot \left(\frac{\frac{1}{n}}{x} \cdot \sqrt[3]{1 + \frac{\frac{-1}{2}}{x}}\right) + \frac{\log x}{\left(x \cdot n\right) \cdot n}\\ \end{array}\]

Derivation

Split input into 2 regimes
if n < -32555399.34096443 or 4231354.817802664 < n
1. Initial program 44.7
  \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
2. Taylor expanded around inf 32.7
  \[\leadsto \color{blue}{\frac{1}{x \cdot n} - \left(\frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}} + \frac{1}{2} \cdot \frac{1}{{x}^{2} \cdot n}\right)}\]
3. Simplified32.7
  \[\leadsto \color{blue}{\frac{\log x}{n \cdot \left(n \cdot x\right)} + \left(\frac{\frac{-1}{2}}{x} + 1\right) \cdot \frac{1}{n \cdot x}}\]
4. Using strategy rm
5. Applied associate-/r*32.1
  \[\leadsto \frac{\log x}{n \cdot \left(n \cdot x\right)} + \left(\frac{\frac{-1}{2}}{x} + 1\right) \cdot \color{blue}{\frac{\frac{1}{n}}{x}}\]
6. Using strategy rm
7. Applied add-cube-cbrt32.1
  \[\leadsto \frac{\log x}{n \cdot \left(n \cdot x\right)} + \color{blue}{\left(\left(\sqrt[3]{\frac{\frac{-1}{2}}{x} + 1} \cdot \sqrt[3]{\frac{\frac{-1}{2}}{x} + 1}\right) \cdot \sqrt[3]{\frac{\frac{-1}{2}}{x} + 1}\right)} \cdot \frac{\frac{1}{n}}{x}\]
8. Applied associate-*l*32.1
  \[\leadsto \frac{\log x}{n \cdot \left(n \cdot x\right)} + \color{blue}{\left(\sqrt[3]{\frac{\frac{-1}{2}}{x} + 1} \cdot \sqrt[3]{\frac{\frac{-1}{2}}{x} + 1}\right) \cdot \left(\sqrt[3]{\frac{\frac{-1}{2}}{x} + 1} \cdot \frac{\frac{1}{n}}{x}\right)}\]
if -32555399.34096443 < n < 4231354.817802664
1. Initial program 8.5
  \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
2. Using strategy rm
3. Applied add-cube-cbrt8.5
  \[\leadsto \color{blue}{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\]
Recombined 2 regimes into one program.
Final simplification22.1
\[\leadsto \begin{array}{l} \mathbf{if}\;n \le -32555399.34096443:\\ \;\;\;\;\left(\sqrt[3]{1 + \frac{\frac{-1}{2}}{x}} \cdot \sqrt[3]{1 + \frac{\frac{-1}{2}}{x}}\right) \cdot \left(\frac{\frac{1}{n}}{x} \cdot \sqrt[3]{1 + \frac{\frac{-1}{2}}{x}}\right) + \frac{\log x}{\left(x \cdot n\right) \cdot n}\\ \mathbf{elif}\;n \le 4231354.817802664:\\ \;\;\;\;\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{1 + \frac{\frac{-1}{2}}{x}} \cdot \sqrt[3]{1 + \frac{\frac{-1}{2}}{x}}\right) \cdot \left(\frac{\frac{1}{n}}{x} \cdot \sqrt[3]{1 + \frac{\frac{-1}{2}}{x}}\right) + \frac{\log x}{\left(x \cdot n\right) \cdot n}\\ \end{array}\]

Error

Derivation

`if n < -32555399.34096443 or 4231354.817802664 < n`

`if -32555399.34096443 < n < 4231354.817802664`

Reproduce