if n < -3266859465.0066943 or 147870895489.3786 < n

1. Started with
   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
   43.8
2. Applied taylor to get
   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leadsto \frac{1}{n \cdot x} - \left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{2} \cdot \frac{1}{n \cdot {x}^2}\right)\]
   9.0
3. Taylor expanded around inf to get
   \[\color{red}{\frac{1}{n \cdot x} - \left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{2} \cdot \frac{1}{n \cdot {x}^2}\right)} \leadsto \color{blue}{\frac{1}{n \cdot x} - \left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{2} \cdot \frac{1}{n \cdot {x}^2}\right)}\]
   9.0
4. Applied taylor to get
   \[\frac{1}{n \cdot x} - \left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{2} \cdot \frac{1}{n \cdot {x}^2}\right) \leadsto \frac{1}{n \cdot x} - \left(\frac{1}{2} \cdot \frac{1}{n \cdot {x}^2} - \frac{\log x}{{n}^2 \cdot x}\right)\]
   8.9
5. Taylor expanded around inf to get
   \[\frac{1}{n \cdot x} - \color{red}{\left(\frac{1}{2} \cdot \frac{1}{n \cdot {x}^2} - \frac{\log x}{{n}^2 \cdot x}\right)} \leadsto \frac{1}{n \cdot x} - \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{n \cdot {x}^2} - \frac{\log x}{{n}^2 \cdot x}\right)}\]
   8.9
6. Applied simplify to get
   \[\frac{1}{n \cdot x} - \left(\frac{1}{2} \cdot \frac{1}{n \cdot {x}^2} - \frac{\log x}{{n}^2 \cdot x}\right) \leadsto \left(\frac{\log x}{\left(n \cdot x\right) \cdot n} + \frac{\frac{1}{n}}{x}\right) - \frac{\frac{1}{2}}{\left(x \cdot x\right) \cdot n}\]
   8.1

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7. Applied final simplification
8. Applied simplify to get
   \[\color{red}{\left(\frac{\log x}{\left(n \cdot x\right) \cdot n} + \frac{\frac{1}{n}}{x}\right) - \frac{\frac{1}{2}}{\left(x \cdot x\right) \cdot n}} \leadsto \color{blue}{\frac{\log x}{x \cdot \left(n \cdot n\right)} + \left(\frac{1}{n \cdot x} - \frac{\frac{\frac{1}{2}}{n}}{{x}^2}\right)}\]
   8.8

if -3266859465.0066943 < n < 147870895489.3786

1. Started with
   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
   3.7
2. Using strategy rm
   3.7
3. Applied add-log-exp to get
   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{red}{{x}^{\left(\frac{1}{n}\right)}} \leadsto {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{\log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right)}\]
   3.9
4. Applied add-log-exp to get
   \[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right) \leadsto \color{blue}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)} - \log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right)\]
   3.9
5. Applied diff-log to get
   \[\color{red}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) - \log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right)} \leadsto \color{blue}{\log \left(\frac{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)}\]
   3.9
6. Applied simplify to get
   \[\log \color{red}{\left(\frac{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)} \leadsto \log \color{blue}{\left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\]
   3.9
7. Using strategy rm
   3.9
8. Applied add-cube-cbrt to get
   \[\log \color{red}{\left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)} \leadsto \log \color{blue}{\left({\left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)}^3\right)}\]
   4.6