if n < -18.617496f0 or 2506.933f0 < n < 4.0715707f+34

1. Started with
   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
   21.3
2. Applied taylor to get
   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leadsto \frac{1}{n \cdot x} - \left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{2} \cdot \frac{1}{n \cdot {x}^2}\right)\]
   5.4
3. Taylor expanded around inf to get
   \[\color{red}{\frac{1}{n \cdot x} - \left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{2} \cdot \frac{1}{n \cdot {x}^2}\right)} \leadsto \color{blue}{\frac{1}{n \cdot x} - \left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{2} \cdot \frac{1}{n \cdot {x}^2}\right)}\]
   5.4
4. Applied taylor to get
   \[\frac{1}{n \cdot x} - \left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{2} \cdot \frac{1}{n \cdot {x}^2}\right) \leadsto \frac{1}{n \cdot x} - \left(\frac{1}{2} \cdot \frac{1}{n \cdot {x}^2} - \frac{\log x}{{n}^2 \cdot x}\right)\]
   4.5
5. Taylor expanded around inf to get
   \[\frac{1}{n \cdot x} - \color{red}{\left(\frac{1}{2} \cdot \frac{1}{n \cdot {x}^2} - \frac{\log x}{{n}^2 \cdot x}\right)} \leadsto \frac{1}{n \cdot x} - \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{n \cdot {x}^2} - \frac{\log x}{{n}^2 \cdot x}\right)}\]
   4.5
6. Applied simplify to get
   \[\frac{1}{n \cdot x} - \left(\frac{1}{2} \cdot \frac{1}{n \cdot {x}^2} - \frac{\log x}{{n}^2 \cdot x}\right) \leadsto \left(\frac{\log x}{\left(n \cdot x\right) \cdot n} + \frac{\frac{1}{n}}{x}\right) - \frac{\frac{1}{2}}{\left(x \cdot x\right) \cdot n}\]
   3.7

---

7. Applied final simplification
8. Applied simplify to get
   \[\color{red}{\left(\frac{\log x}{\left(n \cdot x\right) \cdot n} + \frac{\frac{1}{n}}{x}\right) - \frac{\frac{1}{2}}{\left(x \cdot x\right) \cdot n}} \leadsto \color{blue}{\frac{\log x}{x \cdot \left(n \cdot n\right)} + \left(\frac{1}{n \cdot x} - \frac{\frac{\frac{1}{2}}{n}}{{x}^2}\right)}\]
   4.5

if -18.617496f0 < n < 2506.933f0

1. Started with
   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
   3.8
2. Using strategy rm
   3.8
3. Applied add-cube-cbrt to get
   \[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \leadsto \color{blue}{{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^3}\]
   3.8
4. Using strategy rm
   3.8
5. Applied add-cube-cbrt to get
   \[{\color{red}{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}}^3 \leadsto {\color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)}^3\right)}}^3\]
   3.8
6. Using strategy rm
   3.8
7. Applied pow3 to get
   \[{\color{red}{\left({\left(\sqrt[3]{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)}^3\right)}}^3 \leadsto {\color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)}^{3}\right)}}^3\]
   3.8

if 4.0715707f+34 < n

1. Started with
   \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
   16.7
2. Using strategy rm
   16.7
3. Applied add-cube-cbrt to get
   \[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \leadsto \color{blue}{{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^3}\]
   16.7
4. Using strategy rm
   16.7
5. Applied flip3-- to get
   \[{\left(\sqrt[3]{\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)}^3 \leadsto {\left(\sqrt[3]{\color{blue}{\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{n}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}\right)}}}\right)}^3\]
   16.7
6. Applied cbrt-div to get
   \[{\color{red}{\left(\sqrt[3]{\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{n}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}\right)}}\right)}}^3 \leadsto {\color{blue}{\left(\frac{\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}}}{\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{n}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}\right)}}\right)}}^3\]
   16.7
7. Applied cube-div to get
   \[\color{red}{{\left(\frac{\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}}}{\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{n}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}\right)}}\right)}^3} \leadsto \color{blue}{\frac{{\left(\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}}\right)}^3}{{\left(\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{n}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}\right)}\right)}^3}}\]
   16.7
8. Applied simplify to get
   \[\frac{\color{red}{{\left(\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}}\right)}^3}}{{\left(\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{n}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}\right)}\right)}^3} \leadsto \frac{\color{blue}{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^3 - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^3}}{{\left(\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{n}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}\right)}\right)}^3}\]
   16.7
9. Applied taylor to get
   \[\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^3 - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^3}{{\left(\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{n}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}\right)}\right)}^3} \leadsto \frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^3 - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^3}{\frac{5}{2} \cdot \frac{{\left(\log \left(\frac{1}{x}\right)\right)}^2}{{n}^2} + \left(3 \cdot \frac{\log \left(\frac{1}{x}\right)}{n} + \left(3 + \left(\frac{\log \left(1 + \frac{1}{x}\right) \cdot \log \left(\frac{1}{x}\right)}{{n}^2} + \left(3 \cdot \frac{\log \left(1 + \frac{1}{x}\right)}{n} + \frac{5}{2} \cdot \frac{{\left(\log \left(1 + \frac{1}{x}\right)\right)}^2}{{n}^2}\right)\right)\right)\right)}\]
   16.7
10. Taylor expanded around inf to get
    \[\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^3 - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^3}{\color{red}{\frac{5}{2} \cdot \frac{{\left(\log \left(\frac{1}{x}\right)\right)}^2}{{n}^2} + \left(3 \cdot \frac{\log \left(\frac{1}{x}\right)}{n} + \left(3 + \left(\frac{\log \left(1 + \frac{1}{x}\right) \cdot \log \left(\frac{1}{x}\right)}{{n}^2} + \left(3 \cdot \frac{\log \left(1 + \frac{1}{x}\right)}{n} + \frac{5}{2} \cdot \frac{{\left(\log \left(1 + \frac{1}{x}\right)\right)}^2}{{n}^2}\right)\right)\right)\right)}} \leadsto \frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^3 - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^3}{\color{blue}{\frac{5}{2} \cdot \frac{{\left(\log \left(\frac{1}{x}\right)\right)}^2}{{n}^2} + \left(3 \cdot \frac{\log \left(\frac{1}{x}\right)}{n} + \left(3 + \left(\frac{\log \left(1 + \frac{1}{x}\right) \cdot \log \left(\frac{1}{x}\right)}{{n}^2} + \left(3 \cdot \frac{\log \left(1 + \frac{1}{x}\right)}{n} + \frac{5}{2} \cdot \frac{{\left(\log \left(1 + \frac{1}{x}\right)\right)}^2}{{n}^2}\right)\right)\right)\right)}}\]
    16.7
11. Applied simplify to get
    \[\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^3 - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^3}{\frac{5}{2} \cdot \frac{{\left(\log \left(\frac{1}{x}\right)\right)}^2}{{n}^2} + \left(3 \cdot \frac{\log \left(\frac{1}{x}\right)}{n} + \left(3 + \left(\frac{\log \left(1 + \frac{1}{x}\right) \cdot \log \left(\frac{1}{x}\right)}{{n}^2} + \left(3 \cdot \frac{\log \left(1 + \frac{1}{x}\right)}{n} + \frac{5}{2} \cdot \frac{{\left(\log \left(1 + \frac{1}{x}\right)\right)}^2}{{n}^2}\right)\right)\right)\right)} \leadsto \frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^3 - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^3}{\left(\frac{5}{2} \cdot \frac{{\left(\log x\right)}^2}{n \cdot n} + \frac{\left(-3\right) \cdot \log x}{n}\right) + \left(\left(3 + \frac{\frac{5}{2} \cdot \log \left(\frac{1}{x} + 1\right)}{\frac{n \cdot n}{\log \left(\frac{1}{x} + 1\right)}}\right) + \frac{\log \left(\frac{1}{x} + 1\right)}{n} \cdot \left(\frac{-\log x}{n} + 3\right)\right)}\]
    16.7

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12. Applied final simplification
13. Applied simplify to get
    \[\color{red}{\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^3 - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^3}{\left(\frac{5}{2} \cdot \frac{{\left(\log x\right)}^2}{n \cdot n} + \frac{\left(-3\right) \cdot \log x}{n}\right) + \left(\left(3 + \frac{\frac{5}{2} \cdot \log \left(\frac{1}{x} + 1\right)}{\frac{n \cdot n}{\log \left(\frac{1}{x} + 1\right)}}\right) + \frac{\log \left(\frac{1}{x} + 1\right)}{n} \cdot \left(\frac{-\log x}{n} + 3\right)\right)}} \leadsto \color{blue}{\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^3 - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^3}{\frac{\log \left(\frac{1}{x} + 1\right)}{n} \cdot \left(\left(\left(-\frac{\log x}{n}\right) + 3\right) + \log \left(\frac{1}{x} + 1\right) \cdot \frac{\frac{5}{2}}{n}\right) + \left(\left(\frac{-3}{\frac{n}{\log x}} + \frac{\log x \cdot \frac{5}{2}}{\frac{n \cdot n}{\log x}}\right) + 3\right)}}\]
    16.7