if x < 112873629.87382902

1. Started with
   \[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
   0.4
2. Using strategy rm
   0.4
3. Applied add-exp-log to get
   \[{\color{red}{\left(x + 1\right)}}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)} \leadsto {\color{blue}{\left(e^{\log \left(x + 1\right)}\right)}}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
   0.4
4. Applied pow-exp to get
   \[\color{red}{{\left(e^{\log \left(x + 1\right)}\right)}^{\left(\frac{1}{3}\right)}} - {x}^{\left(\frac{1}{3}\right)} \leadsto \color{blue}{e^{\log \left(x + 1\right) \cdot \frac{1}{3}}} - {x}^{\left(\frac{1}{3}\right)}\]
   0.4
5. Applied simplify to get
   \[e^{\color{red}{\log \left(x + 1\right) \cdot \frac{1}{3}}} - {x}^{\left(\frac{1}{3}\right)} \leadsto e^{\color{blue}{\frac{\log \left(x + 1\right)}{3}}} - {x}^{\left(\frac{1}{3}\right)}\]
   0.4
6. Using strategy rm
   0.4
7. Applied flip3-- to get
   \[\color{red}{e^{\frac{\log \left(x + 1\right)}{3}} - {x}^{\left(\frac{1}{3}\right)}} \leadsto \color{blue}{\frac{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^{3} - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^{3}}{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + e^{\frac{\log \left(x + 1\right)}{3}} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}}\]
   0.5
8. Applied simplify to get
   \[\frac{\color{red}{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^{3} - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^{3}}}{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + e^{\frac{\log \left(x + 1\right)}{3}} \cdot {x}^{\left(\frac{1}{3}\right)}\right)} \leadsto \frac{\color{blue}{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}}{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + e^{\frac{\log \left(x + 1\right)}{3}} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}\]
   0.5
9. Applied taylor to get
   \[\frac{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + e^{\frac{\log \left(x + 1\right)}{3}} \cdot {x}^{\left(\frac{1}{3}\right)}\right)} \leadsto \frac{{\left({\left(1 + \frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3 - {\left({\left(\frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3}{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + e^{\frac{\log \left(x + 1\right)}{3}} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}\]
   57.8
10. Taylor expanded around inf to get
    \[\frac{\color{red}{{\left({\left(1 + \frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3 - {\left({\left(\frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3}}{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + e^{\frac{\log \left(x + 1\right)}{3}} \cdot {x}^{\left(\frac{1}{3}\right)}\right)} \leadsto \frac{\color{blue}{{\left({\left(1 + \frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3 - {\left({\left(\frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3}}{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + e^{\frac{\log \left(x + 1\right)}{3}} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}\]
    57.8
11. Applied simplify to get
    \[\frac{{\left({\left(1 + \frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3 - {\left({\left(\frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3}{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + e^{\frac{\log \left(x + 1\right)}{3}} \cdot {x}^{\left(\frac{1}{3}\right)}\right)} \leadsto \frac{\left(\frac{1}{x} + 1\right) - \frac{1}{x}}{{\left(e^{\frac{\log \left(1 + x\right)}{3}}\right)}^2 + {x}^{\left(\frac{1}{3}\right)} \cdot \left({x}^{\left(\frac{1}{3}\right)} + e^{\frac{\log \left(1 + x\right)}{3}}\right)}\]
    56.7

---

12. Applied final simplification
13. Applied simplify to get
    \[\color{red}{\frac{\left(\frac{1}{x} + 1\right) - \frac{1}{x}}{{\left(e^{\frac{\log \left(1 + x\right)}{3}}\right)}^2 + {x}^{\left(\frac{1}{3}\right)} \cdot \left({x}^{\left(\frac{1}{3}\right)} + e^{\frac{\log \left(1 + x\right)}{3}}\right)}} \leadsto \color{blue}{\frac{1}{{x}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)} + \left(e^{\frac{\log \left(1 + x\right)}{3}} + {x}^{\left(\frac{1}{3}\right)}\right) \cdot e^{\frac{\log \left(1 + x\right)}{3}}}}\]
    0.1

if 112873629.87382902 < x

1. Started with
   \[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
   60.6
2. Using strategy rm
   60.6
3. Applied add-exp-log to get
   \[{\color{red}{\left(x + 1\right)}}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)} \leadsto {\color{blue}{\left(e^{\log \left(x + 1\right)}\right)}}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
   61.1
4. Applied pow-exp to get
   \[\color{red}{{\left(e^{\log \left(x + 1\right)}\right)}^{\left(\frac{1}{3}\right)}} - {x}^{\left(\frac{1}{3}\right)} \leadsto \color{blue}{e^{\log \left(x + 1\right) \cdot \frac{1}{3}}} - {x}^{\left(\frac{1}{3}\right)}\]
   61.0
5. Applied simplify to get
   \[e^{\color{red}{\log \left(x + 1\right) \cdot \frac{1}{3}}} - {x}^{\left(\frac{1}{3}\right)} \leadsto e^{\color{blue}{\frac{\log \left(x + 1\right)}{3}}} - {x}^{\left(\frac{1}{3}\right)}\]
   60.4
6. Using strategy rm
   60.4
7. Applied flip3-- to get
   \[\color{red}{e^{\frac{\log \left(x + 1\right)}{3}} - {x}^{\left(\frac{1}{3}\right)}} \leadsto \color{blue}{\frac{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^{3} - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^{3}}{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + e^{\frac{\log \left(x + 1\right)}{3}} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}}\]
   60.4
8. Applied simplify to get
   \[\frac{\color{red}{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^{3} - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^{3}}}{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + e^{\frac{\log \left(x + 1\right)}{3}} \cdot {x}^{\left(\frac{1}{3}\right)}\right)} \leadsto \frac{\color{blue}{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}}{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + e^{\frac{\log \left(x + 1\right)}{3}} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}\]
   60.4
9. Applied taylor to get
   \[\frac{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + e^{\frac{\log \left(x + 1\right)}{3}} \cdot {x}^{\left(\frac{1}{3}\right)}\right)} \leadsto \frac{{\left({\left(1 + \frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3 - {\left({\left(\frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3}{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + e^{\frac{\log \left(x + 1\right)}{3}} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}\]
   5.0
10. Taylor expanded around inf to get
    \[\frac{\color{red}{{\left({\left(1 + \frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3 - {\left({\left(\frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3}}{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + e^{\frac{\log \left(x + 1\right)}{3}} \cdot {x}^{\left(\frac{1}{3}\right)}\right)} \leadsto \frac{\color{blue}{{\left({\left(1 + \frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3 - {\left({\left(\frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3}}{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + e^{\frac{\log \left(x + 1\right)}{3}} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}\]
    5.0
11. Applied simplify to get
    \[\frac{{\left({\left(1 + \frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3 - {\left({\left(\frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3}{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + e^{\frac{\log \left(x + 1\right)}{3}} \cdot {x}^{\left(\frac{1}{3}\right)}\right)} \leadsto \frac{\left(\frac{1}{x} + 1\right) - \frac{1}{x}}{{\left(e^{\frac{\log \left(1 + x\right)}{3}}\right)}^2 + {x}^{\left(\frac{1}{3}\right)} \cdot \left({x}^{\left(\frac{1}{3}\right)} + e^{\frac{\log \left(1 + x\right)}{3}}\right)}\]
    5.0

---

12. Applied final simplification
13. Applied simplify to get
    \[\color{red}{\frac{\left(\frac{1}{x} + 1\right) - \frac{1}{x}}{{\left(e^{\frac{\log \left(1 + x\right)}{3}}\right)}^2 + {x}^{\left(\frac{1}{3}\right)} \cdot \left({x}^{\left(\frac{1}{3}\right)} + e^{\frac{\log \left(1 + x\right)}{3}}\right)}} \leadsto \color{blue}{\frac{1}{{x}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)} + \left(e^{\frac{\log \left(1 + x\right)}{3}} + {x}^{\left(\frac{1}{3}\right)}\right) \cdot e^{\frac{\log \left(1 + x\right)}{3}}}}\]
    5.0