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Derivation

1. Initial program 29.3

   \[\sqrt[3]{x + 1} - \sqrt[3]{x}\]

2. Initial simplification29.3

   \[\leadsto \sqrt[3]{1 + x} - \sqrt[3]{x}\]
3. Using strategy rm

4. Applied add-cbrt-cube29.4

   \[\leadsto \sqrt[3]{1 + x} - \color{blue}{\sqrt[3]{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}\]
5. Using strategy rm

6. Applied add-cube-cbrt29.4

   \[\leadsto \sqrt[3]{1 + x} - \sqrt[3]{\left(\sqrt[3]{x} \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\right)}\right) \cdot \sqrt[3]{x}}\]
7. Using strategy rm

8. Applied add-exp-log29.4

   \[\leadsto \color{blue}{e^{\log \left(\sqrt[3]{1 + x} - \sqrt[3]{\left(\sqrt[3]{x} \cdot \left(\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right) \cdot \sqrt[3]{x}}\right)}}\]
9. Using strategy rm

10. Applied flip3--29.3

    \[\leadsto e^{\log \color{blue}{\left(\frac{{\left(\sqrt[3]{1 + x}\right)}^{3} - {\left(\sqrt[3]{\left(\sqrt[3]{x} \cdot \left(\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right) \cdot \sqrt[3]{x}}\right)}^{3}}{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x} + \left(\sqrt[3]{\left(\sqrt[3]{x} \cdot \left(\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right) \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\left(\sqrt[3]{x} \cdot \left(\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right) \cdot \sqrt[3]{x}} + \sqrt[3]{1 + x} \cdot \sqrt[3]{\left(\sqrt[3]{x} \cdot \left(\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right) \cdot \sqrt[3]{x}}\right)}\right)}}\]

11. Applied log-div29.3

    \[\leadsto e^{\color{blue}{\log \left({\left(\sqrt[3]{1 + x}\right)}^{3} - {\left(\sqrt[3]{\left(\sqrt[3]{x} \cdot \left(\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right) \cdot \sqrt[3]{x}}\right)}^{3}\right) - \log \left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x} + \left(\sqrt[3]{\left(\sqrt[3]{x} \cdot \left(\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right) \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\left(\sqrt[3]{x} \cdot \left(\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right) \cdot \sqrt[3]{x}} + \sqrt[3]{1 + x} \cdot \sqrt[3]{\left(\sqrt[3]{x} \cdot \left(\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right) \cdot \sqrt[3]{x}}\right)\right)}}\]

12. Applied exp-diff29.3

    \[\leadsto \color{blue}{\frac{e^{\log \left({\left(\sqrt[3]{1 + x}\right)}^{3} - {\left(\sqrt[3]{\left(\sqrt[3]{x} \cdot \left(\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right) \cdot \sqrt[3]{x}}\right)}^{3}\right)}}{e^{\log \left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x} + \left(\sqrt[3]{\left(\sqrt[3]{x} \cdot \left(\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right) \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\left(\sqrt[3]{x} \cdot \left(\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right) \cdot \sqrt[3]{x}} + \sqrt[3]{1 + x} \cdot \sqrt[3]{\left(\sqrt[3]{x} \cdot \left(\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right) \cdot \sqrt[3]{x}}\right)\right)}}}\]

13. Simplified2.6

    \[\leadsto \frac{\color{blue}{1}}{e^{\log \left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x} + \left(\sqrt[3]{\left(\sqrt[3]{x} \cdot \left(\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right) \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\left(\sqrt[3]{x} \cdot \left(\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right) \cdot \sqrt[3]{x}} + \sqrt[3]{1 + x} \cdot \sqrt[3]{\left(\sqrt[3]{x} \cdot \left(\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right) \cdot \sqrt[3]{x}}\right)\right)}}\]

14. Final simplification2.6

    \[\leadsto \frac{1}{e^{\log \left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{\left(\left(\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\left(\left(\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}} + \sqrt[3]{x + 1} \cdot \sqrt[3]{\left(\left(\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}\right)\right)}}\]

Runtime

Time bar (total: 16.1s)Debug logProfile

herbie shell --seed 2018340 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))