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Derivation

1. Initial program 29.3

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

3. Applied add-cbrt-cube29.4

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

5. Applied add-exp-log29.4

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

7. Applied flip3--29.3

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

8. Applied log-div29.3

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

9. Simplified29.3

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

11. Applied diff-log29.3

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

12. Applied rem-exp-log29.3

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

13. Simplified0.6

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

14. Final simplification0.6

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

Runtime

Time bar (total: 8.6m)Debug logProfile

herbie shell --seed 2018290 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))