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Derivation

1. Initial program 0.0

   \[\frac{1}{x - 1} + \frac{x}{x + 1}\]

2. Initial simplification0.0

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

4. Applied flip-+0.0

   \[\leadsto \color{blue}{\frac{\frac{1}{x - 1} \cdot \frac{1}{x - 1} - \frac{x}{x + 1} \cdot \frac{x}{x + 1}}{\frac{1}{x - 1} - \frac{x}{x + 1}}}\]
5. Using strategy rm

6. Applied add-cbrt-cube0.0

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

7. Applied add-cbrt-cube0.0

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

8. Applied cbrt-unprod0.0

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

9. Simplified0.0

   \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(\frac{1}{x + -1}\right)}^{6}}} - \frac{x}{x + 1} \cdot \frac{x}{x + 1}}{\frac{1}{x - 1} - \frac{x}{x + 1}}\]

10. Final simplification0.0

    \[\leadsto \frac{\sqrt[3]{{\left(\frac{1}{x + -1}\right)}^{6}} - \frac{x}{x + 1} \cdot \frac{x}{x + 1}}{\frac{1}{x - 1} - \frac{x}{x + 1}}\]

Runtime

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herbie shell --seed 2018255 +o rules:numerics
(FPCore (x)
  :name "Asymptote B"
  (+ (/ 1 (- x 1)) (/ x (+ x 1))))