Initial program 9.9
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
- Using strategy
rm Applied add-sqr-sqrt25.4
\[\leadsto \left(\color{blue}{\sqrt{\frac{1}{x + 1}} \cdot \sqrt{\frac{1}{x + 1}}} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
- Using strategy
rm Applied sqrt-div24.7
\[\leadsto \left(\sqrt{\frac{1}{x + 1}} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{x + 1}}} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
Applied sqrt-div26.6
\[\leadsto \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{x + 1}}} \cdot \frac{\sqrt{1}}{\sqrt{x + 1}} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
Applied frac-times24.7
\[\leadsto \left(\color{blue}{\frac{\sqrt{1} \cdot \sqrt{1}}{\sqrt{x + 1} \cdot \sqrt{x + 1}}} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
Applied frac-sub29.8
\[\leadsto \color{blue}{\frac{\left(\sqrt{1} \cdot \sqrt{1}\right) \cdot x - \left(\sqrt{x + 1} \cdot \sqrt{x + 1}\right) \cdot 2}{\left(\sqrt{x + 1} \cdot \sqrt{x + 1}\right) \cdot x}} + \frac{1}{x - 1}\]
Applied frac-add28.7
\[\leadsto \color{blue}{\frac{\left(\left(\sqrt{1} \cdot \sqrt{1}\right) \cdot x - \left(\sqrt{x + 1} \cdot \sqrt{x + 1}\right) \cdot 2\right) \cdot \left(x - 1\right) + \left(\left(\sqrt{x + 1} \cdot \sqrt{x + 1}\right) \cdot x\right) \cdot 1}{\left(\left(\sqrt{x + 1} \cdot \sqrt{x + 1}\right) \cdot x\right) \cdot \left(x - 1\right)}}\]
Simplified28.8
\[\leadsto \frac{\color{blue}{(\left(x - (x \cdot 2 + 2)_*\right) \cdot \left(x + -1\right) + \left((x \cdot x + x)_*\right))_*}}{\left(\left(\sqrt{x + 1} \cdot \sqrt{x + 1}\right) \cdot x\right) \cdot \left(x - 1\right)}\]
Simplified25.8
\[\leadsto \frac{(\left(x - (x \cdot 2 + 2)_*\right) \cdot \left(x + -1\right) + \left((x \cdot x + x)_*\right))_*}{\color{blue}{(x \cdot x + x)_* \cdot \left(x + -1\right)}}\]
Taylor expanded around -inf 0.3
\[\leadsto \frac{\color{blue}{2}}{(x \cdot x + x)_* \cdot \left(x + -1\right)}\]
Final simplification0.3
\[\leadsto \frac{2}{\left(x + -1\right) \cdot (x \cdot x + x)_*}\]
