Initial program 9.9
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
- Using strategy
rm Applied frac-sub25.7
\[\leadsto \color{blue}{\frac{1 \cdot x - \left(x + 1\right) \cdot 2}{\left(x + 1\right) \cdot x}} + \frac{1}{x - 1}\]
Applied frac-add25.0
\[\leadsto \color{blue}{\frac{\left(1 \cdot x - \left(x + 1\right) \cdot 2\right) \cdot \left(x - 1\right) + \left(\left(x + 1\right) \cdot x\right) \cdot 1}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}}\]
Taylor expanded around inf 0.2
\[\leadsto \frac{\color{blue}{2}}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}\]
- Using strategy
rm Applied add-sqr-sqrt1.0
\[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}\]
Applied associate-/l*0.7
\[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}{\sqrt{2}}}}\]
- Using strategy
rm Applied div-inv0.8
\[\leadsto \frac{\sqrt{2}}{\color{blue}{\left(\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)\right) \cdot \frac{1}{\sqrt{2}}}}\]
Applied *-un-lft-identity0.8
\[\leadsto \frac{\sqrt{\color{blue}{1 \cdot 2}}}{\left(\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)\right) \cdot \frac{1}{\sqrt{2}}}\]
Applied sqrt-prod0.8
\[\leadsto \frac{\color{blue}{\sqrt{1} \cdot \sqrt{2}}}{\left(\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)\right) \cdot \frac{1}{\sqrt{2}}}\]
Applied times-frac1.0
\[\leadsto \color{blue}{\frac{\sqrt{1}}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)} \cdot \frac{\sqrt{2}}{\frac{1}{\sqrt{2}}}}\]
Simplified0.8
\[\leadsto \color{blue}{\frac{\frac{1}{x}}{-1 + x \cdot x}} \cdot \frac{\sqrt{2}}{\frac{1}{\sqrt{2}}}\]
Simplified0.1
\[\leadsto \frac{\frac{1}{x}}{-1 + x \cdot x} \cdot \color{blue}{2}\]
Final simplification0.1
\[\leadsto 2 \cdot \frac{\frac{1}{x}}{-1 + x \cdot x}\]
