- Started with
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
4.0
- Using strategy
rm 4.0
- Applied add-cube-cbrt to get
\[\color{red}{\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}} \leadsto \color{blue}{{\left(\sqrt[3]{\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}}\right)}^3}\]
4.3
- Using strategy
rm 4.3
- Applied frac-sub to get
\[{\left(\sqrt[3]{\color{red}{\left(\frac{1}{x + 1} - \frac{2}{x}\right)} + \frac{1}{x - 1}}\right)}^3 \leadsto {\left(\sqrt[3]{\color{blue}{\frac{1 \cdot x - \left(x + 1\right) \cdot 2}{\left(x + 1\right) \cdot x}} + \frac{1}{x - 1}}\right)}^3\]
11.6
- Applied frac-add to get
\[{\left(\sqrt[3]{\color{red}{\frac{1 \cdot x - \left(x + 1\right) \cdot 2}{\left(x + 1\right) \cdot x} + \frac{1}{x - 1}}}\right)}^3 \leadsto {\left(\sqrt[3]{\color{blue}{\frac{\left(1 \cdot x - \left(x + 1\right) \cdot 2\right) \cdot \left(x - 1\right) + \left(\left(x + 1\right) \cdot x\right) \cdot 1}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}}}\right)}^3\]
11.5
- Applied cbrt-div to get
\[{\color{red}{\left(\sqrt[3]{\frac{\left(1 \cdot x - \left(x + 1\right) \cdot 2\right) \cdot \left(x - 1\right) + \left(\left(x + 1\right) \cdot x\right) \cdot 1}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}}\right)}}^3 \leadsto {\color{blue}{\left(\frac{\sqrt[3]{\left(1 \cdot x - \left(x + 1\right) \cdot 2\right) \cdot \left(x - 1\right) + \left(\left(x + 1\right) \cdot x\right) \cdot 1}}{\sqrt[3]{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}}\right)}}^3\]
11.6
- Applied cube-div to get
\[\color{red}{{\left(\frac{\sqrt[3]{\left(1 \cdot x - \left(x + 1\right) \cdot 2\right) \cdot \left(x - 1\right) + \left(\left(x + 1\right) \cdot x\right) \cdot 1}}{\sqrt[3]{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}}\right)}^3} \leadsto \color{blue}{\frac{{\left(\sqrt[3]{\left(1 \cdot x - \left(x + 1\right) \cdot 2\right) \cdot \left(x - 1\right) + \left(\left(x + 1\right) \cdot x\right) \cdot 1}\right)}^3}{{\left(\sqrt[3]{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}\right)}^3}}\]
11.6
- Applied simplify to get
\[\frac{\color{red}{{\left(\sqrt[3]{\left(1 \cdot x - \left(x + 1\right) \cdot 2\right) \cdot \left(x - 1\right) + \left(\left(x + 1\right) \cdot x\right) \cdot 1}\right)}^3}}{{\left(\sqrt[3]{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}\right)}^3} \leadsto \frac{\color{blue}{\left(x + x \cdot x\right) + \left(\left(x - 2\right) - x \cdot 2\right) \cdot \left(x - 1\right)}}{{\left(\sqrt[3]{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}\right)}^3}\]
11.6
- Applied simplify to get
\[\frac{\left(x + x \cdot x\right) + \left(\left(x - 2\right) - x \cdot 2\right) \cdot \left(x - 1\right)}{\color{red}{{\left(\sqrt[3]{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}\right)}^3}} \leadsto \frac{\left(x + x \cdot x\right) + \left(\left(x - 2\right) - x \cdot 2\right) \cdot \left(x - 1\right)}{\color{blue}{x \cdot \left(\left(x + 1\right) \cdot \left(x - 1\right)\right)}}\]
11.3
- Applied taylor to get
\[\frac{\left(x + x \cdot x\right) + \left(\left(x - 2\right) - x \cdot 2\right) \cdot \left(x - 1\right)}{x \cdot \left(\left(x + 1\right) \cdot \left(x - 1\right)\right)} \leadsto \frac{2}{x \cdot \left(\left(x + 1\right) \cdot \left(x - 1\right)\right)}\]
0.4
- Taylor expanded around 0 to get
\[\frac{\color{red}{2}}{x \cdot \left(\left(x + 1\right) \cdot \left(x - 1\right)\right)} \leadsto \frac{\color{blue}{2}}{x \cdot \left(\left(x + 1\right) \cdot \left(x - 1\right)\right)}\]
0.4
- Applied simplify to get
\[\frac{2}{x \cdot \left(\left(x + 1\right) \cdot \left(x - 1\right)\right)} \leadsto \frac{\frac{2}{x}}{\left(x - 1\right) \cdot \left(1 + x\right)}\]
0.1
- Applied final simplification
