\(\frac{\frac{4 - (\left(x \cdot 4\right) * x + \left(x \cdot 4\right))_*}{{\left((x * x + x)_*\right)}^2 \cdot {\left(x - 1\right)}^2}}{\left(\frac{1}{1 + x} - \frac{2}{x}\right) - \frac{1}{x - 1}}\)
- Started with
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
4.0
- Using strategy
rm 4.0
- Applied flip-+ to get
\[\color{red}{\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}} \leadsto \color{blue}{\frac{{\left(\frac{1}{x + 1} - \frac{2}{x}\right)}^2 - {\left(\frac{1}{x - 1}\right)}^2}{\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}}}\]
11.3
- Using strategy
rm 11.3
- Applied square-div to get
\[\frac{{\left(\frac{1}{x + 1} - \frac{2}{x}\right)}^2 - \color{red}{{\left(\frac{1}{x - 1}\right)}^2}}{\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}} \leadsto \frac{{\left(\frac{1}{x + 1} - \frac{2}{x}\right)}^2 - \color{blue}{\frac{{1}^2}{{\left(x - 1\right)}^2}}}{\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}}\]
13.6
- Applied frac-sub to get
\[\frac{{\color{red}{\left(\frac{1}{x + 1} - \frac{2}{x}\right)}}^2 - \frac{{1}^2}{{\left(x - 1\right)}^2}}{\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}} \leadsto \frac{{\color{blue}{\left(\frac{1 \cdot x - \left(x + 1\right) \cdot 2}{\left(x + 1\right) \cdot x}\right)}}^2 - \frac{{1}^2}{{\left(x - 1\right)}^2}}{\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}}\]
12.6
- Applied square-div to get
\[\frac{\color{red}{{\left(\frac{1 \cdot x - \left(x + 1\right) \cdot 2}{\left(x + 1\right) \cdot x}\right)}^2} - \frac{{1}^2}{{\left(x - 1\right)}^2}}{\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}} \leadsto \frac{\color{blue}{\frac{{\left(1 \cdot x - \left(x + 1\right) \cdot 2\right)}^2}{{\left(\left(x + 1\right) \cdot x\right)}^2}} - \frac{{1}^2}{{\left(x - 1\right)}^2}}{\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}}\]
21.2
- Applied frac-sub to get
\[\frac{\color{red}{\frac{{\left(1 \cdot x - \left(x + 1\right) \cdot 2\right)}^2}{{\left(\left(x + 1\right) \cdot x\right)}^2} - \frac{{1}^2}{{\left(x - 1\right)}^2}}}{\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}} \leadsto \frac{\color{blue}{\frac{{\left(1 \cdot x - \left(x + 1\right) \cdot 2\right)}^2 \cdot {\left(x - 1\right)}^2 - {\left(\left(x + 1\right) \cdot x\right)}^2 \cdot {1}^2}{{\left(\left(x + 1\right) \cdot x\right)}^2 \cdot {\left(x - 1\right)}^2}}}{\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}}\]
21.5
- Applied taylor to get
\[\frac{\frac{{\left(1 \cdot x - \left(x + 1\right) \cdot 2\right)}^2 \cdot {\left(x - 1\right)}^2 - {\left(\left(x + 1\right) \cdot x\right)}^2 \cdot {1}^2}{{\left(\left(x + 1\right) \cdot x\right)}^2 \cdot {\left(x - 1\right)}^2}}{\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}} \leadsto \frac{\frac{4 - \left(4 \cdot {x}^2 + 4 \cdot x\right)}{{\left(\left(x + 1\right) \cdot x\right)}^2 \cdot {\left(x - 1\right)}^2}}{\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}}\]
17.7
- Taylor expanded around 0 to get
\[\frac{\frac{\color{red}{4 - \left(4 \cdot {x}^2 + 4 \cdot x\right)}}{{\left(\left(x + 1\right) \cdot x\right)}^2 \cdot {\left(x - 1\right)}^2}}{\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}} \leadsto \frac{\frac{\color{blue}{4 - \left(4 \cdot {x}^2 + 4 \cdot x\right)}}{{\left(\left(x + 1\right) \cdot x\right)}^2 \cdot {\left(x - 1\right)}^2}}{\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}}\]
17.7
- Applied simplify to get
\[\frac{\frac{4 - \left(4 \cdot {x}^2 + 4 \cdot x\right)}{{\left(\left(x + 1\right) \cdot x\right)}^2 \cdot {\left(x - 1\right)}^2}}{\left(\frac{1}{x + 1} - \frac{2}{x}\right) - \frac{1}{x - 1}} \leadsto \frac{\frac{4 - (\left(x \cdot 4\right) * x + \left(x \cdot 4\right))_*}{{\left((x * x + x)_*\right)}^2 \cdot {\left(x - 1\right)}^2}}{\left(\frac{1}{1 + x} - \frac{2}{x}\right) - \frac{1}{x - 1}}\]
8.1
- Applied final simplification
