\(\frac{1}{(\left({x}^{\left(\frac{1}{3}\right)}\right) * \left({x}^{\left(\frac{1}{3}\right)} + {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right) + \left({\left(1 + x\right)}^{\left(\frac{1}{3}\right)} \cdot {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right))_*}\)
- Started with
\[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
13.1
- Using strategy
rm 13.1
- Applied add-exp-log to get
\[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}} \leadsto \color{blue}{e^{\log \left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\right)}}\]
13.1
- Using strategy
rm 13.1
- Applied flip3-- to get
\[e^{\log \color{red}{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\right)}} \leadsto e^{\log \color{blue}{\left(\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^{3}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}\right)}}\]
12.9
- Applied log-div to get
\[e^{\color{red}{\log \left(\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^{3}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}\right)}} \leadsto e^{\color{blue}{\log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^{3}\right) - \log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)\right)}}\]
13.0
- Applied exp-diff to get
\[\color{red}{e^{\log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^{3}\right) - \log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)\right)}} \leadsto \color{blue}{\frac{e^{\log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^{3}\right)}}{e^{\log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)\right)}}}\]
13.0
- Applied simplify to get
\[\frac{\color{red}{e^{\log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^{3}\right)}}}{e^{\log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)\right)}} \leadsto \frac{\color{blue}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}}{e^{\log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)\right)}}\]
13.2
- Applied taylor to get
\[\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}{e^{\log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)\right)}} \leadsto \frac{{\left({\left(1 + x\right)}^{\frac{1}{3}}\right)}^3 - {\left({x}^{\frac{1}{3}}\right)}^3}{e^{\log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)\right)}}\]
13.2
- Taylor expanded around 0 to get
\[\frac{\color{red}{{\left({\left(1 + x\right)}^{\frac{1}{3}}\right)}^3 - {\left({x}^{\frac{1}{3}}\right)}^3}}{e^{\log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)\right)}} \leadsto \frac{\color{blue}{{\left({\left(1 + x\right)}^{\frac{1}{3}}\right)}^3 - {\left({x}^{\frac{1}{3}}\right)}^3}}{e^{\log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)\right)}}\]
13.2
- Applied simplify to get
\[\frac{{\left({\left(1 + x\right)}^{\frac{1}{3}}\right)}^3 - {\left({x}^{\frac{1}{3}}\right)}^3}{e^{\log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)\right)}} \leadsto \frac{1 + \left(x - x\right)}{(\left({x}^{\left(\frac{1}{3}\right)}\right) * \left({\left(1 + x\right)}^{\left(\frac{1}{3}\right)} + {x}^{\left(\frac{1}{3}\right)}\right) + \left({\left(1 + x\right)}^{\left(\frac{1}{3}\right)} \cdot {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right))_*}\]
1.8
- Applied final simplification
- Applied simplify to get
\[\color{red}{\frac{1 + \left(x - x\right)}{(\left({x}^{\left(\frac{1}{3}\right)}\right) * \left({\left(1 + x\right)}^{\left(\frac{1}{3}\right)} + {x}^{\left(\frac{1}{3}\right)}\right) + \left({\left(1 + x\right)}^{\left(\frac{1}{3}\right)} \cdot {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right))_*}} \leadsto \color{blue}{\frac{1}{(\left({x}^{\left(\frac{1}{3}\right)}\right) * \left({x}^{\left(\frac{1}{3}\right)} + {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right) + \left({\left(1 + x\right)}^{\left(\frac{1}{3}\right)} \cdot {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right))_*}}\]
1.8
