\(\frac{1}{\left({x}^{\left(\frac{1}{3}\right)} + e^{\frac{\log \left(1 + x\right)}{3}}\right) \cdot e^{\frac{\log \left(1 + x\right)}{3}} + {\left({x}^{\left(\frac{1}{3}\right)}\right)}^2}\)
- Started with
\[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
30.4
- Using strategy
rm 30.4
- Applied add-exp-log to get
\[{\color{red}{\left(x + 1\right)}}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)} \leadsto {\color{blue}{\left(e^{\log \left(x + 1\right)}\right)}}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
30.7
- Applied pow-exp to get
\[\color{red}{{\left(e^{\log \left(x + 1\right)}\right)}^{\left(\frac{1}{3}\right)}} - {x}^{\left(\frac{1}{3}\right)} \leadsto \color{blue}{e^{\log \left(x + 1\right) \cdot \frac{1}{3}}} - {x}^{\left(\frac{1}{3}\right)}\]
30.7
- Applied simplify to get
\[e^{\color{red}{\log \left(x + 1\right) \cdot \frac{1}{3}}} - {x}^{\left(\frac{1}{3}\right)} \leadsto e^{\color{blue}{\frac{\log \left(x + 1\right)}{3}}} - {x}^{\left(\frac{1}{3}\right)}\]
30.4
- Using strategy
rm 30.4
- Applied flip3-- to get
\[\color{red}{e^{\frac{\log \left(x + 1\right)}{3}} - {x}^{\left(\frac{1}{3}\right)}} \leadsto \color{blue}{\frac{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^{3} - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^{3}}{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + e^{\frac{\log \left(x + 1\right)}{3}} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}}\]
30.4
- Applied simplify to get
\[\frac{\color{red}{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^{3} - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^{3}}}{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + e^{\frac{\log \left(x + 1\right)}{3}} \cdot {x}^{\left(\frac{1}{3}\right)}\right)} \leadsto \frac{\color{blue}{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}}{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + e^{\frac{\log \left(x + 1\right)}{3}} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}\]
30.4
- Applied taylor to get
\[\frac{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + e^{\frac{\log \left(x + 1\right)}{3}} \cdot {x}^{\left(\frac{1}{3}\right)}\right)} \leadsto \frac{{\left({\left(1 + x\right)}^{\frac{1}{3}}\right)}^3 - {\left({x}^{\frac{1}{3}}\right)}^3}{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + e^{\frac{\log \left(x + 1\right)}{3}} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}\]
30.3
- Taylor expanded around 0 to get
\[\frac{\color{red}{{\left({\left(1 + x\right)}^{\frac{1}{3}}\right)}^3 - {\left({x}^{\frac{1}{3}}\right)}^3}}{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + e^{\frac{\log \left(x + 1\right)}{3}} \cdot {x}^{\left(\frac{1}{3}\right)}\right)} \leadsto \frac{\color{blue}{{\left({\left(1 + x\right)}^{\frac{1}{3}}\right)}^3 - {\left({x}^{\frac{1}{3}}\right)}^3}}{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + e^{\frac{\log \left(x + 1\right)}{3}} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}\]
30.3
- Applied simplify to get
\[\frac{{\left({\left(1 + x\right)}^{\frac{1}{3}}\right)}^3 - {\left({x}^{\frac{1}{3}}\right)}^3}{{\left(e^{\frac{\log \left(x + 1\right)}{3}}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + e^{\frac{\log \left(x + 1\right)}{3}} \cdot {x}^{\left(\frac{1}{3}\right)}\right)} \leadsto \frac{1 + \left(x - x\right)}{{x}^{\left(\frac{1}{3}\right)} \cdot \left(e^{\frac{\log \left(1 + x\right)}{3}} + {x}^{\left(\frac{1}{3}\right)}\right) + {\left(e^{\frac{\log \left(1 + x\right)}{3}}\right)}^2}\]
2.6
- Applied final simplification
- Applied simplify to get
\[\color{red}{\frac{1 + \left(x - x\right)}{{x}^{\left(\frac{1}{3}\right)} \cdot \left(e^{\frac{\log \left(1 + x\right)}{3}} + {x}^{\left(\frac{1}{3}\right)}\right) + {\left(e^{\frac{\log \left(1 + x\right)}{3}}\right)}^2}} \leadsto \color{blue}{\frac{1}{\left({x}^{\left(\frac{1}{3}\right)} + e^{\frac{\log \left(1 + x\right)}{3}}\right) \cdot e^{\frac{\log \left(1 + x\right)}{3}} + {\left({x}^{\left(\frac{1}{3}\right)}\right)}^2}}\]
2.6
- Removed slow pow expressions
