\(\frac{1}{{\left(1 + x\right)}^{\left(\frac{1}{3}\right)} \cdot \left(e^{\frac{\log x}{3}} + {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right) + e^{\frac{\log x}{3}} \cdot e^{\frac{\log x}{3}}}\)
- 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 pow-to-exp to get
\[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - \color{red}{{x}^{\left(\frac{1}{3}\right)}} \leadsto {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - \color{blue}{e^{\log x \cdot \frac{1}{3}}}\]
13.4
- Applied simplify to get
\[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - e^{\color{red}{\log x \cdot \frac{1}{3}}} \leadsto {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - e^{\color{blue}{\frac{\log x}{3}}}\]
13.1
- Using strategy
rm 13.1
- Applied add-exp-log to get
\[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - e^{\frac{\log x}{3}}} \leadsto \color{blue}{e^{\log \left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - e^{\frac{\log x}{3}}\right)}}\]
13.1
- Using strategy
rm 13.1
- Applied flip3-- to get
\[e^{\log \color{red}{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - e^{\frac{\log x}{3}}\right)}} \leadsto e^{\log \color{blue}{\left(\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^{3} - {\left(e^{\frac{\log x}{3}}\right)}^{3}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left(e^{\frac{\log x}{3}}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot e^{\frac{\log x}{3}}\right)}\right)}}\]
13.1
- Applied log-div to get
\[e^{\color{red}{\log \left(\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^{3} - {\left(e^{\frac{\log x}{3}}\right)}^{3}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left(e^{\frac{\log x}{3}}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot e^{\frac{\log x}{3}}\right)}\right)}} \leadsto e^{\color{blue}{\log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^{3} - {\left(e^{\frac{\log x}{3}}\right)}^{3}\right) - \log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left(e^{\frac{\log x}{3}}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot e^{\frac{\log x}{3}}\right)\right)}}\]
13.2
- Applied simplify to get
\[e^{\color{red}{\log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^{3} - {\left(e^{\frac{\log x}{3}}\right)}^{3}\right)} - \log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left(e^{\frac{\log x}{3}}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot e^{\frac{\log x}{3}}\right)\right)} \leadsto e^{\color{blue}{\log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3 - {\left(e^{\frac{\log x}{3}}\right)}^3\right)} - \log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left(e^{\frac{\log x}{3}}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot e^{\frac{\log x}{3}}\right)\right)}\]
13.2
- Applied taylor to get
\[e^{\log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3 - {\left(e^{\frac{\log x}{3}}\right)}^3\right) - \log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left(e^{\frac{\log x}{3}}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot e^{\frac{\log x}{3}}\right)\right)} \leadsto e^{\log \left({\left({\left(1 + \frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3 - {\left({\left(\frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3\right) - \log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left(e^{\frac{\log x}{3}}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot e^{\frac{\log x}{3}}\right)\right)}\]
14.8
- Taylor expanded around inf to get
\[e^{\log \color{red}{\left({\left({\left(1 + \frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3 - {\left({\left(\frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3\right)} - \log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left(e^{\frac{\log x}{3}}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot e^{\frac{\log x}{3}}\right)\right)} \leadsto e^{\log \color{blue}{\left({\left({\left(1 + \frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3 - {\left({\left(\frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3\right)} - \log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left(e^{\frac{\log x}{3}}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot e^{\frac{\log x}{3}}\right)\right)}\]
14.8
- Applied simplify to get
\[e^{\log \left({\left({\left(1 + \frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3 - {\left({\left(\frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3\right) - \log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left(e^{\frac{\log x}{3}}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot e^{\frac{\log x}{3}}\right)\right)} \leadsto \frac{\left(\frac{1}{x} + 1\right) - \frac{1}{x}}{e^{\frac{\log x}{3}} \cdot \left({\left(1 + x\right)}^{\left(\frac{1}{3}\right)} + e^{\frac{\log x}{3}}\right) + {\left({\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right)}^2}\]
13.5
- Applied final simplification
- Applied simplify to get
\[\color{red}{\frac{\left(\frac{1}{x} + 1\right) - \frac{1}{x}}{e^{\frac{\log x}{3}} \cdot \left({\left(1 + x\right)}^{\left(\frac{1}{3}\right)} + e^{\frac{\log x}{3}}\right) + {\left({\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right)}^2}} \leadsto \color{blue}{\frac{1}{{\left(1 + x\right)}^{\left(\frac{1}{3}\right)} \cdot \left(e^{\frac{\log x}{3}} + {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right) + e^{\frac{\log x}{3}} \cdot e^{\frac{\log x}{3}}}}\]
1.3
- Removed slow pow expressions
