\(\left(\left(\frac{1}{4} + \left(\frac{1}{2} \cdot N\right) \cdot \log N\right) + \left(\frac{\frac{\frac{1}{8}}{N}}{\log N} + \frac{\frac{1}{8}}{N} \cdot \log N\right)\right) - \left(\left(\frac{1}{4} \cdot \log N + \left(\frac{\frac{1}{8}}{N} + 1\right)\right) + \frac{\left(\frac{\frac{1}{2}}{N} + \frac{1}{2}\right) \cdot \log \left(\frac{1}{N} + 1\right)}{\frac{\frac{\log N \cdot N}{\frac{1}{N} + 1}}{\log \left(\frac{1}{N} + 1\right)}}\right)\)
- Started with
\[\left(\left(N + 1\right) \cdot \log \left(N + 1\right) - N \cdot \log N\right) - 1\]
63.3
- Using strategy
rm 63.3
- Applied add-cube-cbrt to get
\[\left(\left(N + 1\right) \cdot \color{red}{\log \left(N + 1\right)} - N \cdot \log N\right) - 1 \leadsto \left(\left(N + 1\right) \cdot \color{blue}{{\left(\sqrt[3]{\log \left(N + 1\right)}\right)}^3} - N \cdot \log N\right) - 1\]
60.8
- Applied add-cube-cbrt to get
\[\left(\color{red}{\left(N + 1\right)} \cdot {\left(\sqrt[3]{\log \left(N + 1\right)}\right)}^3 - N \cdot \log N\right) - 1 \leadsto \left(\color{blue}{{\left(\sqrt[3]{N + 1}\right)}^3} \cdot {\left(\sqrt[3]{\log \left(N + 1\right)}\right)}^3 - N \cdot \log N\right) - 1\]
60.2
- Applied cube-unprod to get
\[\left(\color{red}{{\left(\sqrt[3]{N + 1}\right)}^3 \cdot {\left(\sqrt[3]{\log \left(N + 1\right)}\right)}^3} - N \cdot \log N\right) - 1 \leadsto \left(\color{blue}{{\left(\sqrt[3]{N + 1} \cdot \sqrt[3]{\log \left(N + 1\right)}\right)}^3} - N \cdot \log N\right) - 1\]
60.2
- Using strategy
rm 60.2
- Applied flip-- to get
\[\color{red}{\left({\left(\sqrt[3]{N + 1} \cdot \sqrt[3]{\log \left(N + 1\right)}\right)}^3 - N \cdot \log N\right)} - 1 \leadsto \color{blue}{\frac{{\left({\left(\sqrt[3]{N + 1} \cdot \sqrt[3]{\log \left(N + 1\right)}\right)}^3\right)}^2 - {\left(N \cdot \log N\right)}^2}{{\left(\sqrt[3]{N + 1} \cdot \sqrt[3]{\log \left(N + 1\right)}\right)}^3 + N \cdot \log N}} - 1\]
59.7
- Applied simplify to get
\[\frac{{\left({\left(\sqrt[3]{N + 1} \cdot \sqrt[3]{\log \left(N + 1\right)}\right)}^3\right)}^2 - {\left(N \cdot \log N\right)}^2}{\color{red}{{\left(\sqrt[3]{N + 1} \cdot \sqrt[3]{\log \left(N + 1\right)}\right)}^3 + N \cdot \log N}} - 1 \leadsto \frac{{\left({\left(\sqrt[3]{N + 1} \cdot \sqrt[3]{\log \left(N + 1\right)}\right)}^3\right)}^2 - {\left(N \cdot \log N\right)}^2}{\color{blue}{N \cdot \log N + \left(N + 1\right) \cdot \log \left(N + 1\right)}} - 1\]
59.7
- Applied taylor to get
\[\frac{{\left({\left(\sqrt[3]{N + 1} \cdot \sqrt[3]{\log \left(N + 1\right)}\right)}^3\right)}^2 - {\left(N \cdot \log N\right)}^2}{N \cdot \log N + \left(N + 1\right) \cdot \log \left(N + 1\right)} - 1 \leadsto \left(\left(\frac{1}{2} \cdot \left(\log N \cdot N\right) + \left(\frac{1}{8} \cdot \frac{1}{\log N \cdot N} + \left(\frac{1}{4} + \frac{1}{8} \cdot \frac{\log N}{N}\right)\right)\right) - \left(\frac{1}{2} \cdot \frac{{\left({\left(\sqrt[3]{\log \left(1 + \frac{1}{N}\right)} \cdot \sqrt[3]{1 + \frac{1}{N}}\right)}^3\right)}^2}{\log N \cdot N} + \left(\frac{1}{4} \cdot \log N + \frac{1}{8} \cdot \frac{1}{N}\right)\right)\right) - 1\]
0.3
- Taylor expanded around inf to get
\[\color{red}{\left(\left(\frac{1}{2} \cdot \left(\log N \cdot N\right) + \left(\frac{1}{8} \cdot \frac{1}{\log N \cdot N} + \left(\frac{1}{4} + \frac{1}{8} \cdot \frac{\log N}{N}\right)\right)\right) - \left(\frac{1}{2} \cdot \frac{{\left({\left(\sqrt[3]{\log \left(1 + \frac{1}{N}\right)} \cdot \sqrt[3]{1 + \frac{1}{N}}\right)}^3\right)}^2}{\log N \cdot N} + \left(\frac{1}{4} \cdot \log N + \frac{1}{8} \cdot \frac{1}{N}\right)\right)\right)} - 1 \leadsto \color{blue}{\left(\left(\frac{1}{2} \cdot \left(\log N \cdot N\right) + \left(\frac{1}{8} \cdot \frac{1}{\log N \cdot N} + \left(\frac{1}{4} + \frac{1}{8} \cdot \frac{\log N}{N}\right)\right)\right) - \left(\frac{1}{2} \cdot \frac{{\left({\left(\sqrt[3]{\log \left(1 + \frac{1}{N}\right)} \cdot \sqrt[3]{1 + \frac{1}{N}}\right)}^3\right)}^2}{\log N \cdot N} + \left(\frac{1}{4} \cdot \log N + \frac{1}{8} \cdot \frac{1}{N}\right)\right)\right)} - 1\]
0.3
- Applied simplify to get
\[\left(\left(\frac{1}{2} \cdot \left(\log N \cdot N\right) + \left(\frac{1}{8} \cdot \frac{1}{\log N \cdot N} + \left(\frac{1}{4} + \frac{1}{8} \cdot \frac{\log N}{N}\right)\right)\right) - \left(\frac{1}{2} \cdot \frac{{\left({\left(\sqrt[3]{\log \left(1 + \frac{1}{N}\right)} \cdot \sqrt[3]{1 + \frac{1}{N}}\right)}^3\right)}^2}{\log N \cdot N} + \left(\frac{1}{4} \cdot \log N + \frac{1}{8} \cdot \frac{1}{N}\right)\right)\right) - 1 \leadsto \left(\left(\left(\frac{\log N}{\frac{N}{\frac{1}{8}}} + \frac{1}{4}\right) + \left(\left(\frac{1}{2} \cdot N\right) \cdot \log N + \frac{\frac{\frac{1}{8}}{N}}{\log N}\right)\right) - \frac{\left(\frac{1}{2} \cdot \left(\frac{1}{N} + 1\right)\right) \cdot \log \left(\frac{1}{N} + 1\right)}{\frac{\frac{N \cdot \log N}{\frac{1}{N} + 1}}{\log \left(\frac{1}{N} + 1\right)}}\right) - \left(\frac{1}{4} \cdot \log N + \left(\frac{\frac{1}{8}}{N} + 1\right)\right)\]
0.3
- Applied final simplification
- Applied simplify to get
\[\color{red}{\left(\left(\left(\frac{\log N}{\frac{N}{\frac{1}{8}}} + \frac{1}{4}\right) + \left(\left(\frac{1}{2} \cdot N\right) \cdot \log N + \frac{\frac{\frac{1}{8}}{N}}{\log N}\right)\right) - \frac{\left(\frac{1}{2} \cdot \left(\frac{1}{N} + 1\right)\right) \cdot \log \left(\frac{1}{N} + 1\right)}{\frac{\frac{N \cdot \log N}{\frac{1}{N} + 1}}{\log \left(\frac{1}{N} + 1\right)}}\right) - \left(\frac{1}{4} \cdot \log N + \left(\frac{\frac{1}{8}}{N} + 1\right)\right)} \leadsto \color{blue}{\left(\left(\frac{1}{4} + \left(\frac{1}{2} \cdot N\right) \cdot \log N\right) + \left(\frac{\frac{\frac{1}{8}}{N}}{\log N} + \frac{\frac{1}{8}}{N} \cdot \log N\right)\right) - \left(\left(\frac{1}{4} \cdot \log N + \left(\frac{\frac{1}{8}}{N} + 1\right)\right) + \frac{\left(\frac{\frac{1}{2}}{N} + \frac{1}{2}\right) \cdot \log \left(\frac{1}{N} + 1\right)}{\frac{\frac{\log N \cdot N}{\frac{1}{N} + 1}}{\log \left(\frac{1}{N} + 1\right)}}\right)}\]
0.3
