Initial program 16.0
\[\frac{x}{x \cdot x + 1}\]
Initial simplification16.0
\[\leadsto \frac{x}{(x \cdot x + 1)_*}\]
Taylor expanded around 0 15.5
\[\leadsto \color{blue}{\left(x + {x}^{5}\right) - {x}^{3}}\]
- Using strategy
rm Applied *-un-lft-identity15.5
\[\leadsto \left(x + {x}^{5}\right) - \color{blue}{1 \cdot {x}^{3}}\]
Applied *-un-lft-identity15.5
\[\leadsto \color{blue}{1 \cdot \left(x + {x}^{5}\right)} - 1 \cdot {x}^{3}\]
Applied prod-diff15.5
\[\leadsto \color{blue}{(1 \cdot \left(x + {x}^{5}\right) + \left(-{x}^{3} \cdot 1\right))_* + (\left(-{x}^{3}\right) \cdot 1 + \left({x}^{3} \cdot 1\right))_*}\]
Simplified15.5
\[\leadsto \color{blue}{\left((\left(-x\right) \cdot \left(x \cdot x\right) + \left({x}^{5}\right))_* + x\right)} + (\left(-{x}^{3}\right) \cdot 1 + \left({x}^{3} \cdot 1\right))_*\]
Simplified15.5
\[\leadsto \left((\left(-x\right) \cdot \left(x \cdot x\right) + \left({x}^{5}\right))_* + x\right) + \color{blue}{0}\]
- Using strategy
rm Applied add-cube-cbrt15.5
\[\leadsto \left(\color{blue}{\left(\sqrt[3]{(\left(-x\right) \cdot \left(x \cdot x\right) + \left({x}^{5}\right))_*} \cdot \sqrt[3]{(\left(-x\right) \cdot \left(x \cdot x\right) + \left({x}^{5}\right))_*}\right) \cdot \sqrt[3]{(\left(-x\right) \cdot \left(x \cdot x\right) + \left({x}^{5}\right))_*}} + x\right) + 0\]
Final simplification15.5
\[\leadsto x + \left(\sqrt[3]{(\left(-x\right) \cdot \left(x \cdot x\right) + \left({x}^{5}\right))_*} \cdot \sqrt[3]{(\left(-x\right) \cdot \left(x \cdot x\right) + \left({x}^{5}\right))_*}\right) \cdot \sqrt[3]{(\left(-x\right) \cdot \left(x \cdot x\right) + \left({x}^{5}\right))_*}\]
