\(\left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{8} \cdot x\right) + \left(\frac{1}{48} \cdot x\right) \cdot x\right)\right) \cdot \left(\sqrt{e^{x}} + 1\right)\)
- Started with
\[e^{x} - 1\]
25.9
- Using strategy
rm 25.9
- Applied add-sqr-sqrt to get
\[\color{red}{e^{x}} - 1 \leadsto \color{blue}{{\left(\sqrt{e^{x}}\right)}^2} - 1\]
26.1
- Applied difference-of-sqr-1 to get
\[\color{red}{{\left(\sqrt{e^{x}}\right)}^2 - 1} \leadsto \color{blue}{\left(\sqrt{e^{x}} + 1\right) \cdot \left(\sqrt{e^{x}} - 1\right)}\]
26.1
- Applied taylor to get
\[\left(\sqrt{e^{x}} + 1\right) \cdot \left(\sqrt{e^{x}} - 1\right) \leadsto \left(\sqrt{e^{x}} + 1\right) \cdot \left(\frac{1}{8} \cdot {x}^2 + \left(\frac{1}{48} \cdot {x}^{3} + \frac{1}{2} \cdot x\right)\right)\]
0.2
- Taylor expanded around 0 to get
\[\left(\sqrt{e^{x}} + 1\right) \cdot \color{red}{\left(\frac{1}{8} \cdot {x}^2 + \left(\frac{1}{48} \cdot {x}^{3} + \frac{1}{2} \cdot x\right)\right)} \leadsto \left(\sqrt{e^{x}} + 1\right) \cdot \color{blue}{\left(\frac{1}{8} \cdot {x}^2 + \left(\frac{1}{48} \cdot {x}^{3} + \frac{1}{2} \cdot x\right)\right)}\]
0.2
- Applied simplify to get
\[\left(\sqrt{e^{x}} + 1\right) \cdot \left(\frac{1}{8} \cdot {x}^2 + \left(\frac{1}{48} \cdot {x}^{3} + \frac{1}{2} \cdot x\right)\right) \leadsto \left(\left(x \cdot \frac{1}{8}\right) \cdot x + \left(x \cdot \frac{1}{2} + {x}^3 \cdot \frac{1}{48}\right)\right) \cdot \left(\sqrt{e^{x}} + 1\right)\]
0.2
- Applied final simplification
- Applied simplify to get
\[\color{red}{\left(\left(x \cdot \frac{1}{8}\right) \cdot x + \left(x \cdot \frac{1}{2} + {x}^3 \cdot \frac{1}{48}\right)\right) \cdot \left(\sqrt{e^{x}} + 1\right)} \leadsto \color{blue}{\left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{8} \cdot x\right) + \left(\frac{1}{48} \cdot x\right) \cdot x\right)\right) \cdot \left(\sqrt{e^{x}} + 1\right)}\]
0.2
- Removed slow pow expressions
