Initial program 42.1
\[\frac{e^{x}}{e^{x} - 1}\]
Initial simplification42.1
\[\leadsto \frac{e^{x}}{e^{x} - 1}\]
Taylor expanded around 0 51.8
\[\leadsto \frac{e^{x}}{\color{blue}{x + \left(\frac{1}{6} \cdot {x}^{3} + \frac{1}{2} \cdot {x}^{2}\right)}}\]
Simplified41.5
\[\leadsto \frac{e^{x}}{\color{blue}{x + \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(x \cdot x\right)}}\]
- Using strategy
rm Applied add-cube-cbrt41.5
\[\leadsto \frac{e^{x}}{\color{blue}{\left(\sqrt[3]{x + \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(x \cdot x\right)} \cdot \sqrt[3]{x + \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(x \cdot x\right)}\right) \cdot \sqrt[3]{x + \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(x \cdot x\right)}}}\]
Applied *-un-lft-identity41.5
\[\leadsto \frac{\color{blue}{1 \cdot e^{x}}}{\left(\sqrt[3]{x + \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(x \cdot x\right)} \cdot \sqrt[3]{x + \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(x \cdot x\right)}\right) \cdot \sqrt[3]{x + \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(x \cdot x\right)}}\]
Applied times-frac41.5
\[\leadsto \color{blue}{\frac{1}{\sqrt[3]{x + \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(x \cdot x\right)} \cdot \sqrt[3]{x + \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(x \cdot x\right)}} \cdot \frac{e^{x}}{\sqrt[3]{x + \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(x \cdot x\right)}}}\]
Taylor expanded around inf 61.8
\[\leadsto \frac{1}{\sqrt[3]{x + \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(x \cdot x\right)} \cdot \color{blue}{\left(\frac{e^{\frac{1}{3} \cdot \left(\log \frac{1}{6} - 3 \cdot \log \left(\frac{1}{x}\right)\right)}}{x} + \left(e^{\frac{1}{3} \cdot \left(\log \frac{1}{6} - 3 \cdot \log \left(\frac{1}{x}\right)\right)} + \frac{e^{\frac{1}{3} \cdot \left(\log \frac{1}{6} - 3 \cdot \log \left(\frac{1}{x}\right)\right)}}{{x}^{2}}\right)\right)}} \cdot \frac{e^{x}}{\sqrt[3]{x + \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(x \cdot x\right)}}\]
Simplified40.4
\[\leadsto \frac{1}{\sqrt[3]{x + \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(x \cdot x\right)} \cdot \color{blue}{\left(\sqrt[3]{\frac{1}{6}} \cdot x + \frac{\sqrt[3]{\frac{1}{6}}}{x} \cdot \left(x + \frac{x}{x}\right)\right)}} \cdot \frac{e^{x}}{\sqrt[3]{x + \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(x \cdot x\right)}}\]
Final simplification40.4
\[\leadsto \frac{e^{x}}{\sqrt[3]{\left(\frac{1}{6} \cdot x + \frac{1}{2}\right) \cdot \left(x \cdot x\right) + x}} \cdot \frac{1}{\sqrt[3]{\left(\frac{1}{6} \cdot x + \frac{1}{2}\right) \cdot \left(x \cdot x\right) + x} \cdot \left(\sqrt[3]{\frac{1}{6}} \cdot x + \frac{\sqrt[3]{\frac{1}{6}}}{x} \cdot \left(\frac{x}{x} + x\right)\right)}\]
