- Started with
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
22.6
- Using strategy
rm 22.6
- Applied flip3-- to get
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{red}{\left(\frac{1}{\varepsilon} - 1\right)} \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{{\left(\frac{1}{\varepsilon}\right)}^{3} - {1}^{3}}{{\left(\frac{1}{\varepsilon}\right)}^2 + \left({1}^2 + \frac{1}{\varepsilon} \cdot 1\right)}} \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
23.1
- Applied associate-*l/ to get
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{red}{\frac{{\left(\frac{1}{\varepsilon}\right)}^{3} - {1}^{3}}{{\left(\frac{1}{\varepsilon}\right)}^2 + \left({1}^2 + \frac{1}{\varepsilon} \cdot 1\right)} \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{\left({\left(\frac{1}{\varepsilon}\right)}^{3} - {1}^{3}\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{{\left(\frac{1}{\varepsilon}\right)}^2 + \left({1}^2 + \frac{1}{\varepsilon} \cdot 1\right)}}}{2}\]
23.1
- Applied exp-neg to get
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{red}{e^{-\left(1 - \varepsilon\right) \cdot x}} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^{3} - {1}^{3}\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{{\left(\frac{1}{\varepsilon}\right)}^2 + \left({1}^2 + \frac{1}{\varepsilon} \cdot 1\right)}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\frac{1}{e^{\left(1 - \varepsilon\right) \cdot x}}} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^{3} - {1}^{3}\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{{\left(\frac{1}{\varepsilon}\right)}^2 + \left({1}^2 + \frac{1}{\varepsilon} \cdot 1\right)}}{2}\]
23.1
- Applied un-div-inv to get
\[\frac{\color{red}{\left(1 + \frac{1}{\varepsilon}\right) \cdot \frac{1}{e^{\left(1 - \varepsilon\right) \cdot x}}} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^{3} - {1}^{3}\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{{\left(\frac{1}{\varepsilon}\right)}^2 + \left({1}^2 + \frac{1}{\varepsilon} \cdot 1\right)}}{2} \leadsto \frac{\color{blue}{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}}} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^{3} - {1}^{3}\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{{\left(\frac{1}{\varepsilon}\right)}^2 + \left({1}^2 + \frac{1}{\varepsilon} \cdot 1\right)}}{2}\]
23.1
- Applied frac-sub to get
\[\frac{\color{red}{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^{3} - {1}^{3}\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{{\left(\frac{1}{\varepsilon}\right)}^2 + \left({1}^2 + \frac{1}{\varepsilon} \cdot 1\right)}}}{2} \leadsto \frac{\color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left({\left(\frac{1}{\varepsilon}\right)}^2 + \left({1}^2 + \frac{1}{\varepsilon} \cdot 1\right)\right) - e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\left({\left(\frac{1}{\varepsilon}\right)}^{3} - {1}^{3}\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}{e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left({\left(\frac{1}{\varepsilon}\right)}^2 + \left({1}^2 + \frac{1}{\varepsilon} \cdot 1\right)\right)}}}{2}\]
23.0
- Applied simplify to get
\[\frac{\frac{\color{red}{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left({\left(\frac{1}{\varepsilon}\right)}^2 + \left({1}^2 + \frac{1}{\varepsilon} \cdot 1\right)\right) - e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\left({\left(\frac{1}{\varepsilon}\right)}^{3} - {1}^{3}\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left({\left(\frac{1}{\varepsilon}\right)}^2 + \left({1}^2 + \frac{1}{\varepsilon} \cdot 1\right)\right)}}{2} \leadsto \frac{\frac{\color{blue}{\left(\frac{1}{\varepsilon} + 1\right) \cdot \left(\left(\frac{1}{\varepsilon} + 1\right) + {\left(\frac{1}{\varepsilon}\right)}^2\right) - e^{x \cdot \left(1 - \varepsilon\right) - \varepsilon \cdot x} \cdot \frac{\frac{\frac{1}{\varepsilon}}{\varepsilon \cdot \varepsilon} - 1}{e^{x}}}}{e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left({\left(\frac{1}{\varepsilon}\right)}^2 + \left({1}^2 + \frac{1}{\varepsilon} \cdot 1\right)\right)}}{2}\]
23.1
- Applied taylor to get
\[\frac{\frac{\left(\frac{1}{\varepsilon} + 1\right) \cdot \left(\left(\frac{1}{\varepsilon} + 1\right) + {\left(\frac{1}{\varepsilon}\right)}^2\right) - e^{x \cdot \left(1 - \varepsilon\right) - \varepsilon \cdot x} \cdot \frac{\frac{\frac{1}{\varepsilon}}{\varepsilon \cdot \varepsilon} - 1}{e^{x}}}{e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left({\left(\frac{1}{\varepsilon}\right)}^2 + \left({1}^2 + \frac{1}{\varepsilon} \cdot 1\right)\right)}}{2} \leadsto \frac{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^2}{2}\]
0.2
- Taylor expanded around 0 to get
\[\frac{\color{red}{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^2}}{2} \leadsto \frac{\color{blue}{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^2}}{2}\]
0.2
- Applied simplify to get
\[\frac{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^2}{2} \leadsto \frac{\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left(2 - x \cdot x\right)}{2}\]
0.2
- Applied final simplification
- Applied simplify to get
\[\color{red}{\frac{\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left(2 - x \cdot x\right)}{2}} \leadsto \color{blue}{\frac{{x}^3 \cdot \frac{2}{3} + \left(2 - x \cdot x\right)}{2}}\]
0.2
