Initial program 29.6
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
Simplified29.6
\[\leadsto \color{blue}{\frac{\left(e^{x \cdot \left(\varepsilon + -1\right)} + \frac{e^{x \cdot \left(\varepsilon + -1\right)}}{\varepsilon}\right) - \left(\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon} - e^{x \cdot \left(-1 - \varepsilon\right)}\right)}{2}}\]
- Using strategy
rm Applied add-log-exp31.3
\[\leadsto \frac{\left(e^{x \cdot \left(\varepsilon + -1\right)} + \frac{e^{x \cdot \left(\varepsilon + -1\right)}}{\varepsilon}\right) - \color{blue}{\log \left(e^{\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon} - e^{x \cdot \left(-1 - \varepsilon\right)}}\right)}}{2}\]
Applied add-log-exp31.0
\[\leadsto \frac{\left(e^{x \cdot \left(\varepsilon + -1\right)} + \color{blue}{\log \left(e^{\frac{e^{x \cdot \left(\varepsilon + -1\right)}}{\varepsilon}}\right)}\right) - \log \left(e^{\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon} - e^{x \cdot \left(-1 - \varepsilon\right)}}\right)}{2}\]
Applied add-log-exp31.1
\[\leadsto \frac{\left(\color{blue}{\log \left(e^{e^{x \cdot \left(\varepsilon + -1\right)}}\right)} + \log \left(e^{\frac{e^{x \cdot \left(\varepsilon + -1\right)}}{\varepsilon}}\right)\right) - \log \left(e^{\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon} - e^{x \cdot \left(-1 - \varepsilon\right)}}\right)}{2}\]
Applied sum-log31.1
\[\leadsto \frac{\color{blue}{\log \left(e^{e^{x \cdot \left(\varepsilon + -1\right)}} \cdot e^{\frac{e^{x \cdot \left(\varepsilon + -1\right)}}{\varepsilon}}\right)} - \log \left(e^{\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon} - e^{x \cdot \left(-1 - \varepsilon\right)}}\right)}{2}\]
Applied diff-log31.1
\[\leadsto \frac{\color{blue}{\log \left(\frac{e^{e^{x \cdot \left(\varepsilon + -1\right)}} \cdot e^{\frac{e^{x \cdot \left(\varepsilon + -1\right)}}{\varepsilon}}}{e^{\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon} - e^{x \cdot \left(-1 - \varepsilon\right)}}}\right)}}{2}\]
Simplified1.2
\[\leadsto \frac{\log \color{blue}{\left(e^{\left(e^{\left(\varepsilon + -1\right) \cdot x} + e^{x \cdot \left(-1 - \varepsilon\right)}\right) - \left(\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon} - \frac{e^{\left(\varepsilon + -1\right) \cdot x}}{\varepsilon}\right)}\right)}}{2}\]
- Using strategy
rm Applied add-sqr-sqrt1.9
\[\leadsto \frac{\log \left(e^{\color{blue}{\sqrt{\left(e^{\left(\varepsilon + -1\right) \cdot x} + e^{x \cdot \left(-1 - \varepsilon\right)}\right) - \left(\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon} - \frac{e^{\left(\varepsilon + -1\right) \cdot x}}{\varepsilon}\right)} \cdot \sqrt{\left(e^{\left(\varepsilon + -1\right) \cdot x} + e^{x \cdot \left(-1 - \varepsilon\right)}\right) - \left(\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon} - \frac{e^{\left(\varepsilon + -1\right) \cdot x}}{\varepsilon}\right)}}}\right)}{2}\]
Applied exp-prod1.2
\[\leadsto \frac{\log \color{blue}{\left({\left(e^{\sqrt{\left(e^{\left(\varepsilon + -1\right) \cdot x} + e^{x \cdot \left(-1 - \varepsilon\right)}\right) - \left(\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon} - \frac{e^{\left(\varepsilon + -1\right) \cdot x}}{\varepsilon}\right)}}\right)}^{\left(\sqrt{\left(e^{\left(\varepsilon + -1\right) \cdot x} + e^{x \cdot \left(-1 - \varepsilon\right)}\right) - \left(\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon} - \frac{e^{\left(\varepsilon + -1\right) \cdot x}}{\varepsilon}\right)}\right)}\right)}}{2}\]
Taylor expanded around 0 1.2
\[\leadsto \frac{\log \left({\left(e^{\sqrt{\left(e^{\left(\varepsilon + -1\right) \cdot x} + e^{x \cdot \left(-1 - \varepsilon\right)}\right) - \left(\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon} - \frac{e^{\left(\varepsilon + -1\right) \cdot x}}{\varepsilon}\right)}}\right)}^{\left(\sqrt{\left(e^{\left(\varepsilon + -1\right) \cdot x} + e^{x \cdot \left(-1 - \varepsilon\right)}\right) - \color{blue}{\left(2 \cdot {x}^{2} - \left(2 \cdot x + {x}^{3}\right)\right)}}\right)}\right)}{2}\]
Simplified1.2
\[\leadsto \frac{\log \left({\left(e^{\sqrt{\left(e^{\left(\varepsilon + -1\right) \cdot x} + e^{x \cdot \left(-1 - \varepsilon\right)}\right) - \left(\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon} - \frac{e^{\left(\varepsilon + -1\right) \cdot x}}{\varepsilon}\right)}}\right)}^{\left(\sqrt{\left(e^{\left(\varepsilon + -1\right) \cdot x} + e^{x \cdot \left(-1 - \varepsilon\right)}\right) - \color{blue}{x \cdot \left(-2 - x \cdot \left(x + -2\right)\right)}}\right)}\right)}{2}\]
- Using strategy
rm Applied *-un-lft-identity1.2
\[\leadsto \frac{\color{blue}{1 \cdot \log \left({\left(e^{\sqrt{\left(e^{\left(\varepsilon + -1\right) \cdot x} + e^{x \cdot \left(-1 - \varepsilon\right)}\right) - \left(\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon} - \frac{e^{\left(\varepsilon + -1\right) \cdot x}}{\varepsilon}\right)}}\right)}^{\left(\sqrt{\left(e^{\left(\varepsilon + -1\right) \cdot x} + e^{x \cdot \left(-1 - \varepsilon\right)}\right) - x \cdot \left(-2 - x \cdot \left(x + -2\right)\right)}\right)}\right)}}{2}\]
Final simplification1.2
\[\leadsto \frac{\log \left({\left(e^{\sqrt{\left(e^{\left(\varepsilon + -1\right) \cdot x} + e^{\left(-1 - \varepsilon\right) \cdot x}\right) - \left(\frac{e^{\left(-1 - \varepsilon\right) \cdot x}}{\varepsilon} - \frac{e^{\left(\varepsilon + -1\right) \cdot x}}{\varepsilon}\right)}}\right)}^{\left(\sqrt{\left(e^{\left(\varepsilon + -1\right) \cdot x} + e^{\left(-1 - \varepsilon\right) \cdot x}\right) - x \cdot \left(-2 - x \cdot \left(-2 + x\right)\right)}\right)}\right)}{2}\]
