Initial program 3.9
\[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c_n}}\]
- Using strategy
rm Applied add-exp-log3.9
\[\leadsto \frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c_p} \cdot {\color{blue}{\left(e^{\log \left(1 - \frac{1}{1 + e^{-t}}\right)}\right)}}^{c_n}}\]
Applied pow-exp4.0
\[\leadsto \frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c_p} \cdot \color{blue}{e^{\log \left(1 - \frac{1}{1 + e^{-t}}\right) \cdot c_n}}}\]
Applied add-exp-log4.0
\[\leadsto \frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{{\left(\frac{1}{\color{blue}{e^{\log \left(1 + e^{-t}\right)}}}\right)}^{c_p} \cdot e^{\log \left(1 - \frac{1}{1 + e^{-t}}\right) \cdot c_n}}\]
Applied rec-exp4.0
\[\leadsto \frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{{\color{blue}{\left(e^{-\log \left(1 + e^{-t}\right)}\right)}}^{c_p} \cdot e^{\log \left(1 - \frac{1}{1 + e^{-t}}\right) \cdot c_n}}\]
Applied pow-exp4.0
\[\leadsto \frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{\color{blue}{e^{\left(-\log \left(1 + e^{-t}\right)\right) \cdot c_p}} \cdot e^{\log \left(1 - \frac{1}{1 + e^{-t}}\right) \cdot c_n}}\]
Applied prod-exp4.0
\[\leadsto \frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c_n}}{\color{blue}{e^{\left(-\log \left(1 + e^{-t}\right)\right) \cdot c_p + \log \left(1 - \frac{1}{1 + e^{-t}}\right) \cdot c_n}}}\]
Applied add-exp-log4.0
\[\leadsto \frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p} \cdot {\color{blue}{\left(e^{\log \left(1 - \frac{1}{1 + e^{-s}}\right)}\right)}}^{c_n}}{e^{\left(-\log \left(1 + e^{-t}\right)\right) \cdot c_p + \log \left(1 - \frac{1}{1 + e^{-t}}\right) \cdot c_n}}\]
Applied pow-exp4.0
\[\leadsto \frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c_p} \cdot \color{blue}{e^{\log \left(1 - \frac{1}{1 + e^{-s}}\right) \cdot c_n}}}{e^{\left(-\log \left(1 + e^{-t}\right)\right) \cdot c_p + \log \left(1 - \frac{1}{1 + e^{-t}}\right) \cdot c_n}}\]
Applied add-exp-log4.0
\[\leadsto \frac{{\left(\frac{1}{\color{blue}{e^{\log \left(1 + e^{-s}\right)}}}\right)}^{c_p} \cdot e^{\log \left(1 - \frac{1}{1 + e^{-s}}\right) \cdot c_n}}{e^{\left(-\log \left(1 + e^{-t}\right)\right) \cdot c_p + \log \left(1 - \frac{1}{1 + e^{-t}}\right) \cdot c_n}}\]
Applied rec-exp4.0
\[\leadsto \frac{{\color{blue}{\left(e^{-\log \left(1 + e^{-s}\right)}\right)}}^{c_p} \cdot e^{\log \left(1 - \frac{1}{1 + e^{-s}}\right) \cdot c_n}}{e^{\left(-\log \left(1 + e^{-t}\right)\right) \cdot c_p + \log \left(1 - \frac{1}{1 + e^{-t}}\right) \cdot c_n}}\]
Applied pow-exp3.9
\[\leadsto \frac{\color{blue}{e^{\left(-\log \left(1 + e^{-s}\right)\right) \cdot c_p}} \cdot e^{\log \left(1 - \frac{1}{1 + e^{-s}}\right) \cdot c_n}}{e^{\left(-\log \left(1 + e^{-t}\right)\right) \cdot c_p + \log \left(1 - \frac{1}{1 + e^{-t}}\right) \cdot c_n}}\]
Applied prod-exp3.9
\[\leadsto \frac{\color{blue}{e^{\left(-\log \left(1 + e^{-s}\right)\right) \cdot c_p + \log \left(1 - \frac{1}{1 + e^{-s}}\right) \cdot c_n}}}{e^{\left(-\log \left(1 + e^{-t}\right)\right) \cdot c_p + \log \left(1 - \frac{1}{1 + e^{-t}}\right) \cdot c_n}}\]
Applied div-exp1.7
\[\leadsto \color{blue}{e^{\left(\left(-\log \left(1 + e^{-s}\right)\right) \cdot c_p + \log \left(1 - \frac{1}{1 + e^{-s}}\right) \cdot c_n\right) - \left(\left(-\log \left(1 + e^{-t}\right)\right) \cdot c_p + \log \left(1 - \frac{1}{1 + e^{-t}}\right) \cdot c_n\right)}}\]
Applied simplify1.7
\[\leadsto e^{\color{blue}{\left(\log \left(e^{-t} + 1\right) - \log \left(1 + e^{-s}\right)\right) \cdot c_p - c_n \cdot \left(\log \left(1 - \frac{1}{e^{-t} + 1}\right) - \log \left(1 - \frac{1}{1 + e^{-s}}\right)\right)}}\]
- Using strategy
rm Applied flip3--1.7
\[\leadsto e^{\left(\log \left(e^{-t} + 1\right) - \log \left(1 + e^{-s}\right)\right) \cdot c_p - c_n \cdot \left(\log \color{blue}{\left(\frac{{1}^{3} - {\left(\frac{1}{e^{-t} + 1}\right)}^{3}}{1 \cdot 1 + \left(\frac{1}{e^{-t} + 1} \cdot \frac{1}{e^{-t} + 1} + 1 \cdot \frac{1}{e^{-t} + 1}\right)}\right)} - \log \left(1 - \frac{1}{1 + e^{-s}}\right)\right)}\]
Applied log-div1.7
\[\leadsto e^{\left(\log \left(e^{-t} + 1\right) - \log \left(1 + e^{-s}\right)\right) \cdot c_p - c_n \cdot \left(\color{blue}{\left(\log \left({1}^{3} - {\left(\frac{1}{e^{-t} + 1}\right)}^{3}\right) - \log \left(1 \cdot 1 + \left(\frac{1}{e^{-t} + 1} \cdot \frac{1}{e^{-t} + 1} + 1 \cdot \frac{1}{e^{-t} + 1}\right)\right)\right)} - \log \left(1 - \frac{1}{1 + e^{-s}}\right)\right)}\]
Applied associate--l-1.7
\[\leadsto e^{\left(\log \left(e^{-t} + 1\right) - \log \left(1 + e^{-s}\right)\right) \cdot c_p - c_n \cdot \color{blue}{\left(\log \left({1}^{3} - {\left(\frac{1}{e^{-t} + 1}\right)}^{3}\right) - \left(\log \left(1 \cdot 1 + \left(\frac{1}{e^{-t} + 1} \cdot \frac{1}{e^{-t} + 1} + 1 \cdot \frac{1}{e^{-t} + 1}\right)\right) + \log \left(1 - \frac{1}{1 + e^{-s}}\right)\right)\right)}}\]
Applied simplify1.7
\[\leadsto e^{\left(\log \left(e^{-t} + 1\right) - \log \left(1 + e^{-s}\right)\right) \cdot c_p - c_n \cdot \left(\log \left({1}^{3} - {\left(\frac{1}{e^{-t} + 1}\right)}^{3}\right) - \color{blue}{\left(\log \left(\left(\frac{1}{e^{-t} + 1} + 1\right) + \frac{1}{e^{-t} + 1} \cdot \frac{1}{e^{-t} + 1}\right) + \log \left(1 - \frac{1}{e^{-s} + 1}\right)\right)}\right)}\]
