Initial program 0.6
\[\log \left(1 + e^{x}\right) - x \cdot y\]
- Using strategy
rm Applied add-sqr-sqrt1.4
\[\leadsto \log \color{blue}{\left(\sqrt{1 + e^{x}} \cdot \sqrt{1 + e^{x}}\right)} - x \cdot y\]
Applied log-prod1.1
\[\leadsto \color{blue}{\left(\log \left(\sqrt{1 + e^{x}}\right) + \log \left(\sqrt{1 + e^{x}}\right)\right)} - x \cdot y\]
- Using strategy
rm Applied pow1/21.1
\[\leadsto \left(\log \left(\sqrt{1 + e^{x}}\right) + \log \color{blue}{\left({\left(1 + e^{x}\right)}^{\frac{1}{2}}\right)}\right) - x \cdot y\]
Applied log-pow1.1
\[\leadsto \left(\log \left(\sqrt{1 + e^{x}}\right) + \color{blue}{\frac{1}{2} \cdot \log \left(1 + e^{x}\right)}\right) - x \cdot y\]
- Using strategy
rm Applied flip3-+1.2
\[\leadsto \left(\log \left(\sqrt{\color{blue}{\frac{{1}^{3} + {\left(e^{x}\right)}^{3}}{1 \cdot 1 + \left(e^{x} \cdot e^{x} - 1 \cdot e^{x}\right)}}}\right) + \frac{1}{2} \cdot \log \left(1 + e^{x}\right)\right) - x \cdot y\]
Applied sqrt-div1.2
\[\leadsto \left(\log \color{blue}{\left(\frac{\sqrt{{1}^{3} + {\left(e^{x}\right)}^{3}}}{\sqrt{1 \cdot 1 + \left(e^{x} \cdot e^{x} - 1 \cdot e^{x}\right)}}\right)} + \frac{1}{2} \cdot \log \left(1 + e^{x}\right)\right) - x \cdot y\]
Applied log-div1.2
\[\leadsto \left(\color{blue}{\left(\log \left(\sqrt{{1}^{3} + {\left(e^{x}\right)}^{3}}\right) - \log \left(\sqrt{1 \cdot 1 + \left(e^{x} \cdot e^{x} - 1 \cdot e^{x}\right)}\right)\right)} + \frac{1}{2} \cdot \log \left(1 + e^{x}\right)\right) - x \cdot y\]
Applied simplify1.2
\[\leadsto \left(\left(\color{blue}{\log \left(\sqrt{1 + {\left(e^{x}\right)}^{3}}\right)} - \log \left(\sqrt{1 \cdot 1 + \left(e^{x} \cdot e^{x} - 1 \cdot e^{x}\right)}\right)\right) + \frac{1}{2} \cdot \log \left(1 + e^{x}\right)\right) - x \cdot y\]
Applied simplify1.2
\[\leadsto \left(\left(\log \left(\sqrt{1 + {\left(e^{x}\right)}^{3}}\right) - \color{blue}{\log \left(\sqrt{e^{x + x} + \left(1 - e^{x}\right)}\right)}\right) + \frac{1}{2} \cdot \log \left(1 + e^{x}\right)\right) - x \cdot y\]
- Using strategy
rm Applied flip3-+1.2
\[\leadsto \left(\left(\log \left(\sqrt{1 + {\left(e^{x}\right)}^{3}}\right) - \log \left(\sqrt{\color{blue}{\frac{{\left(e^{x + x}\right)}^{3} + {\left(1 - e^{x}\right)}^{3}}{e^{x + x} \cdot e^{x + x} + \left(\left(1 - e^{x}\right) \cdot \left(1 - e^{x}\right) - e^{x + x} \cdot \left(1 - e^{x}\right)\right)}}}\right)\right) + \frac{1}{2} \cdot \log \left(1 + e^{x}\right)\right) - x \cdot y\]
Applied sqrt-div1.2
\[\leadsto \left(\left(\log \left(\sqrt{1 + {\left(e^{x}\right)}^{3}}\right) - \log \color{blue}{\left(\frac{\sqrt{{\left(e^{x + x}\right)}^{3} + {\left(1 - e^{x}\right)}^{3}}}{\sqrt{e^{x + x} \cdot e^{x + x} + \left(\left(1 - e^{x}\right) \cdot \left(1 - e^{x}\right) - e^{x + x} \cdot \left(1 - e^{x}\right)\right)}}\right)}\right) + \frac{1}{2} \cdot \log \left(1 + e^{x}\right)\right) - x \cdot y\]
Applied log-div1.2
\[\leadsto \left(\left(\log \left(\sqrt{1 + {\left(e^{x}\right)}^{3}}\right) - \color{blue}{\left(\log \left(\sqrt{{\left(e^{x + x}\right)}^{3} + {\left(1 - e^{x}\right)}^{3}}\right) - \log \left(\sqrt{e^{x + x} \cdot e^{x + x} + \left(\left(1 - e^{x}\right) \cdot \left(1 - e^{x}\right) - e^{x + x} \cdot \left(1 - e^{x}\right)\right)}\right)\right)}\right) + \frac{1}{2} \cdot \log \left(1 + e^{x}\right)\right) - x \cdot y\]
Applied simplify1.2
\[\leadsto \left(\left(\log \left(\sqrt{1 + {\left(e^{x}\right)}^{3}}\right) - \left(\log \left(\sqrt{{\left(e^{x + x}\right)}^{3} + {\left(1 - e^{x}\right)}^{3}}\right) - \color{blue}{\log \left(\sqrt{\left(1 - e^{x}\right) \cdot \left(1 - e^{x}\right) - \left(\left(1 - e^{x}\right) - e^{x} \cdot e^{x}\right) \cdot \left(e^{x} \cdot e^{x}\right)}\right)}\right)\right) + \frac{1}{2} \cdot \log \left(1 + e^{x}\right)\right) - x \cdot y\]
