Initial program 0.6
\[\log \left(1 + e^{x}\right) - x \cdot y\]
Initial simplification0.6
\[\leadsto \log \left(1 + e^{x}\right) - y \cdot x\]
- Using strategy
rm Applied flip3-+0.6
\[\leadsto \log \color{blue}{\left(\frac{{1}^{3} + {\left(e^{x}\right)}^{3}}{1 \cdot 1 + \left(e^{x} \cdot e^{x} - 1 \cdot e^{x}\right)}\right)} - y \cdot x\]
Applied log-div0.6
\[\leadsto \color{blue}{\left(\log \left({1}^{3} + {\left(e^{x}\right)}^{3}\right) - \log \left(1 \cdot 1 + \left(e^{x} \cdot e^{x} - 1 \cdot e^{x}\right)\right)\right)} - y \cdot x\]
Applied associate--l-0.6
\[\leadsto \color{blue}{\log \left({1}^{3} + {\left(e^{x}\right)}^{3}\right) - \left(\log \left(1 \cdot 1 + \left(e^{x} \cdot e^{x} - 1 \cdot e^{x}\right)\right) + y \cdot x\right)}\]
Simplified0.6
\[\leadsto \color{blue}{\log \left({\left(e^{x}\right)}^{3} + 1\right)} - \left(\log \left(1 \cdot 1 + \left(e^{x} \cdot e^{x} - 1 \cdot e^{x}\right)\right) + y \cdot x\right)\]
- Using strategy
rm Applied add-sqr-sqrt1.4
\[\leadsto \log \color{blue}{\left(\sqrt{{\left(e^{x}\right)}^{3} + 1} \cdot \sqrt{{\left(e^{x}\right)}^{3} + 1}\right)} - \left(\log \left(1 \cdot 1 + \left(e^{x} \cdot e^{x} - 1 \cdot e^{x}\right)\right) + y \cdot x\right)\]
Applied log-prod1.1
\[\leadsto \color{blue}{\left(\log \left(\sqrt{{\left(e^{x}\right)}^{3} + 1}\right) + \log \left(\sqrt{{\left(e^{x}\right)}^{3} + 1}\right)\right)} - \left(\log \left(1 \cdot 1 + \left(e^{x} \cdot e^{x} - 1 \cdot e^{x}\right)\right) + y \cdot x\right)\]
- Using strategy
rm Applied add-sqr-sqrt1.1
\[\leadsto \left(\log \left(\sqrt{{\left(e^{x}\right)}^{3} + 1}\right) + \log \left(\sqrt{\color{blue}{\sqrt{{\left(e^{x}\right)}^{3} + 1} \cdot \sqrt{{\left(e^{x}\right)}^{3} + 1}}}\right)\right) - \left(\log \left(1 \cdot 1 + \left(e^{x} \cdot e^{x} - 1 \cdot e^{x}\right)\right) + y \cdot x\right)\]
Applied sqrt-prod0.6
\[\leadsto \left(\log \left(\sqrt{{\left(e^{x}\right)}^{3} + 1}\right) + \log \color{blue}{\left(\sqrt{\sqrt{{\left(e^{x}\right)}^{3} + 1}} \cdot \sqrt{\sqrt{{\left(e^{x}\right)}^{3} + 1}}\right)}\right) - \left(\log \left(1 \cdot 1 + \left(e^{x} \cdot e^{x} - 1 \cdot e^{x}\right)\right) + y \cdot x\right)\]
Applied log-prod0.6
\[\leadsto \left(\log \left(\sqrt{{\left(e^{x}\right)}^{3} + 1}\right) + \color{blue}{\left(\log \left(\sqrt{\sqrt{{\left(e^{x}\right)}^{3} + 1}}\right) + \log \left(\sqrt{\sqrt{{\left(e^{x}\right)}^{3} + 1}}\right)\right)}\right) - \left(\log \left(1 \cdot 1 + \left(e^{x} \cdot e^{x} - 1 \cdot e^{x}\right)\right) + y \cdot x\right)\]
Final simplification0.6
\[\leadsto \left(\log \left(\sqrt{{\left(e^{x}\right)}^{3} + 1}\right) + \left(\log \left(\sqrt{\sqrt{{\left(e^{x}\right)}^{3} + 1}}\right) + \log \left(\sqrt{\sqrt{{\left(e^{x}\right)}^{3} + 1}}\right)\right)\right) - \left(\log \left(1 + \left(e^{x} \cdot e^{x} - e^{x}\right)\right) + x \cdot y\right)\]
