Initial program 0.5
\[\log \left(1 + e^{x}\right) - x \cdot y\]
Initial simplification0.5
\[\leadsto \log \left(1 + e^{x}\right) - y \cdot x\]
Taylor expanded around inf 0.5
\[\leadsto \log \color{blue}{\left(e^{x} + 1\right)} - y \cdot x\]
- Using strategy
rm Applied add-cbrt-cube0.5
\[\leadsto \color{blue}{\sqrt[3]{\left(\log \left(e^{x} + 1\right) \cdot \log \left(e^{x} + 1\right)\right) \cdot \log \left(e^{x} + 1\right)}} - y \cdot x\]
- Using strategy
rm Applied flip3-+0.5
\[\leadsto \sqrt[3]{\left(\log \left(e^{x} + 1\right) \cdot \log \color{blue}{\left(\frac{{\left(e^{x}\right)}^{3} + {1}^{3}}{e^{x} \cdot e^{x} + \left(1 \cdot 1 - e^{x} \cdot 1\right)}\right)}\right) \cdot \log \left(e^{x} + 1\right)} - y \cdot x\]
Applied log-div0.5
\[\leadsto \sqrt[3]{\left(\log \left(e^{x} + 1\right) \cdot \color{blue}{\left(\log \left({\left(e^{x}\right)}^{3} + {1}^{3}\right) - \log \left(e^{x} \cdot e^{x} + \left(1 \cdot 1 - e^{x} \cdot 1\right)\right)\right)}\right) \cdot \log \left(e^{x} + 1\right)} - y \cdot x\]
Simplified0.5
\[\leadsto \sqrt[3]{\left(\log \left(e^{x} + 1\right) \cdot \left(\color{blue}{\log \left(1 + {\left(e^{x}\right)}^{3}\right)} - \log \left(e^{x} \cdot e^{x} + \left(1 \cdot 1 - e^{x} \cdot 1\right)\right)\right)\right) \cdot \log \left(e^{x} + 1\right)} - y \cdot x\]
- Using strategy
rm Applied pow1/30.5
\[\leadsto \color{blue}{{\left(\left(\log \left(e^{x} + 1\right) \cdot \left(\log \left(1 + {\left(e^{x}\right)}^{3}\right) - \log \left(e^{x} \cdot e^{x} + \left(1 \cdot 1 - e^{x} \cdot 1\right)\right)\right)\right) \cdot \log \left(e^{x} + 1\right)\right)}^{\frac{1}{3}}} - y \cdot x\]
Final simplification0.5
\[\leadsto {\left(\left(\log \left(e^{x} + 1\right) \cdot \left(\log \left(1 + {\left(e^{x}\right)}^{3}\right) - \log \left(\left(1 - e^{x}\right) + e^{x} \cdot e^{x}\right)\right)\right) \cdot \log \left(e^{x} + 1\right)\right)}^{\frac{1}{3}} - x \cdot y\]
