Initial program 4.2
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
Simplified0.1
\[\leadsto \color{blue}{\sqrt{e^{x} + 1}}\]
- Using strategy
rm Applied flip3-+0.2
\[\leadsto \sqrt{\color{blue}{\frac{{\left(e^{x}\right)}^{3} + {1}^{3}}{e^{x} \cdot e^{x} + \left(1 \cdot 1 - e^{x} \cdot 1\right)}}}\]
- Using strategy
rm Applied add-sqr-sqrt0.2
\[\leadsto \sqrt{\frac{\color{blue}{\sqrt{{\left(e^{x}\right)}^{3} + {1}^{3}} \cdot \sqrt{{\left(e^{x}\right)}^{3} + {1}^{3}}}}{e^{x} \cdot e^{x} + \left(1 \cdot 1 - e^{x} \cdot 1\right)}}\]
- Using strategy
rm Applied sum-cubes0.2
\[\leadsto \sqrt{\frac{\sqrt{\color{blue}{\left(e^{x} \cdot e^{x} + \left(1 \cdot 1 - e^{x} \cdot 1\right)\right) \cdot \left(e^{x} + 1\right)}} \cdot \sqrt{{\left(e^{x}\right)}^{3} + {1}^{3}}}{e^{x} \cdot e^{x} + \left(1 \cdot 1 - e^{x} \cdot 1\right)}}\]
Applied sqrt-prod0.2
\[\leadsto \sqrt{\frac{\color{blue}{\left(\sqrt{e^{x} \cdot e^{x} + \left(1 \cdot 1 - e^{x} \cdot 1\right)} \cdot \sqrt{e^{x} + 1}\right)} \cdot \sqrt{{\left(e^{x}\right)}^{3} + {1}^{3}}}{e^{x} \cdot e^{x} + \left(1 \cdot 1 - e^{x} \cdot 1\right)}}\]
Applied associate-*l*0.2
\[\leadsto \sqrt{\frac{\color{blue}{\sqrt{e^{x} \cdot e^{x} + \left(1 \cdot 1 - e^{x} \cdot 1\right)} \cdot \left(\sqrt{e^{x} + 1} \cdot \sqrt{{\left(e^{x}\right)}^{3} + {1}^{3}}\right)}}{e^{x} \cdot e^{x} + \left(1 \cdot 1 - e^{x} \cdot 1\right)}}\]
Final simplification0.2
\[\leadsto \sqrt{\frac{\sqrt{e^{x} \cdot e^{x} + \left(1 - e^{x}\right)} \cdot \left(\sqrt{e^{x} + 1} \cdot \sqrt{{\left(e^{x}\right)}^{3} + 1}\right)}{e^{x} \cdot e^{x} + \left(1 - e^{x}\right)}}\]
