Initial program 52.3
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
Initial simplification52.3
\[\leadsto \frac{\frac{\left(\left(\alpha + \beta\right) \cdot i + \left(i \cdot i + \beta \cdot \alpha\right)\right) \cdot \left(\left(\alpha + \beta\right) \cdot i + i \cdot i\right)}{\left(2 \cdot i + \left(\alpha + \beta\right)\right) \cdot \left(2 \cdot i + \left(\alpha + \beta\right)\right) - 1.0}}{\left(2 \cdot i + \left(\alpha + \beta\right)\right) \cdot \left(2 \cdot i + \left(\alpha + \beta\right)\right)}\]
- Using strategy
rm Applied *-un-lft-identity52.3
\[\leadsto \frac{\frac{\left(\left(\alpha + \beta\right) \cdot i + \left(i \cdot i + \beta \cdot \alpha\right)\right) \cdot \left(\left(\alpha + \beta\right) \cdot i + i \cdot i\right)}{\color{blue}{1 \cdot \left(\left(2 \cdot i + \left(\alpha + \beta\right)\right) \cdot \left(2 \cdot i + \left(\alpha + \beta\right)\right) - 1.0\right)}}}{\left(2 \cdot i + \left(\alpha + \beta\right)\right) \cdot \left(2 \cdot i + \left(\alpha + \beta\right)\right)}\]
Applied times-frac39.0
\[\leadsto \frac{\color{blue}{\frac{\left(\alpha + \beta\right) \cdot i + \left(i \cdot i + \beta \cdot \alpha\right)}{1} \cdot \frac{\left(\alpha + \beta\right) \cdot i + i \cdot i}{\left(2 \cdot i + \left(\alpha + \beta\right)\right) \cdot \left(2 \cdot i + \left(\alpha + \beta\right)\right) - 1.0}}}{\left(2 \cdot i + \left(\alpha + \beta\right)\right) \cdot \left(2 \cdot i + \left(\alpha + \beta\right)\right)}\]
Applied times-frac39.0
\[\leadsto \color{blue}{\frac{\frac{\left(\alpha + \beta\right) \cdot i + \left(i \cdot i + \beta \cdot \alpha\right)}{1}}{2 \cdot i + \left(\alpha + \beta\right)} \cdot \frac{\frac{\left(\alpha + \beta\right) \cdot i + i \cdot i}{\left(2 \cdot i + \left(\alpha + \beta\right)\right) \cdot \left(2 \cdot i + \left(\alpha + \beta\right)\right) - 1.0}}{2 \cdot i + \left(\alpha + \beta\right)}}\]
Simplified39.0
\[\leadsto \color{blue}{\frac{\left(i + \alpha\right) \cdot \left(\beta + i\right)}{i \cdot 2 + \left(\beta + \alpha\right)}} \cdot \frac{\frac{\left(\alpha + \beta\right) \cdot i + i \cdot i}{\left(2 \cdot i + \left(\alpha + \beta\right)\right) \cdot \left(2 \cdot i + \left(\alpha + \beta\right)\right) - 1.0}}{2 \cdot i + \left(\alpha + \beta\right)}\]
Simplified39.0
\[\leadsto \frac{\left(i + \alpha\right) \cdot \left(\beta + i\right)}{i \cdot 2 + \left(\beta + \alpha\right)} \cdot \color{blue}{\frac{\frac{i}{i \cdot 2 + \left(\beta + \alpha\right)} \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) \cdot \left(i \cdot 2 + \left(\beta + \alpha\right)\right) - 1.0}}\]
- Using strategy
rm Applied *-un-lft-identity39.0
\[\leadsto \frac{\left(i + \alpha\right) \cdot \left(\beta + i\right)}{\color{blue}{1 \cdot \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}} \cdot \frac{\frac{i}{i \cdot 2 + \left(\beta + \alpha\right)} \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) \cdot \left(i \cdot 2 + \left(\beta + \alpha\right)\right) - 1.0}\]
Applied times-frac36.4
\[\leadsto \color{blue}{\left(\frac{i + \alpha}{1} \cdot \frac{\beta + i}{i \cdot 2 + \left(\beta + \alpha\right)}\right)} \cdot \frac{\frac{i}{i \cdot 2 + \left(\beta + \alpha\right)} \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) \cdot \left(i \cdot 2 + \left(\beta + \alpha\right)\right) - 1.0}\]
Applied associate-*l*36.6
\[\leadsto \color{blue}{\frac{i + \alpha}{1} \cdot \left(\frac{\beta + i}{i \cdot 2 + \left(\beta + \alpha\right)} \cdot \frac{\frac{i}{i \cdot 2 + \left(\beta + \alpha\right)} \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) \cdot \left(i \cdot 2 + \left(\beta + \alpha\right)\right) - 1.0}\right)}\]
Simplified36.6
\[\leadsto \color{blue}{\left(\alpha + i\right)} \cdot \left(\frac{\beta + i}{i \cdot 2 + \left(\beta + \alpha\right)} \cdot \frac{\frac{i}{i \cdot 2 + \left(\beta + \alpha\right)} \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) \cdot \left(i \cdot 2 + \left(\beta + \alpha\right)\right) - 1.0}\right)\]
Final simplification36.6
\[\leadsto \left(\frac{i + \beta}{\left(\beta + \alpha\right) + 2 \cdot i} \cdot \frac{\frac{i}{\left(\beta + \alpha\right) + 2 \cdot i} \cdot \left(\left(\beta + \alpha\right) + i\right)}{\left(\left(\beta + \alpha\right) + 2 \cdot i\right) \cdot \left(\left(\beta + \alpha\right) + 2 \cdot i\right) - 1.0}\right) \cdot \left(i + \alpha\right)\]
