Initial program 31.3
\[\frac{1 + \sqrt{{x}^{2} - 1}}{\sqrt{{x}^{2} - 1}}\]
- Using strategy
rm
Applied add-sqr-sqrt31.3
\[\leadsto \frac{1 + \sqrt{\color{blue}{\sqrt{{x}^{2}} \cdot \sqrt{{x}^{2}}} - 1}}{\sqrt{{x}^{2} - 1}}\]
Applied difference-of-sqr-131.3
\[\leadsto \frac{1 + \sqrt{\color{blue}{\left(\sqrt{{x}^{2}} + 1\right) \cdot \left(\sqrt{{x}^{2}} - 1\right)}}}{\sqrt{{x}^{2} - 1}}\]
Applied sqrt-prod31.6
\[\leadsto \frac{1 + \color{blue}{\sqrt{\sqrt{{x}^{2}} + 1} \cdot \sqrt{\sqrt{{x}^{2}} - 1}}}{\sqrt{{x}^{2} - 1}}\]
Applied simplify31.6
\[\leadsto \frac{1 + \color{blue}{\sqrt{\left|x\right| + 1}} \cdot \sqrt{\sqrt{{x}^{2}} - 1}}{\sqrt{{x}^{2} - 1}}\]
Applied simplify31.6
\[\leadsto \frac{1 + \sqrt{\left|x\right| + 1} \cdot \color{blue}{\sqrt{\left|x\right| - 1}}}{\sqrt{{x}^{2} - 1}}\]
- Using strategy
rm
Applied add-sqr-sqrt31.6
\[\leadsto \frac{1 + \sqrt{\left|x\right| + 1} \cdot \sqrt{\left|x\right| - 1}}{\sqrt{\color{blue}{\sqrt{{x}^{2}} \cdot \sqrt{{x}^{2}}} - 1}}\]
Applied difference-of-sqr-131.6
\[\leadsto \frac{1 + \sqrt{\left|x\right| + 1} \cdot \sqrt{\left|x\right| - 1}}{\sqrt{\color{blue}{\left(\sqrt{{x}^{2}} + 1\right) \cdot \left(\sqrt{{x}^{2}} - 1\right)}}}\]
Applied sqrt-prod31.3
\[\leadsto \frac{1 + \sqrt{\left|x\right| + 1} \cdot \sqrt{\left|x\right| - 1}}{\color{blue}{\sqrt{\sqrt{{x}^{2}} + 1} \cdot \sqrt{\sqrt{{x}^{2}} - 1}}}\]
Applied simplify31.3
\[\leadsto \frac{1 + \sqrt{\left|x\right| + 1} \cdot \sqrt{\left|x\right| - 1}}{\color{blue}{\sqrt{\left|x\right| + 1}} \cdot \sqrt{\sqrt{{x}^{2}} - 1}}\]
Applied simplify0.0
\[\leadsto \frac{1 + \sqrt{\left|x\right| + 1} \cdot \sqrt{\left|x\right| - 1}}{\sqrt{\left|x\right| + 1} \cdot \color{blue}{\sqrt{\left|x\right| - 1}}}\]