if beta < 4.731717689114216e+137

1. Started with
   \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
   50.6
2. Applied simplify to get
   \[\color{red}{\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}} \leadsto \color{blue}{\frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}}\]
   35.3
3. Applied taylor to get
   \[\frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0} \leadsto \frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{\left(2 \cdot \left(\beta \cdot (i * 2 + \alpha)_*\right) + \left({\left((i * 2 + \alpha)_*\right)}^2 + {\beta}^2\right)\right) - 1.0}\]
   35.3
4. Taylor expanded around 0 to get
   \[\frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{\color{red}{\left(2 \cdot \left(\beta \cdot (i * 2 + \alpha)_*\right) + \left({\left((i * 2 + \alpha)_*\right)}^2 + {\beta}^2\right)\right)} - 1.0} \leadsto \frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{\color{blue}{\left(2 \cdot \left(\beta \cdot (i * 2 + \alpha)_*\right) + \left({\left((i * 2 + \alpha)_*\right)}^2 + {\beta}^2\right)\right)} - 1.0}\]
   35.3
5. Applied simplify to get
   \[\color{red}{\frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{\left(2 \cdot \left(\beta \cdot (i * 2 + \alpha)_*\right) + \left({\left((i * 2 + \alpha)_*\right)}^2 + {\beta}^2\right)\right) - 1.0}} \leadsto \color{blue}{\frac{\frac{i}{(i * 2 + \alpha)_* + \beta} \cdot \frac{\left(\alpha + i\right) + \beta}{(i * 2 + \alpha)_* + \beta}}{\frac{(\left((i * 2 + \alpha)_*\right) * \left((\beta * 2 + \left((i * 2 + \alpha)_*\right))_*\right) + \left({\beta}^2 - 1.0\right))_*}{(i * \left(\left(\alpha + i\right) + \beta\right) + \left(\alpha \cdot \beta\right))_*}}}\]
   35.3

if 4.731717689114216e+137 < beta

1. Started with
   \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
   63.0
2. Applied simplify to get
   \[\color{red}{\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}} \leadsto \color{blue}{\frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}}\]
   53.8
3. Using strategy rm
   53.8
4. Applied div-inv to get
   \[\color{red}{\frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}} \leadsto \color{blue}{\left(\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}\right) \cdot \frac{1}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}}\]
   53.8
5. Applied taylor to get
   \[\left(\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}\right) \cdot \frac{1}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0} \leadsto \left(\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}\right) \cdot \left(\left(\frac{1}{{\beta}^2} + \left(1.0 \cdot \frac{1}{{\beta}^{4}} + 3 \cdot \frac{{\left((\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*\right)}^2}{{\beta}^{4}}\right)\right) - 2 \cdot \frac{(\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*}{{\beta}^{3}}\right)\]
   56.2
6. Taylor expanded around -inf to get
   \[\left(\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}\right) \cdot \color{red}{\left(\left(\frac{1}{{\beta}^2} + \left(1.0 \cdot \frac{1}{{\beta}^{4}} + 3 \cdot \frac{{\left((\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*\right)}^2}{{\beta}^{4}}\right)\right) - 2 \cdot \frac{(\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*}{{\beta}^{3}}\right)} \leadsto \left(\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}\right) \cdot \color{blue}{\left(\left(\frac{1}{{\beta}^2} + \left(1.0 \cdot \frac{1}{{\beta}^{4}} + 3 \cdot \frac{{\left((\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*\right)}^2}{{\beta}^{4}}\right)\right) - 2 \cdot \frac{(\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*}{{\beta}^{3}}\right)}\]
   56.2
7. Applied taylor to get
   \[\left(\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}\right) \cdot \left(\left(\frac{1}{{\beta}^2} + \left(1.0 \cdot \frac{1}{{\beta}^{4}} + 3 \cdot \frac{{\left((\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*\right)}^2}{{\beta}^{4}}\right)\right) - 2 \cdot \frac{(\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*}{{\beta}^{3}}\right) \leadsto \left(\frac{(\left(\frac{-1}{i}\right) * \left(-\left(\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_* \cdot i}{\beta} + \left(\frac{(\left(\frac{-1}{i}\right) * \left(-\left(\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_* \cdot {i}^2}{{\beta}^2} + \frac{(\left(\frac{-1}{i}\right) * \left(-\left(\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_* \cdot \left(\alpha \cdot i\right)}{{\beta}^2}\right)\right) \cdot \left(\left(\frac{1}{{\beta}^2} + \left(1.0 \cdot \frac{1}{{\beta}^{4}} + 3 \cdot \frac{{\left((\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*\right)}^2}{{\beta}^{4}}\right)\right) - 2 \cdot \frac{(\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*}{{\beta}^{3}}\right)\]
   44.4
8. Taylor expanded around -inf to get
   \[\color{red}{\left(\frac{(\left(\frac{-1}{i}\right) * \left(-\left(\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_* \cdot i}{\beta} + \left(\frac{(\left(\frac{-1}{i}\right) * \left(-\left(\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_* \cdot {i}^2}{{\beta}^2} + \frac{(\left(\frac{-1}{i}\right) * \left(-\left(\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_* \cdot \left(\alpha \cdot i\right)}{{\beta}^2}\right)\right)} \cdot \left(\left(\frac{1}{{\beta}^2} + \left(1.0 \cdot \frac{1}{{\beta}^{4}} + 3 \cdot \frac{{\left((\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*\right)}^2}{{\beta}^{4}}\right)\right) - 2 \cdot \frac{(\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*}{{\beta}^{3}}\right) \leadsto \color{blue}{\left(\frac{(\left(\frac{-1}{i}\right) * \left(-\left(\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_* \cdot i}{\beta} + \left(\frac{(\left(\frac{-1}{i}\right) * \left(-\left(\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_* \cdot {i}^2}{{\beta}^2} + \frac{(\left(\frac{-1}{i}\right) * \left(-\left(\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_* \cdot \left(\alpha \cdot i\right)}{{\beta}^2}\right)\right)} \cdot \left(\left(\frac{1}{{\beta}^2} + \left(1.0 \cdot \frac{1}{{\beta}^{4}} + 3 \cdot \frac{{\left((\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*\right)}^2}{{\beta}^{4}}\right)\right) - 2 \cdot \frac{(\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*}{{\beta}^{3}}\right)\]
   44.4
9. Applied simplify to get
   \[\left(\frac{(\left(\frac{-1}{i}\right) * \left(-\left(\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_* \cdot i}{\beta} + \left(\frac{(\left(\frac{-1}{i}\right) * \left(-\left(\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_* \cdot {i}^2}{{\beta}^2} + \frac{(\left(\frac{-1}{i}\right) * \left(-\left(\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_* \cdot \left(\alpha \cdot i\right)}{{\beta}^2}\right)\right) \cdot \left(\left(\frac{1}{{\beta}^2} + \left(1.0 \cdot \frac{1}{{\beta}^{4}} + 3 \cdot \frac{{\left((\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*\right)}^2}{{\beta}^{4}}\right)\right) - 2 \cdot \frac{(\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*}{{\beta}^{3}}\right) \leadsto (\left((\left(\frac{(\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*}{{\beta}^{4}} \cdot (\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*\right) * 3 + \left(\frac{1.0}{{\beta}^{4}} + \frac{\frac{1}{\beta}}{\beta}\right))_* - \frac{(\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*}{\frac{{\beta}^3}{2}}\right) * \left(\frac{(\left(\frac{-1}{i}\right) * \left(-\left(\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)\right)\right) + \left(\frac{1}{\alpha \cdot \beta}\right))_*}{\frac{\beta}{i} \cdot \frac{\beta}{i}} + \frac{(\left(\frac{-1}{i}\right) * \left(-\left(\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)\right)\right) + \left(\frac{1}{\alpha \cdot \beta}\right))_*}{\frac{\beta}{i}}\right) + \left(\frac{(\left(\frac{-1}{i}\right) * \left(-\left(\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)\right)\right) + \left(\frac{1}{\alpha \cdot \beta}\right))_*}{\frac{\beta}{i} \cdot \frac{\beta}{\alpha}} \cdot \left((\left(\frac{(\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*}{{\beta}^{4}} \cdot (\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*\right) * 3 + \left(\frac{1.0}{{\beta}^{4}} + \frac{\frac{1}{\beta}}{\beta}\right))_* - \frac{(\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*}{\frac{{\beta}^3}{2}}\right)\right))_*\]
   9.4

---

10. Applied final simplification
11. Applied simplify to get
    \[\color{red}{(\left((\left(\frac{(\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*}{{\beta}^{4}} \cdot (\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*\right) * 3 + \left(\frac{1.0}{{\beta}^{4}} + \frac{\frac{1}{\beta}}{\beta}\right))_* - \frac{(\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*}{\frac{{\beta}^3}{2}}\right) * \left(\frac{(\left(\frac{-1}{i}\right) * \left(-\left(\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)\right)\right) + \left(\frac{1}{\alpha \cdot \beta}\right))_*}{\frac{\beta}{i} \cdot \frac{\beta}{i}} + \frac{(\left(\frac{-1}{i}\right) * \left(-\left(\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)\right)\right) + \left(\frac{1}{\alpha \cdot \beta}\right))_*}{\frac{\beta}{i}}\right) + \left(\frac{(\left(\frac{-1}{i}\right) * \left(-\left(\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)\right)\right) + \left(\frac{1}{\alpha \cdot \beta}\right))_*}{\frac{\beta}{i} \cdot \frac{\beta}{\alpha}} \cdot \left((\left(\frac{(\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*}{{\beta}^{4}} \cdot (\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*\right) * 3 + \left(\frac{1.0}{{\beta}^{4}} + \frac{\frac{1}{\beta}}{\beta}\right))_* - \frac{(\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*}{\frac{{\beta}^3}{2}}\right)\right))_*} \leadsto \color{blue}{(\left((\left(\frac{(\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*}{\frac{{\beta}^{4}}{(\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*}}\right) * 3 + \left(\frac{\frac{1}{\beta}}{\beta} + \frac{1.0}{{\beta}^{4}}\right))_* - \frac{(\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*}{\frac{{\beta}^3}{2}}\right) * \left(\frac{(\left(\frac{-1}{i}\right) * \left(-\left(\left(\frac{1}{i} + \frac{1}{\beta}\right) + \frac{1}{\alpha}\right)\right) + \left(\frac{\frac{1}{\alpha}}{\beta}\right))_*}{{\left(\frac{\beta}{i}\right)}^2} + \frac{(\left(\frac{-1}{i}\right) * \left(-\left(\left(\frac{1}{i} + \frac{1}{\beta}\right) + \frac{1}{\alpha}\right)\right) + \left(\frac{\frac{1}{\alpha}}{\beta}\right))_*}{\frac{\beta}{i}}\right) + \left(\frac{(\left(\frac{(\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*}{\frac{{\beta}^{4}}{(\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*}}\right) * 3 + \left(\frac{\frac{1}{\beta}}{\beta} + \frac{1.0}{{\beta}^{4}}\right))_* - \frac{(\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*}{\frac{{\beta}^3}{2}}}{\frac{\frac{\frac{\beta}{\alpha}}{\frac{i}{\beta}}}{(\left(\frac{-1}{i}\right) * \left(-\left(\left(\frac{1}{i} + \frac{1}{\beta}\right) + \frac{1}{\alpha}\right)\right) + \left(\frac{\frac{1}{\alpha}}{\beta}\right))_*}}\right))_*}\]
    9.4