\(\frac{\frac{{\left({\left(\tan \left(x + \varepsilon\right)\right)}^3\right)}^2 - {\left({\left(\tan x\right)}^3\right)}^2}{{\left({\left(\tan \left(x + \varepsilon\right)\right)}^2\right)}^2 + \left({\left({\left(\log_* (1 + (e^{\tan x} - 1)^*)\right)}^2\right)}^2 + {\left(\tan \left(x + \varepsilon\right)\right)}^2 \cdot {\left(\log_* (1 + (e^{\tan x} - 1)^*)\right)}^2\right)}}{\tan x + \tan \left(x + \varepsilon\right)}\)
- Started with
\[\tan \left(x + \varepsilon\right) - \tan x\]
16.8
- Using strategy
rm 16.8
- Applied log1p-expm1-u to get
\[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\log_* (1 + (e^{\tan x} - 1)^*)}\]
16.9
- Using strategy
rm 16.9
- Applied flip-- to get
\[\color{red}{\tan \left(x + \varepsilon\right) - \log_* (1 + (e^{\tan x} - 1)^*)} \leadsto \color{blue}{\frac{{\left(\tan \left(x + \varepsilon\right)\right)}^2 - {\left(\log_* (1 + (e^{\tan x} - 1)^*)\right)}^2}{\tan \left(x + \varepsilon\right) + \log_* (1 + (e^{\tan x} - 1)^*)}}\]
16.9
- Applied simplify to get
\[\frac{{\left(\tan \left(x + \varepsilon\right)\right)}^2 - {\left(\log_* (1 + (e^{\tan x} - 1)^*)\right)}^2}{\color{red}{\tan \left(x + \varepsilon\right) + \log_* (1 + (e^{\tan x} - 1)^*)}} \leadsto \frac{{\left(\tan \left(x + \varepsilon\right)\right)}^2 - {\left(\log_* (1 + (e^{\tan x} - 1)^*)\right)}^2}{\color{blue}{\tan x + \tan \left(x + \varepsilon\right)}}\]
16.9
- Using strategy
rm 16.9
- Applied flip3-- to get
\[\frac{\color{red}{{\left(\tan \left(x + \varepsilon\right)\right)}^2 - {\left(\log_* (1 + (e^{\tan x} - 1)^*)\right)}^2}}{\tan x + \tan \left(x + \varepsilon\right)} \leadsto \frac{\color{blue}{\frac{{\left({\left(\tan \left(x + \varepsilon\right)\right)}^2\right)}^{3} - {\left({\left(\log_* (1 + (e^{\tan x} - 1)^*)\right)}^2\right)}^{3}}{{\left({\left(\tan \left(x + \varepsilon\right)\right)}^2\right)}^2 + \left({\left({\left(\log_* (1 + (e^{\tan x} - 1)^*)\right)}^2\right)}^2 + {\left(\tan \left(x + \varepsilon\right)\right)}^2 \cdot {\left(\log_* (1 + (e^{\tan x} - 1)^*)\right)}^2\right)}}}{\tan x + \tan \left(x + \varepsilon\right)}\]
20.1
- Applied simplify to get
\[\frac{\frac{\color{red}{{\left({\left(\tan \left(x + \varepsilon\right)\right)}^2\right)}^{3} - {\left({\left(\log_* (1 + (e^{\tan x} - 1)^*)\right)}^2\right)}^{3}}}{{\left({\left(\tan \left(x + \varepsilon\right)\right)}^2\right)}^2 + \left({\left({\left(\log_* (1 + (e^{\tan x} - 1)^*)\right)}^2\right)}^2 + {\left(\tan \left(x + \varepsilon\right)\right)}^2 \cdot {\left(\log_* (1 + (e^{\tan x} - 1)^*)\right)}^2\right)}}{\tan x + \tan \left(x + \varepsilon\right)} \leadsto \frac{\frac{\color{blue}{{\left({\left(\tan \left(x + \varepsilon\right)\right)}^3\right)}^2 - {\left({\left(\tan x\right)}^3\right)}^2}}{{\left({\left(\tan \left(x + \varepsilon\right)\right)}^2\right)}^2 + \left({\left({\left(\log_* (1 + (e^{\tan x} - 1)^*)\right)}^2\right)}^2 + {\left(\tan \left(x + \varepsilon\right)\right)}^2 \cdot {\left(\log_* (1 + (e^{\tan x} - 1)^*)\right)}^2\right)}}{\tan x + \tan \left(x + \varepsilon\right)}\]
16.9
