if N < 112873629.87382902

1. Started with
   \[\log \left(N + 1\right) - \log N\]
   31.1
2. Applied simplify to get
   \[\color{red}{\log \left(N + 1\right) - \log N} \leadsto \color{blue}{\log_* (1 + N) - \log N}\]
   31.1
3. Using strategy rm
   31.1
4. Applied log1p-udef to get
   \[\color{red}{\log_* (1 + N)} - \log N \leadsto \color{blue}{\log \left(1 + N\right)} - \log N\]
   31.1
5. Applied diff-log to get
   \[\color{red}{\log \left(1 + N\right) - \log N} \leadsto \color{blue}{\log \left(\frac{1 + N}{N}\right)}\]
   28.7

if 112873629.87382902 < N

1. Started with
   \[\log \left(N + 1\right) - \log N\]
   59.8
2. Applied simplify to get
   \[\color{red}{\log \left(N + 1\right) - \log N} \leadsto \color{blue}{\log_* (1 + N) - \log N}\]
   59.8
3. Using strategy rm
   59.8
4. Applied log1p-udef to get
   \[\color{red}{\log_* (1 + N)} - \log N \leadsto \color{blue}{\log \left(1 + N\right)} - \log N\]
   59.8
5. Applied diff-log to get
   \[\color{red}{\log \left(1 + N\right) - \log N} \leadsto \color{blue}{\log \left(\frac{1 + N}{N}\right)}\]
   59.6
6. Applied taylor to get
   \[\log \left(\frac{1 + N}{N}\right) \leadsto \left(\frac{1}{3} \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) - \frac{1}{2} \cdot \frac{1}{{N}^2}\]
   0.0
7. Taylor expanded around inf to get
   \[\color{red}{\left(\frac{1}{3} \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) - \frac{1}{2} \cdot \frac{1}{{N}^2}} \leadsto \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) - \frac{1}{2} \cdot \frac{1}{{N}^2}}\]
   0.0
8. Applied simplify to get
   \[\color{red}{\left(\frac{1}{3} \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) - \frac{1}{2} \cdot \frac{1}{{N}^2}} \leadsto \color{blue}{\frac{1}{N} - \frac{1}{N \cdot N} \cdot \left(\frac{1}{2} - \frac{\frac{1}{3}}{N}\right)}\]
   0.0
9. Applied simplify to get
   \[\frac{1}{N} - \color{red}{\frac{1}{N \cdot N} \cdot \left(\frac{1}{2} - \frac{\frac{1}{3}}{N}\right)} \leadsto \frac{1}{N} - \color{blue}{\frac{\frac{1}{2} - \frac{\frac{1}{3}}{N}}{N \cdot N}}\]
   0.0