Initial program 59.6
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
Initial simplification59.6
\[\leadsto \log \left(\frac{e^{\frac{-\pi}{\frac{4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{\frac{-\pi}{\frac{4}{f}}}}\right) \cdot \frac{-4}{\pi}\]
Taylor expanded around 0 0.8
\[\leadsto \log \left(\frac{e^{\frac{-\pi}{\frac{4}{f}}} + e^{\frac{\pi}{4} \cdot f}}{\color{blue}{\frac{1}{2} \cdot \left(f \cdot \pi\right) + \left(\frac{1}{192} \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \frac{1}{61440} \cdot \left({f}^{5} \cdot {\pi}^{5}\right)\right)}}\right) \cdot \frac{-4}{\pi}\]
Taylor expanded around 0 0.8
\[\leadsto \color{blue}{\left(4 \cdot \frac{\log f}{\pi} + \frac{7}{5760} \cdot \left({f}^{4} \cdot {\pi}^{3}\right)\right) - \left(4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi} + \frac{1}{12} \cdot \left({f}^{2} \cdot \pi\right)\right)}\]
Simplified0.8
\[\leadsto \color{blue}{\frac{4}{\pi} \cdot \left(\log f - \log \left(\frac{4}{\pi}\right)\right) - \left(\left(\pi \cdot f\right) \cdot \left(\frac{1}{12} \cdot f\right) - \left(\pi \cdot \pi\right) \cdot \left({f}^{4} \cdot \left(\pi \cdot \frac{7}{5760}\right)\right)\right)}\]
- Using strategy
rm Applied log-div0.8
\[\leadsto \frac{4}{\pi} \cdot \left(\log f - \color{blue}{\left(\log 4 - \log \pi\right)}\right) - \left(\left(\pi \cdot f\right) \cdot \left(\frac{1}{12} \cdot f\right) - \left(\pi \cdot \pi\right) \cdot \left({f}^{4} \cdot \left(\pi \cdot \frac{7}{5760}\right)\right)\right)\]
Applied associate--r-0.9
\[\leadsto \frac{4}{\pi} \cdot \color{blue}{\left(\left(\log f - \log 4\right) + \log \pi\right)} - \left(\left(\pi \cdot f\right) \cdot \left(\frac{1}{12} \cdot f\right) - \left(\pi \cdot \pi\right) \cdot \left({f}^{4} \cdot \left(\pi \cdot \frac{7}{5760}\right)\right)\right)\]
Applied distribute-lft-in0.9
\[\leadsto \color{blue}{\left(\frac{4}{\pi} \cdot \left(\log f - \log 4\right) + \frac{4}{\pi} \cdot \log \pi\right)} - \left(\left(\pi \cdot f\right) \cdot \left(\frac{1}{12} \cdot f\right) - \left(\pi \cdot \pi\right) \cdot \left({f}^{4} \cdot \left(\pi \cdot \frac{7}{5760}\right)\right)\right)\]
Final simplification0.9
\[\leadsto \left(\frac{4}{\pi} \cdot \log \pi + \left(\log f - \log 4\right) \cdot \frac{4}{\pi}\right) - \left(\left(f \cdot \pi\right) \cdot \left(\frac{1}{12} \cdot f\right) - \left(\pi \cdot \pi\right) \cdot \left({f}^{4} \cdot \left(\frac{7}{5760} \cdot \pi\right)\right)\right)\]
