Initial program 59.5
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
Taylor expanded around 0 2.3
\[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{\frac{1}{2} \cdot \left(f \cdot \pi\right) + \left(\frac{1}{192} \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \frac{1}{61440} \cdot \left({f}^{5} \cdot {\pi}^{5}\right)\right)}}\right)\]
- Using strategy
rm Applied *-un-lft-identity2.3
\[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(1 \cdot \frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\frac{1}{2} \cdot \left(f \cdot \pi\right) + \left(\frac{1}{192} \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \frac{1}{61440} \cdot \left({f}^{5} \cdot {\pi}^{5}\right)\right)}\right)}\]
Applied log-prod2.3
\[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(\log 1 + \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\frac{1}{2} \cdot \left(f \cdot \pi\right) + \left(\frac{1}{192} \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \frac{1}{61440} \cdot \left({f}^{5} \cdot {\pi}^{5}\right)\right)}\right)\right)}\]
Applied distribute-lft-in2.3
\[\leadsto -\color{blue}{\left(\frac{1}{\frac{\pi}{4}} \cdot \log 1 + \frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\frac{1}{2} \cdot \left(f \cdot \pi\right) + \left(\frac{1}{192} \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \frac{1}{61440} \cdot \left({f}^{5} \cdot {\pi}^{5}\right)\right)}\right)\right)}\]
Simplified2.2
\[\leadsto -\left(\frac{1}{\frac{\pi}{4}} \cdot \log 1 + \color{blue}{\frac{\log \left(\frac{e^{\left(-f\right) \cdot \frac{\pi}{4}} + e^{\frac{f \cdot \pi}{4}}}{(\pi \cdot \left((\left(\pi \cdot \pi\right) \cdot \left(\left(\frac{1}{192} \cdot f\right) \cdot \left(f \cdot f\right)\right) + \left(f \cdot \frac{1}{2}\right))_*\right) + \left({f}^{5} \cdot \left({\pi}^{5} \cdot \frac{1}{61440}\right)\right))_*}\right)}{\frac{\pi}{4}}}\right)\]
Taylor expanded around 0 2.2
\[\leadsto -\left(\frac{1}{\frac{\pi}{4}} \cdot \log 1 + \frac{\color{blue}{\left(\log \left(\frac{4}{\pi}\right) + \frac{1}{48} \cdot \left({f}^{2} \cdot {\pi}^{2}\right)\right) - \left(\frac{7}{23040} \cdot \left({f}^{4} \cdot {\pi}^{4}\right) + \log f\right)}}{\frac{\pi}{4}}\right)\]
Simplified2.2
\[\leadsto -\left(\frac{1}{\frac{\pi}{4}} \cdot \log 1 + \frac{\color{blue}{(\frac{1}{48} \cdot \left(\left(f \cdot \pi\right) \cdot \left(f \cdot \pi\right)\right) + \left(\log \left(\frac{4}{\pi}\right)\right))_* - (\left({f}^{4} \cdot \frac{7}{23040}\right) \cdot \left({\pi}^{4}\right) + \left(\log f\right))_*}}{\frac{\pi}{4}}\right)\]
Final simplification2.2
\[\leadsto \left(-\frac{(\frac{1}{48} \cdot \left(\left(f \cdot \pi\right) \cdot \left(f \cdot \pi\right)\right) + \left(\log \left(\frac{4}{\pi}\right)\right))_* - (\left(\frac{7}{23040} \cdot {f}^{4}\right) \cdot \left({\pi}^{4}\right) + \left(\log f\right))_*}{\frac{\pi}{4}}\right) + \frac{-1}{\frac{\pi}{4}} \cdot 0\]
