Initial program 59.7
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
Taylor expanded around 0 2.2
\[\leadsto -\color{blue}{\left(\left(4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi} + \frac{1}{12} \cdot \left({f}^{2} \cdot \pi\right)\right) - \left(4 \cdot \frac{\log f}{\pi} + \frac{7}{5760} \cdot \left({f}^{4} \cdot {\pi}^{3}\right)\right)\right)}\]
Simplified2.2
\[\leadsto -\color{blue}{\left(\left(\left(\frac{1}{12} \cdot \pi\right) \cdot \left(f \cdot f\right) + {f}^{4} \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\frac{-7}{5760} \cdot \pi\right)\right)\right) - \frac{4}{\pi} \cdot \left(\log f - \log \left(\frac{4}{\pi}\right)\right)\right)}\]
- Using strategy
rm Applied log-div2.2
\[\leadsto -\left(\left(\left(\frac{1}{12} \cdot \pi\right) \cdot \left(f \cdot f\right) + {f}^{4} \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\frac{-7}{5760} \cdot \pi\right)\right)\right) - \frac{4}{\pi} \cdot \left(\log f - \color{blue}{\left(\log 4 - \log \pi\right)}\right)\right)\]
Applied associate--r-2.3
\[\leadsto -\left(\left(\left(\frac{1}{12} \cdot \pi\right) \cdot \left(f \cdot f\right) + {f}^{4} \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\frac{-7}{5760} \cdot \pi\right)\right)\right) - \frac{4}{\pi} \cdot \color{blue}{\left(\left(\log f - \log 4\right) + \log \pi\right)}\right)\]
Applied distribute-lft-in2.3
\[\leadsto -\left(\left(\left(\frac{1}{12} \cdot \pi\right) \cdot \left(f \cdot f\right) + {f}^{4} \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\frac{-7}{5760} \cdot \pi\right)\right)\right) - \color{blue}{\left(\frac{4}{\pi} \cdot \left(\log f - \log 4\right) + \frac{4}{\pi} \cdot \log \pi\right)}\right)\]
- Using strategy
rm Applied *-commutative2.3
\[\leadsto -\left(\left(\left(\frac{1}{12} \cdot \pi\right) \cdot \left(f \cdot f\right) + {f}^{4} \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\frac{-7}{5760} \cdot \pi\right)\right)\right) - \left(\color{blue}{\left(\log f - \log 4\right) \cdot \frac{4}{\pi}} + \frac{4}{\pi} \cdot \log \pi\right)\right)\]
- Using strategy
rm Applied *-commutative2.3
\[\leadsto -\left(\left(\left(\frac{1}{12} \cdot \pi\right) \cdot \left(f \cdot f\right) + {f}^{4} \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\frac{-7}{5760} \cdot \pi\right)\right)\right) - \left(\color{blue}{\frac{4}{\pi} \cdot \left(\log f - \log 4\right)} + \frac{4}{\pi} \cdot \log \pi\right)\right)\]
Final simplification2.3
\[\leadsto -\left(\left(\left(f \cdot f\right) \cdot \left(\frac{1}{12} \cdot \pi\right) + \left(\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \frac{-7}{5760}\right)\right) \cdot {f}^{4}\right) - \left(\frac{4}{\pi} \cdot \left(\log f - \log 4\right) + \frac{4}{\pi} \cdot \log \pi\right)\right)\]
