Initial program 59.6
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
Taylor expanded around 0 2.3
\[\leadsto -\color{blue}{\left(\left(4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi} + \frac{1}{12} \cdot \left({f}^{2} \cdot \pi\right)\right) - \left(4 \cdot \frac{\log f}{\pi} + \frac{7}{5760} \cdot \left({f}^{4} \cdot {\pi}^{3}\right)\right)\right)}\]
Simplified2.3
\[\leadsto -\color{blue}{\left(\left(\left(\pi \cdot f\right) \cdot \left(f \cdot \frac{1}{12}\right) - \left(\left(\frac{7}{5760} \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot {f}^{4}\right) + \frac{4}{\pi} \cdot \left(\log \left(\frac{4}{\pi}\right) - \log f\right)\right)}\]
- Using strategy
rm Applied div-inv2.3
\[\leadsto -\left(\left(\left(\pi \cdot f\right) \cdot \left(f \cdot \frac{1}{12}\right) - \left(\left(\frac{7}{5760} \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot {f}^{4}\right) + \frac{4}{\pi} \cdot \left(\log \color{blue}{\left(4 \cdot \frac{1}{\pi}\right)} - \log f\right)\right)\]
Applied log-prod2.3
\[\leadsto -\left(\left(\left(\pi \cdot f\right) \cdot \left(f \cdot \frac{1}{12}\right) - \left(\left(\frac{7}{5760} \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot {f}^{4}\right) + \frac{4}{\pi} \cdot \left(\color{blue}{\left(\log 4 + \log \left(\frac{1}{\pi}\right)\right)} - \log f\right)\right)\]
Applied associate--l+2.4
\[\leadsto -\left(\left(\left(\pi \cdot f\right) \cdot \left(f \cdot \frac{1}{12}\right) - \left(\left(\frac{7}{5760} \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot {f}^{4}\right) + \frac{4}{\pi} \cdot \color{blue}{\left(\log 4 + \left(\log \left(\frac{1}{\pi}\right) - \log f\right)\right)}\right)\]
Applied distribute-lft-in2.4
\[\leadsto -\left(\left(\left(\pi \cdot f\right) \cdot \left(f \cdot \frac{1}{12}\right) - \left(\left(\frac{7}{5760} \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot {f}^{4}\right) + \color{blue}{\left(\frac{4}{\pi} \cdot \log 4 + \frac{4}{\pi} \cdot \left(\log \left(\frac{1}{\pi}\right) - \log f\right)\right)}\right)\]
Simplified2.4
\[\leadsto -\left(\left(\left(\pi \cdot f\right) \cdot \left(f \cdot \frac{1}{12}\right) - \left(\left(\frac{7}{5760} \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot {f}^{4}\right) + \left(\frac{4}{\pi} \cdot \log 4 + \color{blue}{\frac{-4}{\pi} \cdot \left(\log f + \log \pi\right)}\right)\right)\]
Taylor expanded around 0 2.3
\[\leadsto -\left(\left(\left(\pi \cdot f\right) \cdot \left(f \cdot \frac{1}{12}\right) - \left(\left(\frac{7}{5760} \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot {f}^{4}\right) + \left(\frac{4}{\pi} \cdot \log 4 + \color{blue}{-4 \cdot \frac{\log f + \log \pi}{\pi}}\right)\right)\]
Final simplification2.3
\[\leadsto \left(\log 4 \cdot \frac{-4}{\pi} + 4 \cdot \frac{\log \pi + \log f}{\pi}\right) + \left(-\left(\left(\frac{1}{12} \cdot f\right) \cdot \left(\pi \cdot f\right) - {f}^{4} \cdot \left(\left(\pi \cdot \frac{7}{5760}\right) \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\]
