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 0.8
\[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(\left(\log \left(\frac{4}{\pi}\right) + \frac{1}{48} \cdot \left({f}^{2} \cdot {\pi}^{2}\right)\right) - \left(\frac{7}{23040} \cdot \left({f}^{4} \cdot {\pi}^{4}\right) + \log f\right)\right)}\]
- Using strategy
rm Applied add-sqr-sqrt1.3
\[\leadsto -\color{blue}{\left(\sqrt{\frac{1}{\frac{\pi}{4}}} \cdot \sqrt{\frac{1}{\frac{\pi}{4}}}\right)} \cdot \left(\left(\log \left(\frac{4}{\pi}\right) + \frac{1}{48} \cdot \left({f}^{2} \cdot {\pi}^{2}\right)\right) - \left(\frac{7}{23040} \cdot \left({f}^{4} \cdot {\pi}^{4}\right) + \log f\right)\right)\]
Applied associate-*l*0.9
\[\leadsto -\color{blue}{\sqrt{\frac{1}{\frac{\pi}{4}}} \cdot \left(\sqrt{\frac{1}{\frac{\pi}{4}}} \cdot \left(\left(\log \left(\frac{4}{\pi}\right) + \frac{1}{48} \cdot \left({f}^{2} \cdot {\pi}^{2}\right)\right) - \left(\frac{7}{23040} \cdot \left({f}^{4} \cdot {\pi}^{4}\right) + \log f\right)\right)\right)}\]
Simplified0.9
\[\leadsto -\color{blue}{\sqrt{\frac{4}{\pi}}} \cdot \left(\sqrt{\frac{1}{\frac{\pi}{4}}} \cdot \left(\left(\log \left(\frac{4}{\pi}\right) + \frac{1}{48} \cdot \left({f}^{2} \cdot {\pi}^{2}\right)\right) - \left(\frac{7}{23040} \cdot \left({f}^{4} \cdot {\pi}^{4}\right) + \log f\right)\right)\right)\]
Final simplification0.9
\[\leadsto \sqrt{\frac{4}{\pi}} \cdot \left(\left(-\sqrt{\frac{1}{\frac{\pi}{4}}}\right) \cdot \left(\left(\left({\pi}^{2} \cdot {f}^{2}\right) \cdot \frac{1}{48} + \log \left(\frac{4}{\pi}\right)\right) - \left(\log f + \left({f}^{4} \cdot {\pi}^{4}\right) \cdot \frac{7}{23040}\right)\right)\right)\]
