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 2.2
\[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{\frac{1}{2} \cdot \left(f \cdot \pi\right) + \left(\frac{1}{192} \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \frac{1}{61440} \cdot \left({f}^{5} \cdot {\pi}^{5}\right)\right)}}\right)\]
- Using strategy
rm Applied *-un-lft-identity2.2
\[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(1 \cdot \frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\frac{1}{2} \cdot \left(f \cdot \pi\right) + \left(\frac{1}{192} \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \frac{1}{61440} \cdot \left({f}^{5} \cdot {\pi}^{5}\right)\right)}\right)}\]
Applied log-prod2.2
\[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(\log 1 + \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\frac{1}{2} \cdot \left(f \cdot \pi\right) + \left(\frac{1}{192} \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \frac{1}{61440} \cdot \left({f}^{5} \cdot {\pi}^{5}\right)\right)}\right)\right)}\]
Applied distribute-lft-in2.2
\[\leadsto -\color{blue}{\left(\frac{1}{\frac{\pi}{4}} \cdot \log 1 + \frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\frac{1}{2} \cdot \left(f \cdot \pi\right) + \left(\frac{1}{192} \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \frac{1}{61440} \cdot \left({f}^{5} \cdot {\pi}^{5}\right)\right)}\right)\right)}\]
Simplified2.1
\[\leadsto -\left(\frac{1}{\frac{\pi}{4}} \cdot \log 1 + \color{blue}{\frac{\log \left(\frac{e^{\left(-f\right) \cdot \frac{\pi}{4}} + e^{\frac{f \cdot \pi}{4}}}{(\pi \cdot \left((\left(\pi \cdot \pi\right) \cdot \left(\left(\frac{1}{192} \cdot f\right) \cdot \left(f \cdot f\right)\right) + \left(f \cdot \frac{1}{2}\right))_*\right) + \left({f}^{5} \cdot \left({\pi}^{5} \cdot \frac{1}{61440}\right)\right))_*}\right)}{\frac{\pi}{4}}}\right)\]
- Using strategy
rm Applied *-un-lft-identity2.1
\[\leadsto -\left(\frac{1}{\frac{\pi}{4}} \cdot \log 1 + \frac{\log \left(\frac{e^{\left(-f\right) \cdot \frac{\pi}{4}} + e^{\frac{f \cdot \pi}{4}}}{\color{blue}{1 \cdot (\pi \cdot \left((\left(\pi \cdot \pi\right) \cdot \left(\left(\frac{1}{192} \cdot f\right) \cdot \left(f \cdot f\right)\right) + \left(f \cdot \frac{1}{2}\right))_*\right) + \left({f}^{5} \cdot \left({\pi}^{5} \cdot \frac{1}{61440}\right)\right))_*}}\right)}{\frac{\pi}{4}}\right)\]
Applied add-sqr-sqrt2.1
\[\leadsto -\left(\frac{1}{\frac{\pi}{4}} \cdot \log 1 + \frac{\log \left(\frac{\color{blue}{\sqrt{e^{\left(-f\right) \cdot \frac{\pi}{4}} + e^{\frac{f \cdot \pi}{4}}} \cdot \sqrt{e^{\left(-f\right) \cdot \frac{\pi}{4}} + e^{\frac{f \cdot \pi}{4}}}}}{1 \cdot (\pi \cdot \left((\left(\pi \cdot \pi\right) \cdot \left(\left(\frac{1}{192} \cdot f\right) \cdot \left(f \cdot f\right)\right) + \left(f \cdot \frac{1}{2}\right))_*\right) + \left({f}^{5} \cdot \left({\pi}^{5} \cdot \frac{1}{61440}\right)\right))_*}\right)}{\frac{\pi}{4}}\right)\]
Applied times-frac2.1
\[\leadsto -\left(\frac{1}{\frac{\pi}{4}} \cdot \log 1 + \frac{\log \color{blue}{\left(\frac{\sqrt{e^{\left(-f\right) \cdot \frac{\pi}{4}} + e^{\frac{f \cdot \pi}{4}}}}{1} \cdot \frac{\sqrt{e^{\left(-f\right) \cdot \frac{\pi}{4}} + e^{\frac{f \cdot \pi}{4}}}}{(\pi \cdot \left((\left(\pi \cdot \pi\right) \cdot \left(\left(\frac{1}{192} \cdot f\right) \cdot \left(f \cdot f\right)\right) + \left(f \cdot \frac{1}{2}\right))_*\right) + \left({f}^{5} \cdot \left({\pi}^{5} \cdot \frac{1}{61440}\right)\right))_*}\right)}}{\frac{\pi}{4}}\right)\]
Applied log-prod2.1
\[\leadsto -\left(\frac{1}{\frac{\pi}{4}} \cdot \log 1 + \frac{\color{blue}{\log \left(\frac{\sqrt{e^{\left(-f\right) \cdot \frac{\pi}{4}} + e^{\frac{f \cdot \pi}{4}}}}{1}\right) + \log \left(\frac{\sqrt{e^{\left(-f\right) \cdot \frac{\pi}{4}} + e^{\frac{f \cdot \pi}{4}}}}{(\pi \cdot \left((\left(\pi \cdot \pi\right) \cdot \left(\left(\frac{1}{192} \cdot f\right) \cdot \left(f \cdot f\right)\right) + \left(f \cdot \frac{1}{2}\right))_*\right) + \left({f}^{5} \cdot \left({\pi}^{5} \cdot \frac{1}{61440}\right)\right))_*}\right)}}{\frac{\pi}{4}}\right)\]
Simplified2.1
\[\leadsto -\left(\frac{1}{\frac{\pi}{4}} \cdot \log 1 + \frac{\color{blue}{\log \left(\sqrt{e^{\left(-f\right) \cdot \frac{\pi}{4}} + {\left(e^{f}\right)}^{\left(\frac{\pi}{4}\right)}}\right)} + \log \left(\frac{\sqrt{e^{\left(-f\right) \cdot \frac{\pi}{4}} + e^{\frac{f \cdot \pi}{4}}}}{(\pi \cdot \left((\left(\pi \cdot \pi\right) \cdot \left(\left(\frac{1}{192} \cdot f\right) \cdot \left(f \cdot f\right)\right) + \left(f \cdot \frac{1}{2}\right))_*\right) + \left({f}^{5} \cdot \left({\pi}^{5} \cdot \frac{1}{61440}\right)\right))_*}\right)}{\frac{\pi}{4}}\right)\]
Final simplification2.1
\[\leadsto 0 \cdot \frac{1}{\frac{\pi}{4}} + \left(-\frac{\log \left(\frac{\sqrt{e^{\frac{\pi \cdot f}{4}} + e^{\left(-f\right) \cdot \frac{\pi}{4}}}}{(\pi \cdot \left((\left(\pi \cdot \pi\right) \cdot \left(\left(f \cdot f\right) \cdot \left(\frac{1}{192} \cdot f\right)\right) + \left(\frac{1}{2} \cdot f\right))_*\right) + \left({f}^{5} \cdot \left({\pi}^{5} \cdot \frac{1}{61440}\right)\right))_*}\right) + \log \left(\sqrt{{\left(e^{f}\right)}^{\left(\frac{\pi}{4}\right)} + e^{\left(-f\right) \cdot \frac{\pi}{4}}}\right)}{\frac{\pi}{4}}\right)\]
