Initial program 47.0
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
- Using strategy
rm Applied add-cbrt-cube48.6
\[\leadsto \frac{2}{\color{blue}{\sqrt[3]{\left(\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right) \cdot \left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)\right) \cdot \left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}}}\]
Applied simplify35.3
\[\leadsto \frac{2}{\sqrt[3]{\color{blue}{{\left(\left(\tan k \cdot \left(\frac{k}{t} \cdot \frac{k}{t}\right)\right) \cdot \frac{\sin k \cdot t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}\right)}^{3}}}}\]
Taylor expanded around -inf 59.8
\[\leadsto \frac{2}{\color{blue}{e^{\left(\log \left(-1 \cdot \frac{{\left(\sin k\right)}^{2}}{\cos k}\right) + 2 \cdot \log \left(\frac{-1}{\ell}\right)\right) - \left(\log \left(\frac{-1}{t}\right) + 2 \cdot \log \left(\frac{-1}{k}\right)\right)}}}\]
Applied simplify9.8
\[\leadsto \color{blue}{\frac{-2 \cdot \frac{\frac{\cos k}{\sin k}}{\sin k}}{\frac{\frac{\frac{-1}{\ell}}{\frac{-1}{k}} \cdot \frac{\frac{-1}{\ell}}{\frac{-1}{k}}}{\frac{-1}{t}}}}\]
- Using strategy
rm Applied *-un-lft-identity9.8
\[\leadsto \frac{-2 \cdot \frac{\frac{\cos k}{\sin k}}{\sin k}}{\frac{\frac{\frac{-1}{\ell}}{\frac{-1}{k}} \cdot \frac{\frac{-1}{\ell}}{\frac{-1}{k}}}{\color{blue}{1 \cdot \frac{-1}{t}}}}\]
Applied times-frac4.6
\[\leadsto \frac{-2 \cdot \frac{\frac{\cos k}{\sin k}}{\sin k}}{\color{blue}{\frac{\frac{\frac{-1}{\ell}}{\frac{-1}{k}}}{1} \cdot \frac{\frac{\frac{-1}{\ell}}{\frac{-1}{k}}}{\frac{-1}{t}}}}\]
Applied times-frac3.5
\[\leadsto \color{blue}{\frac{-2}{\frac{\frac{\frac{-1}{\ell}}{\frac{-1}{k}}}{1}} \cdot \frac{\frac{\frac{\cos k}{\sin k}}{\sin k}}{\frac{\frac{\frac{-1}{\ell}}{\frac{-1}{k}}}{\frac{-1}{t}}}}\]
Applied simplify3.4
\[\leadsto \color{blue}{\frac{-2}{\frac{k}{\ell}}} \cdot \frac{\frac{\frac{\cos k}{\sin k}}{\sin k}}{\frac{\frac{\frac{-1}{\ell}}{\frac{-1}{k}}}{\frac{-1}{t}}}\]
Applied simplify1.4
\[\leadsto \frac{-2}{\frac{k}{\ell}} \cdot \color{blue}{\frac{\frac{\frac{\cos k}{\sin k}}{\frac{k}{-\ell}}}{\sin k \cdot t}}\]
- Using strategy
rm Applied *-un-lft-identity1.4
\[\leadsto \frac{-2}{\frac{k}{\ell}} \cdot \frac{\frac{\frac{\cos k}{\sin k}}{\color{blue}{1 \cdot \frac{k}{-\ell}}}}{\sin k \cdot t}\]
Applied *-un-lft-identity1.4
\[\leadsto \frac{-2}{\frac{k}{\ell}} \cdot \frac{\frac{\color{blue}{1 \cdot \frac{\cos k}{\sin k}}}{1 \cdot \frac{k}{-\ell}}}{\sin k \cdot t}\]
Applied times-frac1.4
\[\leadsto \frac{-2}{\frac{k}{\ell}} \cdot \frac{\color{blue}{\frac{1}{1} \cdot \frac{\frac{\cos k}{\sin k}}{\frac{k}{-\ell}}}}{\sin k \cdot t}\]
Applied times-frac1.2
\[\leadsto \frac{-2}{\frac{k}{\ell}} \cdot \color{blue}{\left(\frac{\frac{1}{1}}{\sin k} \cdot \frac{\frac{\frac{\cos k}{\sin k}}{\frac{k}{-\ell}}}{t}\right)}\]
Applied associate-*r*0.5
\[\leadsto \color{blue}{\left(\frac{-2}{\frac{k}{\ell}} \cdot \frac{\frac{1}{1}}{\sin k}\right) \cdot \frac{\frac{\frac{\cos k}{\sin k}}{\frac{k}{-\ell}}}{t}}\]
Applied simplify0.4
\[\leadsto \color{blue}{\frac{-2}{\sin k \cdot \frac{k}{\ell}}} \cdot \frac{\frac{\frac{\cos k}{\sin k}}{\frac{k}{-\ell}}}{t}\]
