Initial program 47.3
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
- Using strategy
rm Applied add-cbrt-cube48.9
\[\leadsto \frac{2}{\color{blue}{\sqrt[3]{\left(\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right) \cdot \left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)\right) \cdot \left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}}}\]
Applied simplify35.9
\[\leadsto \frac{2}{\sqrt[3]{\color{blue}{{\left(\left(\tan k \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*\right) \cdot \frac{\sin k \cdot t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}\right)}^{3}}}}\]
Taylor expanded around inf 29.5
\[\leadsto \frac{2}{\sqrt[3]{{\color{blue}{\left(\frac{{k}^{2} \cdot \left(t \cdot {\left(\sin k\right)}^{2}\right)}{{\ell}^{2} \cdot \cos k}\right)}}^{3}}}\]
Applied simplify9.5
\[\leadsto \color{blue}{\frac{\frac{\frac{2}{\frac{k}{\ell}}}{\frac{k}{\ell}}}{\frac{t}{\cos k} \cdot \left(\sin k \cdot \sin k\right)}}\]
- Using strategy
rm Applied div-inv9.5
\[\leadsto \frac{\color{blue}{\frac{2}{\frac{k}{\ell}} \cdot \frac{1}{\frac{k}{\ell}}}}{\frac{t}{\cos k} \cdot \left(\sin k \cdot \sin k\right)}\]
Applied times-frac3.3
\[\leadsto \color{blue}{\frac{\frac{2}{\frac{k}{\ell}}}{\frac{t}{\cos k}} \cdot \frac{\frac{1}{\frac{k}{\ell}}}{\sin k \cdot \sin k}}\]
Applied simplify3.2
\[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\frac{t}{\cos k}} \cdot \color{blue}{\frac{\frac{\ell}{k}}{\sin k \cdot \sin k}}\]
- Using strategy
rm Applied add-cube-cbrt3.6
\[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\frac{t}{\cos k}} \cdot \frac{\color{blue}{\left(\sqrt[3]{\frac{\ell}{k}} \cdot \sqrt[3]{\frac{\ell}{k}}\right) \cdot \sqrt[3]{\frac{\ell}{k}}}}{\sin k \cdot \sin k}\]
Applied times-frac3.1
\[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\frac{t}{\cos k}} \cdot \color{blue}{\left(\frac{\sqrt[3]{\frac{\ell}{k}} \cdot \sqrt[3]{\frac{\ell}{k}}}{\sin k} \cdot \frac{\sqrt[3]{\frac{\ell}{k}}}{\sin k}\right)}\]
Applied associate-*r*1.4
\[\leadsto \color{blue}{\left(\frac{\frac{2}{\frac{k}{\ell}}}{\frac{t}{\cos k}} \cdot \frac{\sqrt[3]{\frac{\ell}{k}} \cdot \sqrt[3]{\frac{\ell}{k}}}{\sin k}\right) \cdot \frac{\sqrt[3]{\frac{\ell}{k}}}{\sin k}}\]
