Initial program 55.8
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
- Using strategy
rm Applied add-cbrt-cube57.6
\[\leadsto \frac{2}{\color{blue}{\sqrt[3]{\left(\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right) \cdot \left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)\right) \cdot \left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}}}\]
Applied simplify47.6
\[\leadsto \frac{2}{\sqrt[3]{\color{blue}{{\left(\left(\tan k \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*\right) \cdot \frac{\sin k \cdot t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}\right)}^{3}}}}\]
Taylor expanded around inf 50.6
\[\leadsto \frac{2}{\sqrt[3]{{\color{blue}{\left(\frac{{k}^{2} \cdot \left(t \cdot {\left(\sin k\right)}^{2}\right)}{{\ell}^{2} \cdot \cos k}\right)}}^{3}}}\]
Applied simplify7.8
\[\leadsto \color{blue}{\frac{\frac{\frac{2}{\frac{k}{\ell}}}{\frac{k}{\ell}}}{\frac{t}{\cos k} \cdot \left(\sin k \cdot \sin k\right)}}\]
- Using strategy
rm Applied add-cube-cbrt8.3
\[\leadsto \frac{\frac{\frac{2}{\frac{k}{\ell}}}{\color{blue}{\left(\sqrt[3]{\frac{k}{\ell}} \cdot \sqrt[3]{\frac{k}{\ell}}\right) \cdot \sqrt[3]{\frac{k}{\ell}}}}}{\frac{t}{\cos k} \cdot \left(\sin k \cdot \sin k\right)}\]
Applied *-un-lft-identity8.3
\[\leadsto \frac{\frac{\color{blue}{1 \cdot \frac{2}{\frac{k}{\ell}}}}{\left(\sqrt[3]{\frac{k}{\ell}} \cdot \sqrt[3]{\frac{k}{\ell}}\right) \cdot \sqrt[3]{\frac{k}{\ell}}}}{\frac{t}{\cos k} \cdot \left(\sin k \cdot \sin k\right)}\]
Applied times-frac8.3
\[\leadsto \frac{\color{blue}{\frac{1}{\sqrt[3]{\frac{k}{\ell}} \cdot \sqrt[3]{\frac{k}{\ell}}} \cdot \frac{\frac{2}{\frac{k}{\ell}}}{\sqrt[3]{\frac{k}{\ell}}}}}{\frac{t}{\cos k} \cdot \left(\sin k \cdot \sin k\right)}\]
Applied times-frac3.0
\[\leadsto \color{blue}{\frac{\frac{1}{\sqrt[3]{\frac{k}{\ell}} \cdot \sqrt[3]{\frac{k}{\ell}}}}{\frac{t}{\cos k}} \cdot \frac{\frac{\frac{2}{\frac{k}{\ell}}}{\sqrt[3]{\frac{k}{\ell}}}}{\sin k \cdot \sin k}}\]
Applied simplify3.0
\[\leadsto \color{blue}{\frac{\frac{\cos k}{\sqrt[3]{\frac{k}{\ell}}}}{t \cdot \sqrt[3]{\frac{k}{\ell}}}} \cdot \frac{\frac{\frac{2}{\frac{k}{\ell}}}{\sqrt[3]{\frac{k}{\ell}}}}{\sin k \cdot \sin k}\]
Initial program 44.3
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
- Using strategy
rm Applied add-cbrt-cube45.9
\[\leadsto \frac{2}{\color{blue}{\sqrt[3]{\left(\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right) \cdot \left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)\right) \cdot \left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}}}\]
Applied simplify31.5
\[\leadsto \frac{2}{\sqrt[3]{\color{blue}{{\left(\left(\tan k \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*\right) \cdot \frac{\sin k \cdot t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}\right)}^{3}}}}\]
Taylor expanded around inf 22.2
\[\leadsto \frac{2}{\sqrt[3]{{\color{blue}{\left(\frac{{k}^{2} \cdot \left(t \cdot {\left(\sin k\right)}^{2}\right)}{{\ell}^{2} \cdot \cos k}\right)}}^{3}}}\]
Applied simplify9.6
\[\leadsto \color{blue}{\frac{\frac{\frac{2}{\frac{k}{\ell}}}{\frac{k}{\ell}}}{\frac{t}{\cos k} \cdot \left(\sin k \cdot \sin k\right)}}\]
- Using strategy
rm Applied div-inv9.6
\[\leadsto \frac{\frac{\frac{2}{\frac{k}{\ell}}}{\color{blue}{k \cdot \frac{1}{\ell}}}}{\frac{t}{\cos k} \cdot \left(\sin k \cdot \sin k\right)}\]
Applied div-inv9.6
\[\leadsto \frac{\frac{\color{blue}{2 \cdot \frac{1}{\frac{k}{\ell}}}}{k \cdot \frac{1}{\ell}}}{\frac{t}{\cos k} \cdot \left(\sin k \cdot \sin k\right)}\]
Applied times-frac10.8
\[\leadsto \frac{\color{blue}{\frac{2}{k} \cdot \frac{\frac{1}{\frac{k}{\ell}}}{\frac{1}{\ell}}}}{\frac{t}{\cos k} \cdot \left(\sin k \cdot \sin k\right)}\]
Applied times-frac8.7
\[\leadsto \color{blue}{\frac{\frac{2}{k}}{\frac{t}{\cos k}} \cdot \frac{\frac{\frac{1}{\frac{k}{\ell}}}{\frac{1}{\ell}}}{\sin k \cdot \sin k}}\]
Applied simplify3.9
\[\leadsto \frac{\frac{2}{k}}{\frac{t}{\cos k}} \cdot \color{blue}{\frac{\frac{\frac{\ell}{k}}{\sin k}}{\frac{\sin k}{\ell}}}\]
Initial program 59.3
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
- Using strategy
rm Applied add-cbrt-cube59.8
\[\leadsto \frac{2}{\color{blue}{\sqrt[3]{\left(\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right) \cdot \left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)\right) \cdot \left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}}}\]
Applied simplify49.7
\[\leadsto \frac{2}{\sqrt[3]{\color{blue}{{\left(\left(\tan k \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*\right) \cdot \frac{\sin k \cdot t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}\right)}^{3}}}}\]
Taylor expanded around inf 58.7
\[\leadsto \frac{2}{\sqrt[3]{{\color{blue}{\left(\frac{{k}^{2} \cdot \left(t \cdot {\left(\sin k\right)}^{2}\right)}{{\ell}^{2} \cdot \cos k}\right)}}^{3}}}\]
Applied simplify10.3
\[\leadsto \color{blue}{\frac{\frac{\frac{2}{\frac{k}{\ell}}}{\frac{k}{\ell}}}{\frac{t}{\cos k} \cdot \left(\sin k \cdot \sin k\right)}}\]
- Using strategy
rm Applied *-un-lft-identity10.3
\[\leadsto \frac{\frac{\frac{2}{\frac{k}{\ell}}}{\color{blue}{1 \cdot \frac{k}{\ell}}}}{\frac{t}{\cos k} \cdot \left(\sin k \cdot \sin k\right)}\]
Applied add-cube-cbrt10.8
\[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{\frac{2}{\frac{k}{\ell}}} \cdot \sqrt[3]{\frac{2}{\frac{k}{\ell}}}\right) \cdot \sqrt[3]{\frac{2}{\frac{k}{\ell}}}}}{1 \cdot \frac{k}{\ell}}}{\frac{t}{\cos k} \cdot \left(\sin k \cdot \sin k\right)}\]
Applied times-frac10.8
\[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{\frac{2}{\frac{k}{\ell}}} \cdot \sqrt[3]{\frac{2}{\frac{k}{\ell}}}}{1} \cdot \frac{\sqrt[3]{\frac{2}{\frac{k}{\ell}}}}{\frac{k}{\ell}}}}{\frac{t}{\cos k} \cdot \left(\sin k \cdot \sin k\right)}\]
Applied times-frac3.1
\[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{\frac{2}{\frac{k}{\ell}}} \cdot \sqrt[3]{\frac{2}{\frac{k}{\ell}}}}{1}}{\frac{t}{\cos k}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\frac{k}{\ell}}}}{\frac{k}{\ell}}}{\sin k \cdot \sin k}}\]
Applied simplify3.1
\[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{2}{\frac{k}{\ell}}} \cdot \sqrt[3]{\frac{2}{\frac{k}{\ell}}}}{\frac{t}{\cos k}}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\frac{k}{\ell}}}}{\frac{k}{\ell}}}{\sin k \cdot \sin k}\]
