Initial program 47.0
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
Initial simplification31.1
\[\leadsto \frac{\frac{\frac{2}{\tan k}}{\frac{\sin k \cdot t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}{\frac{k}{t} \cdot \frac{k}{t}}\]
- Using strategy
rm Applied times-frac30.2
\[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{\frac{\sin k}{\frac{\ell}{t}} \cdot \frac{t}{\frac{\ell}{t}}}}}{\frac{k}{t} \cdot \frac{k}{t}}\]
Applied add-cube-cbrt30.3
\[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}}}{\frac{\sin k}{\frac{\ell}{t}} \cdot \frac{t}{\frac{\ell}{t}}}}{\frac{k}{t} \cdot \frac{k}{t}}\]
Applied times-frac29.8
\[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{\ell}{t}}} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\frac{\ell}{t}}}}}{\frac{k}{t} \cdot \frac{k}{t}}\]
Applied times-frac18.8
\[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{\ell}{t}}}}{\frac{k}{t}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\frac{\ell}{t}}}}{\frac{k}{t}}}\]
Simplified11.3
\[\leadsto \frac{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{\ell}{t}}}}{\frac{k}{t}} \cdot \color{blue}{\left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)}\]
- Using strategy
rm Applied div-inv11.3
\[\leadsto \frac{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{\ell}{t}}}}{\color{blue}{k \cdot \frac{1}{t}}} \cdot \left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\]
Applied div-inv11.3
\[\leadsto \frac{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\color{blue}{\ell \cdot \frac{1}{t}}}}}{k \cdot \frac{1}{t}} \cdot \left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\]
Applied add-cube-cbrt11.5
\[\leadsto \frac{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\color{blue}{\left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{\sin k}}}{\ell \cdot \frac{1}{t}}}}{k \cdot \frac{1}{t}} \cdot \left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\]
Applied times-frac11.6
\[\leadsto \frac{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\color{blue}{\frac{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}{\ell} \cdot \frac{\sqrt[3]{\sin k}}{\frac{1}{t}}}}}{k \cdot \frac{1}{t}} \cdot \left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\]
Applied times-frac11.9
\[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}{\ell}} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{\sqrt[3]{\sin k}}{\frac{1}{t}}}}}{k \cdot \frac{1}{t}} \cdot \left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\]
Applied times-frac8.1
\[\leadsto \color{blue}{\left(\frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}{\ell}}}{k} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{\sqrt[3]{\sin k}}{\frac{1}{t}}}}{\frac{1}{t}}\right)} \cdot \left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\]
Simplified7.7
\[\leadsto \left(\frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}{\ell}}}{k} \cdot \color{blue}{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\sqrt[3]{\sin k}}}\right) \cdot \left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\]
- Using strategy
rm Applied cbrt-div7.8
\[\leadsto \left(\frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}{\ell}}}{k} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\sqrt[3]{\sin k}}\right) \cdot \left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \color{blue}{\frac{\sqrt[3]{2}}{\sqrt[3]{\tan k}}}\right)\]
Applied associate-*r/7.8
\[\leadsto \left(\frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}{\ell}}}{k} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\sqrt[3]{\sin k}}\right) \cdot \color{blue}{\frac{\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{2}}{\sqrt[3]{\tan k}}}\]
Applied associate-*r/7.8
\[\leadsto \color{blue}{\frac{\frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}{\ell}}}{k} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\sqrt[3]{\sin k}}} \cdot \frac{\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{2}}{\sqrt[3]{\tan k}}\]
Applied frac-times7.7
\[\leadsto \color{blue}{\frac{\left(\frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}{\ell}}}{k} \cdot \sqrt[3]{\frac{2}{\tan k}}\right) \cdot \left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{2}\right)}{\sqrt[3]{\sin k} \cdot \sqrt[3]{\tan k}}}\]
Simplified2.2
\[\leadsto \frac{\color{blue}{\frac{\left(\sqrt[3]{\frac{2}{\tan k}} \cdot \frac{\frac{\ell}{k} \cdot \sqrt[3]{2}}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\sqrt[3]{\sin k}}{\ell} \cdot \left(\sqrt[3]{\sin k} \cdot k\right)}}}{\sqrt[3]{\sin k} \cdot \sqrt[3]{\tan k}}\]
Final simplification2.2
\[\leadsto \frac{\frac{\left(\sqrt[3]{\frac{2}{\tan k}} \cdot \frac{\sqrt[3]{2} \cdot \frac{\ell}{k}}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\sqrt[3]{\sin k}}{\ell} \cdot \left(k \cdot \sqrt[3]{\sin k}\right)}}{\sqrt[3]{\sin k} \cdot \sqrt[3]{\tan k}}\]
