Initial program 27.0
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
- Using strategy
rm Applied unpow327.0
\[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
Applied times-frac18.6
\[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
Applied associate-*l*16.5
\[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
- Using strategy
rm Applied *-un-lft-identity16.5
\[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \color{blue}{\left(1 \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}\]
Applied associate-*r*16.5
\[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot 1\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}\]
Applied simplify9.3
\[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\tan k \cdot \frac{t}{\ell}\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \frac{t}{\ell}\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
- Using strategy
rm Applied add-cube-cbrt9.5
\[\leadsto \frac{2}{\color{blue}{\left(\sqrt[3]{\left(\left(\left(\tan k \cdot \frac{t}{\ell}\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \frac{t}{\ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \cdot \sqrt[3]{\left(\left(\left(\tan k \cdot \frac{t}{\ell}\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \frac{t}{\ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\right) \cdot \sqrt[3]{\left(\left(\left(\tan k \cdot \frac{t}{\ell}\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \frac{t}{\ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}}\]
Applied simplify9.6
\[\leadsto \frac{2}{\color{blue}{\left(\sqrt[3]{\frac{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}{\frac{\ell}{t}} \cdot \left(\frac{\tan k}{\frac{\ell}{t}} \cdot \left(t \cdot \sin k\right)\right)} \cdot \sqrt[3]{\frac{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}{\frac{\ell}{t}} \cdot \left(\frac{\tan k}{\frac{\ell}{t}} \cdot \left(t \cdot \sin k\right)\right)}\right)} \cdot \sqrt[3]{\left(\left(\left(\tan k \cdot \frac{t}{\ell}\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \frac{t}{\ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}\]
Applied simplify8.5
\[\leadsto \frac{2}{\left(\sqrt[3]{\frac{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}{\frac{\ell}{t}} \cdot \left(\frac{\tan k}{\frac{\ell}{t}} \cdot \left(t \cdot \sin k\right)\right)} \cdot \sqrt[3]{\frac{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}{\frac{\ell}{t}} \cdot \left(\frac{\tan k}{\frac{\ell}{t}} \cdot \left(t \cdot \sin k\right)\right)}\right) \cdot \color{blue}{\sqrt[3]{\left(\left(\tan k \cdot \frac{t}{\ell}\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \frac{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}{\frac{\ell}{t}}}}}\]
Initial program 62.6
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
- Using strategy
rm Applied unpow362.6
\[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
Applied times-frac59.4
\[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
Applied associate-*l*59.4
\[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
- Using strategy
rm Applied *-un-lft-identity59.4
\[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \color{blue}{\left(1 \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}\]
Applied associate-*r*59.4
\[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot 1\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}\]
Applied simplify49.5
\[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\tan k \cdot \frac{t}{\ell}\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \frac{t}{\ell}\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
Taylor expanded around 0 40.1
\[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{1}{3} \cdot \frac{{k}^{3} \cdot t}{\ell} + \frac{k \cdot t}{\ell}\right)} \cdot \left(t \cdot \sin k\right)\right) \cdot \frac{t}{\ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
Applied simplify40.7
\[\leadsto \color{blue}{\frac{\frac{\frac{2}{\sin k \cdot t}}{(\frac{1}{3} \cdot \left(\frac{{k}^{3}}{\frac{\ell}{t}}\right) + \left(\frac{k}{\frac{\ell}{t}}\right))_*}}{\frac{t}{\ell} \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}\]
Initial program 24.9
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
- Using strategy
rm Applied unpow324.9
\[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
Applied times-frac17.9
\[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
Applied associate-*l*15.6
\[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
- Using strategy
rm Applied *-un-lft-identity15.6
\[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \color{blue}{\left(1 \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}\]
Applied associate-*r*15.6
\[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot 1\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}\]
Applied simplify8.0
\[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\tan k \cdot \frac{t}{\ell}\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \frac{t}{\ell}\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
- Using strategy
rm Applied add-cube-cbrt8.2
\[\leadsto \frac{2}{\color{blue}{\left(\sqrt[3]{\left(\left(\left(\tan k \cdot \frac{t}{\ell}\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \frac{t}{\ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \cdot \sqrt[3]{\left(\left(\left(\tan k \cdot \frac{t}{\ell}\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \frac{t}{\ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\right) \cdot \sqrt[3]{\left(\left(\left(\tan k \cdot \frac{t}{\ell}\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \frac{t}{\ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}}\]
Applied simplify8.2
\[\leadsto \frac{2}{\color{blue}{\left(\sqrt[3]{\frac{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}{\frac{\ell}{t}} \cdot \left(\frac{\tan k}{\frac{\ell}{t}} \cdot \left(t \cdot \sin k\right)\right)} \cdot \sqrt[3]{\frac{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}{\frac{\ell}{t}} \cdot \left(\frac{\tan k}{\frac{\ell}{t}} \cdot \left(t \cdot \sin k\right)\right)}\right)} \cdot \sqrt[3]{\left(\left(\left(\tan k \cdot \frac{t}{\ell}\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \frac{t}{\ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}\]
Applied simplify7.0
\[\leadsto \frac{2}{\left(\sqrt[3]{\frac{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}{\frac{\ell}{t}} \cdot \left(\frac{\tan k}{\frac{\ell}{t}} \cdot \left(t \cdot \sin k\right)\right)} \cdot \sqrt[3]{\frac{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}{\frac{\ell}{t}} \cdot \left(\frac{\tan k}{\frac{\ell}{t}} \cdot \left(t \cdot \sin k\right)\right)}\right) \cdot \color{blue}{\sqrt[3]{\left(\left(\tan k \cdot \frac{t}{\ell}\right) \cdot \left(t \cdot \sin k\right)\right) \cdot \frac{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}{\frac{\ell}{t}}}}}\]
