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Derivation

1. Initial program 47.4

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]

2. Initial simplification30.5

   \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\frac{\sin k \cdot t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}{\frac{k}{t} \cdot \frac{k}{t}}\]
3. Using strategy rm

4. Applied times-frac29.5

   \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{\frac{\sin k}{\frac{\ell}{t}} \cdot \frac{t}{\frac{\ell}{t}}}}}{\frac{k}{t} \cdot \frac{k}{t}}\]

5. Applied add-cube-cbrt29.6

   \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}}}{\frac{\sin k}{\frac{\ell}{t}} \cdot \frac{t}{\frac{\ell}{t}}}}{\frac{k}{t} \cdot \frac{k}{t}}\]

6. Applied times-frac29.3

   \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{\ell}{t}}} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\frac{\ell}{t}}}}}{\frac{k}{t} \cdot \frac{k}{t}}\]

7. Applied times-frac18.4

   \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{\ell}{t}}}}{\frac{k}{t}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\frac{\ell}{t}}}}{\frac{k}{t}}}\]

8. Simplified11.6

   \[\leadsto \frac{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{\ell}{t}}}}{\frac{k}{t}} \cdot \color{blue}{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{1 \cdot \ell} \cdot k}}\]
9. Using strategy rm

10. Applied *-un-lft-identity11.6

    \[\leadsto \frac{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{\ell}{t}}}}{\color{blue}{1 \cdot \frac{k}{t}}} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{1 \cdot \ell} \cdot k}\]

11. Applied associate-/r/11.7

    \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\sin k} \cdot \frac{\ell}{t}}}{1 \cdot \frac{k}{t}} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{1 \cdot \ell} \cdot k}\]

12. Applied times-frac11.7

    \[\leadsto \color{blue}{\left(\frac{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\sin k}}{1} \cdot \frac{\frac{\ell}{t}}{\frac{k}{t}}\right)} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{1 \cdot \ell} \cdot k}\]

13. Simplified11.7

    \[\leadsto \left(\color{blue}{\left(\sqrt[3]{\frac{2}{\tan k}} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\sin k}\right)} \cdot \frac{\frac{\ell}{t}}{\frac{k}{t}}\right) \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{1 \cdot \ell} \cdot k}\]

14. Simplified7.4

    \[\leadsto \left(\left(\sqrt[3]{\frac{2}{\tan k}} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\sin k}\right) \cdot \color{blue}{\left(1 \cdot \frac{\ell}{k}\right)}\right) \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{1 \cdot \ell} \cdot k}\]
15. Using strategy rm

16. Applied cbrt-div7.4

    \[\leadsto \left(\left(\color{blue}{\frac{\sqrt[3]{2}}{\sqrt[3]{\tan k}}} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\sin k}\right) \cdot \left(1 \cdot \frac{\ell}{k}\right)\right) \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{1 \cdot \ell} \cdot k}\]

17. Applied frac-times7.4

    \[\leadsto \left(\color{blue}{\frac{\sqrt[3]{2} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\sqrt[3]{\tan k} \cdot \sin k}} \cdot \left(1 \cdot \frac{\ell}{k}\right)\right) \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{1 \cdot \ell} \cdot k}\]

18. Applied associate-*l/7.3

    \[\leadsto \color{blue}{\frac{\left(\sqrt[3]{2} \cdot \sqrt[3]{\frac{2}{\tan k}}\right) \cdot \left(1 \cdot \frac{\ell}{k}\right)}{\sqrt[3]{\tan k} \cdot \sin k}} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{1 \cdot \ell} \cdot k}\]

19. Applied frac-times6.9

    \[\leadsto \color{blue}{\frac{\left(\left(\sqrt[3]{2} \cdot \sqrt[3]{\frac{2}{\tan k}}\right) \cdot \left(1 \cdot \frac{\ell}{k}\right)\right) \cdot \sqrt[3]{\frac{2}{\tan k}}}{\left(\sqrt[3]{\tan k} \cdot \sin k\right) \cdot \left(\frac{t}{1 \cdot \ell} \cdot k\right)}}\]

20. Simplified6.9

    \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{2} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\frac{k}{\ell}}{\sqrt[3]{\frac{2}{\tan k}}}}}}{\left(\sqrt[3]{\tan k} \cdot \sin k\right) \cdot \left(\frac{t}{1 \cdot \ell} \cdot k\right)}\]

21. Simplified1.7

    \[\leadsto \frac{\frac{\sqrt[3]{2} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\frac{k}{\ell}}{\sqrt[3]{\frac{2}{\tan k}}}}}{\color{blue}{\frac{k}{\ell} \cdot \left(\left(\sin k \cdot t\right) \cdot \sqrt[3]{\tan k}\right)}}\]
22. Using strategy rm

23. Applied tan-quot1.7

    \[\leadsto \frac{\frac{\sqrt[3]{2} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\frac{k}{\ell}}{\sqrt[3]{\frac{2}{\color{blue}{\frac{\sin k}{\cos k}}}}}}}{\frac{k}{\ell} \cdot \left(\left(\sin k \cdot t\right) \cdot \sqrt[3]{\tan k}\right)}\]

24. Applied associate-/r/1.7

    \[\leadsto \frac{\frac{\sqrt[3]{2} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\frac{k}{\ell}}{\sqrt[3]{\color{blue}{\frac{2}{\sin k} \cdot \cos k}}}}}{\frac{k}{\ell} \cdot \left(\left(\sin k \cdot t\right) \cdot \sqrt[3]{\tan k}\right)}\]

25. Applied cbrt-prod1.7

    \[\leadsto \frac{\frac{\sqrt[3]{2} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\frac{k}{\ell}}{\color{blue}{\sqrt[3]{\frac{2}{\sin k}} \cdot \sqrt[3]{\cos k}}}}}{\frac{k}{\ell} \cdot \left(\left(\sin k \cdot t\right) \cdot \sqrt[3]{\tan k}\right)}\]

26. Final simplification1.7

    \[\leadsto \frac{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{2}}{\frac{\frac{k}{\ell}}{\sqrt[3]{\frac{2}{\sin k}} \cdot \sqrt[3]{\cos k}}}}{\left(\left(t \cdot \sin k\right) \cdot \sqrt[3]{\tan k}\right) \cdot \frac{k}{\ell}}\]

Reproduce

herbie shell --seed 2018360 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (- (+ 1 (pow (/ k t) 2)) 1))))

Details

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