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Derivation

1. Initial program 31.8

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

2. Initial simplification24.5

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

4. Applied *-un-lft-identity24.5

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

5. Applied tan-quot24.5

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

6. Applied associate-*r/24.5

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

7. Applied associate-/r/24.5

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

8. Applied times-frac24.5

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

9. Simplified15.9

   \[\leadsto \color{blue}{\frac{\frac{2}{t}}{\frac{\sin k}{\frac{\ell}{t}} \cdot \frac{\sin k}{\frac{\ell}{t}}}} \cdot \frac{\cos k}{\frac{k}{t} \cdot \frac{k}{t} + 2}\]
10. Using strategy rm

11. Applied add-cube-cbrt16.0

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

12. Applied times-frac14.3

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

13. Applied associate-*l*11.8

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

15. Applied associate-*l/11.5

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

17. Applied add-cube-cbrt11.6

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

18. Applied add-cube-cbrt11.6

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

19. Applied cbrt-prod11.6

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

20. Applied times-frac11.6

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

21. Applied associate-*l*11.5

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

22. Final simplification11.5

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

Runtime

Time bar (total: 2.6m)Debug logProfile

herbie shell --seed 2018274 
(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))))