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Derivation

1. Initial program 32.3

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

2. Initial simplification24.8

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

4. Applied *-un-lft-identity24.8

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

5. Applied tan-quot24.8

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

6. Applied associate-*l/24.8

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

7. Applied associate-/r/24.8

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

8. Applied times-frac24.8

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

9. Simplified16.3

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

11. Applied add-cube-cbrt16.5

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

12. Applied times-frac14.6

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

13. Applied associate-*l*12.0

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

15. Applied add-cube-cbrt12.1

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

16. Applied *-un-lft-identity12.1

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

17. Applied times-frac12.1

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

18. Applied associate-*l*12.1

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

19. Final simplification12.1

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

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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