Initial program 1.1
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
- Using strategy
rm Applied add-sqr-sqrt1.1
\[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \color{blue}{\left(\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}} \cdot \sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}\right)}}}\right)}\]
Applied associate-*r*1.1
\[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + \color{blue}{\left({\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}\right) \cdot \sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}}\right)}\]
Applied simplify0.8
\[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + \color{blue}{\left(\left(\left(\frac{\ell}{Om} \cdot 2\right) \cdot \sqrt{\left(\sin ky\right)^2 + \left(\sin kx\right)^2}^*\right) \cdot \left(\frac{\ell}{Om} \cdot 2\right)\right)} \cdot \sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}\right)}\]
- Using strategy
rm Applied log1p-expm1-u1.2
\[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\color{blue}{\log_* (1 + (e^{\sqrt{1 + \left(\left(\left(\frac{\ell}{Om} \cdot 2\right) \cdot \sqrt{\left(\sin ky\right)^2 + \left(\sin kx\right)^2}^*\right) \cdot \left(\frac{\ell}{Om} \cdot 2\right)\right) \cdot \sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} - 1)^*)}}\right)}\]
Applied simplify0.4
\[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\log_* (1 + \color{blue}{(e^{\sqrt{1^2 + \left(\frac{\ell}{\frac{Om}{2}} \cdot \sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*\right)^2}^*} - 1)^*})}\right)}\]
