I would like a direct change of variable proof of the identity

$$\int_0^\arctan\frac{sqrt{2}}{\sqrt{\sqrt{3}} 1/sqrt{1-\frac{2+\sqrt{3}}{4}\sin^2\phi d\phi}=\int_0^\arctan\frac{1}{\sqrt{\sqrt{3}} 1/sqrt{1-\frac{2+\sqrt{3}}{4}\sin^2\2phi d\phi}=\,.$$

I need it as part of a paper on Legendre's proof of the "third singular modulus."