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I have an equation of the type $$ p=\int_0^b\sqrt{\left(a^2-x^2\right)\left(b^2-x^2\right)}dx, $$ in which $a$ and $b$ (with $a>b>0$) are (known) functions of some parameter $H$ (such that it is possible to invert for $H\left(a\right)$ and $H\left(b\right)$). Is there a procedure to express $H$ as a function of $p$ in closed form? The integral on the right hand side can be calculated in terms of elliptic integrals of the first and second kind (Gradshteĭn 3.155.8), $$ p=\int_0^b\sqrt{\left(a^2-x^2\right)\left(b^2-x^2\right)}dx=\frac{a}{3}\left[ \left( a^2+b^2 \right)E\left(\frac{b}{a}\right)-\left(a^2-b^2\right)K\left(\frac{b}{a}\right) \right], $$ but then one would need to invert this relation in order to finally arrive at $H\left(p\right)$. Any hints on how one could proceed to solve this?

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