  The definite integrals $ \int_{-1}^1(1-x^2)^{(\nu-1)/2}[P_\nu(x)]^3\D x$, $
\int_{-1}^1(1-x^2)^{(\nu-1)/2}[P_\nu(x)]^2P_{\nu}(-x)\D x$,
$\int_{-1}^1x(1-x^2)^{(\nu-1)/2}[P_{\nu+1}(x)]^3\D x $ and
$\int_{-1}^1x(1-x^2)^{(\nu-1)/2}[P_{\nu+1}(x)]^2P_{\nu+1}(-x)\D x $ are
evaluated in closed form, where $P_\nu$ is the Legendre function of degree
$\nu$, and $ \R\nu>-1$. Special cases of these formulae are related to certain
integrals over elliptic integrals that have arithmetic interest.
