mathematical physics - Integral of the product of three spherical harmonics

Does anyone know how to derive the following identity for the integral of the product of three spherical harmonics?:

$$\begin{align}\int_0^{2\pi}\int_0^\pi Y_{l_1}^{m_1}(\theta,\phi)Y_{l_2}^{m_2}(\theta,\phi)&Y_{l_3}^{m_3}(\theta,\phi)\sin(\theta)d\theta d\phi =\\ &\sqrt{\frac{(2l_1+1)(2l_2+1)(2l_3+1)}{4\pi}} \left( {\begin{array}{ccc} l_1 & l_2 & l_3 \\ 0 & 0 & 0 \\ \end{array} } \right) \left( {\begin{array}{ccc} l_1 & l_2 & l_3 \\ m_1 & m_2 & m_3 \\ \end{array} } \right) \end{align}$$

Where the $Y_{l}^{m}(\theta,\phi)$ are spherical harmonics. Or does anyone know of a reference where the derivation is given?

Answer

Sakurai, Modern Quantum Mechanics, 2nd Ed. p.216

In his derivation the product of the first two spherical harmonics is expanded using the Clebsch-Gordan Series (which is also proved) to get the following equation.

$Y_{l_1}^{m_1}(\theta,\phi)Y_{l_2}^{m_2}(\theta,\phi)\ =$

$\displaystyle\sum\limits_{l} \displaystyle\sum\limits_{m} \sqrt{\frac{(2l_1+1)(2l_2+1)(2l+1)}{4\pi}} \left( {\begin{array}{ccc} l_1 & l_2 & l \\ 0 & 0 & 0 \\ \end{array} } \right) \left( {\begin{array}{ccc} l_1 & l_2 & l \\ m_1 & m_2 & -m \\ \end{array} } \right)(-1)^m Y_{l}^{m}(\theta,\phi)$

Which makes the integral much easier.

Final Note: Sakurai writes his derivation in Clebsch-Gordan coefficients so the equation was changed to fit with the question asked.