There are several important and useful formulas for the integration of spherical harmonic. We simplify the notation,

The first one is the average of spherical harmonic.

The 2nd one is the orthonormal condition.

The 3rd one is triplet integral, we use the product of spherical harmonic,

The 4th one is another triple integral,

Notice that

using Wigner 3-j symbol,

For other integral, we can use