result

Now, it is time to express in terms of the φ derivatives.

(dim*Log[2*Pi])/2 - psi[0] + (ry[l1]*ry[u1])/2 - 
 (o*ry[l1]*tp3[l2, u1, u2])/2 + 
 (o*ry[l1]*ry[l2]*ry[l3]*tp3[u1, u2, u3])/6 - 
 (o^2*tp3[l1, l2, l3]*tp3[u1, u2, u3])/6 + 
 (o^2*ry[l1]*ry[l2]*tp3[l3, l4, u1]*tp3[u2, u3, u4])/4 + 
 (o^2*tp4[l1, l2, u1, u2])/8 - 
 (o^2*ry[l1]*ry[l2]*tp4[l3, u1, u2, u3])/4 + 
 (o^2*ry[l1]*ry[l2]*ry[l3]*ry[l4]*tp4[u1, u2, u3, u4])/24

The coefficients of as a polynomial of are as follows.