# Question:Integration involving dot products of 3-d vectors

## Question:Integration involving dot products of 3-d vectors

Maple

I am trying to get Maple to understand, and evaluate the following integral analytically AND numerically.
The original integrand to be evaluated is given by

(w + m2)(1/4) (p2 + mu)(1/4) / ( (w2 +a2)2 (p2 +a2) (q2 + b2 2)2 ((p - q + w)2 + b12) )

which is to be integrated simultaneously with respect to dw dp dq (where w, p, q are 3-d vectors).
Limits of integration are (in cartesian co-ordinates) from -Infinity to + Infinity

All variables contained in the integrand are Real and >0.
a, m, mu, b1, b2 are constants and >0.
dp = d3p, dq = d3q, dw = d3w
p2=p2, w2=w2, q2=q2

It is of course possible to help Maple along by noting that
(p - q + w)= p2 + q2 + w2 - (2 p q Cos(\theta_pq) ) + (2 p w Cos(\theta_pw) ) - (2 q w Cos(\theta_wq) ) + b12
where \theta_pq is the angle between vectors p and q, theta_pw is the angle between vectors p and w, etc.

However, using the expanded form of (p - q + w)2 would imply that the integral would now need to be written out in tems of