The high power of p4 in the denominator must confirm that the integral is convergent with p4>0.
Maple seems to be able to do it with the following commands (assume p1=q1, p4=q4, Maple output converted to LaTeX format)
interface(showassumed=0); assume(q1>0,q4>0);
A:= simplify(int(1/((q4^2 + 1)^2 *(q1+q4)^2),q1,q4));
A :=
$\frac{1}{2}\,\frac{(q4^2\,+\,1) \, ln(q4^2\,+\,1)\,+\,(-2 \, q4^2\,-\,2) \, ln(q1\,+\,q4)\,-\,q1\, (q4^2\,+\,1)\,(q1^2+3)\,arctan(q4)\,-\,(q1^2\,+\,1)\,(q1\,q4\,+\,1))}{((q1^2\,+\,1)^2\,(q4^2\,+\,1))}$