The problem here is that Mathematica doesn't recognize {x,y,z} as some kind of a vector object that should be treated as grouped together; instead, it substitutes in three independent variables, and probably starts integrating them one by one. The result is a very complicated integral.

If you do the coordinate transformation yourself, you can reproduce the result. Simply use $\mathrm dp = \mathrm d\|p\|\|p\|^2\mathrm d\vartheta\sin(\vartheta)$ to transform to spherical, the resulting integral can be calculated:

Assuming[pAbs >= 0 && m > 0 && r > 0,
    (1/(2*Pi)^3)*Integrate[ (* phi integral *)
        Integrate[ (* |p| integral *)
            Integrate[ (* theta integral *)
                (Exp[I*r*pAbs*Cos[pTheta]]/(pAbs^2 + m^2))*pAbs^2*Sin[pTheta],
                {pTheta, 0, Pi}
            ],
            {pAbs, 0, Infinity}
        ],
        {pPhi, 0, 2*Pi}
    ]
]

$\displaystyle\frac{e^{-mr}}{4\pi r}$

The problem here is that Mathematica doesn't recognize {x, y, z} as some kind of a vector object that should be treated as grouped together; instead, it substitutes in three independent variables, and probably starts integrating them one by one. The result is a very complicated integral.

If you do the coordinate transformation yourself, you can reproduce the result. Simply use $\mathrm dp = \mathrm d\|p\|\|p\|^2\mathrm d\vartheta\sin(\vartheta)$ to transform to spherical. The resulting integral can be calculated:

Assuming[pAbs >= 0 && m > 0 && r > 0,
    (1/(2*Pi)^3)*Integrate[ (* phi integral *)
        Integrate[ (* |p| integral *)
            Integrate[ (* theta integral *)
                (Exp[I*r*pAbs*Cos[pTheta]]/(pAbs^2 + m^2))*pAbs^2*Sin[pTheta],
                {pTheta, 0, Pi}
            ],
            {pAbs, 0, Infinity}
        ],
        {pPhi, 0, 2*Pi}
    ]
]

$\dfrac{e^{-mr}}{4\pi r}$

The problem here is that Mathematica doesn't recognize {x,y,z} as some kind of a vector object that should be treated as grouped together; instead, it substitutes in three independent variables, and probably starts integrating them one by one. The result is a very complicated integral.

If you do the coordinate transformation yourself, you can reproduce the result. Simply use $\mathrm dp = \mathrm d\|p\|\|p\|^2\mathrm d\vartheta\sin(\vartheta)$ to transform to spherical, the resulting integral can be calculated:

Assuming[pAbs >= 0 && m > 0 && r > 0,
    (1/(2*Pi)^3)*Integrate[ (* phi integral *)
        Integrate[ (* |p| integral *)
            Integrate[ (* theta integral *)
                (Exp[I*r*pAbs*Cos[pTheta]]/(pAbs^2 + m^2))*pAbs^2*Sin[pTheta],
                {pTheta, 0, Pi}
            ],
            {pAbs, 0, Infinity}
        ],
        {pPhi, 0, 2*Pi}
    ]
]

$\frac{e^{-mr}}{4\pi r}$

The problem here is that Mathematica doesn't recognize {x,y,z} as some kind of a vector object that should be treated as grouped together; instead, it substitutes in three independent variables, and probably starts integrating them one by one. The result is a very complicated integral.

If you do the coordinate transformation yourself, you can reproduce the result. Simply use $\mathrm dp = \mathrm d\|p\|\|p\|^2\mathrm d\vartheta\sin(\vartheta)$ to transform to spherical, the resulting integral can be calculated:

Assuming[pAbs >= 0 && m > 0 && r > 0,
    (1/(2*Pi)^3)*Integrate[ (* phi integral *)
        Integrate[ (* |p| integral *)
            Integrate[ (* theta integral *)
                (Exp[I*r*pAbs*Cos[pTheta]]/(pAbs^2 + m^2))*pAbs^2*Sin[pTheta],
                {pTheta, 0, Pi}
            ],
            {pAbs, 0, Infinity}
        ],
        {pPhi, 0, 2*Pi}
    ]
]

$\displaystyle\frac{e^{-mr}}{4\pi r}$