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I need to integrate the expression G given below:

R = {Rx, Ry, Rz};
b = {bx, by, bz};
r = Sqrt[(s*b - R).(s*b - R)];
K = Exp[-r/L]/r;
G = Simplify[-K*(1/r + 1/L)*1/r*b.(s*b - R)];

If I use:

Integrate[G,{s,0,1}]

it takes quite a long time; instead if I use:

int = Integrate[G, s]
Simplify[(int /. s -> 1) - (int /. s -> 0)]

it takes less than a second.

Why is it so?

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When calculating indefinite integrals Mathematica does not care about the convergence in a domain {x_min,x_max}. In case of definite integrals, at times it is necessary to provide information on the constants in order to obtain the proper result. Check the tutorial on definite integrals. The example with 1/(1 + a Sin[x]) is very similar to your problem.

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  • $\begingroup$ very clear thank you $\endgroup$ – mattiav27 Apr 29 '14 at 14:16
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    $\begingroup$ Since this is a fine answer (and I upvoted) I'll post this as a comment. One can often get improvements by giving assumptions and/or requesting that no conditions be generated. In this example that might be something like Integrate[G, {s, 0, 1}, GenerateConditions -> False, Assumptions -> {Thread[Variables[G] > 0]}] $\endgroup$ – Daniel Lichtblau Apr 29 '14 at 15:07

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