# Integrating exponential function

Posted 10 years ago
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 I have tried to solve this integral:Integrate[E^(-((a^2 b c^2)/(a^2 + b)))/(a^2 + b)^2,a]Mathematica is not able to solve it, I have tried the integration by parts and it did not work, as well as some substitutions. Any idea how to tackle this problem?
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Posted 10 years ago
 Do you need a symbolic solution? Or would a numerical approximation be usable as well?
Posted 10 years ago
 Of course most integrals that you can write down cannot be solved in finite terms.  Have you seen this one done in a table of integrals?
Posted 10 years ago
 Mathematica finds a solution if b is 0In[79]:= Integrate[(E^(-((a^2 b c^2)/(a^2 + b)))/(a^2 + b)^2) /.   b -> 0, a]Out[79]= -(1/(3 a^3))An usual way to proceed consists in taking a Taylor Series around b = 0 and integrate the series; it gives a nasty expression which nevertheless might tell you something In[82]:= Integrate[  Series[E^(-((a^2 b c^2)/(a^2 + b)))/(a^2 + b)^2, {b, 0, 5}], a]  Out[82]= SeriesData[b, 0, {  Rational[-1, 3] a^(-3), Rational[1, 15] a^(-5) (6 + 5 a^2 c^2),    Rational[1, 210] a^(-7) (-90 - 126 a^2 c^2 - 35 a^4 c^4),    Rational[1, 630] a^(-9) (    280 + 540 a^2 c^2 + 252 a^4 c^4 + 35 a^6 c^6),    Rational[1, 5544]    a^(-11) (-\2520 - 6160 a^2 c^2 - 3960 a^4 c^4 - 924 a^6 c^6 - 77 a^8 c^8),   Rational[1, 360360] a^(-13) (   166320 + 491400 a^2 c^2 + 400400 a^4 c^4 + 128700 a^6 c^6 + 18018 a\^8 c^8 + 1001 a^10 c^10)}, 0, 6, 1]
Posted 10 years ago
 Mathematica finds also a solution for c = 0 In[84]:= Integrate[   E^(-((a^2 b c^2)/(a^2 + b)))/(a^2 + b)^2 /. c -> 0, a]    Out[84]= a/(2 b (a^2 + b)) + ArcTan[a/Sqrt[b]]/(2 b^(3/2))  In[86]:= Integrate[  Series[E^(-((a^2 b c^2)/(a^2 + b)))/(a^2 + b)^2, {c, 0, 12}],   a] // FullSimplifyOut[86]= SeriesData[c, 0, { Rational[1, 2] b^Rational[-3, 2] (   a b^Rational[1, 2]/(a^2 + b) + ArcTan[a b^Rational[-1, 2]]), 0,   Rational[1,     8] ((a^2 + b)^(-2) (-a^3 + a b) - b^Rational[-1, 2] ArcTan[    a b^Rational[-1, 2]]), 0,   Rational[1, 96] a (a^2 - 3 b) b (a^2 + b)^(-3) (    3 a^2 + b) + Rational[1, 32] b^Rational[1, 2] ArcTan[    a b^Rational[-1, 2]], 0,   Rational[1, 2304]     a b^2 (a^2 + b)^(-4) ((-15)       a^6 + 73 a^4 b + 55 a^2 b^2 + 15 b^3) + Rational[-5, 768]     b^Rational[3, 2] ArcTan[a b^Rational[-1, 2]], 0,   Rational[-1, 92160]     a b^3 (a^2 + b)^(-5) ((-105)       a^8 + 790 a^6 b + 896 a^4 b^2 + 490 a^2 b^3 + 105 b^4)\ + Rational[7, 6144] b^Rational[5, 2] ArcTan[a b^Rational[-1, 2]], 0,   Rational[1, 1843200] b^Rational[7, 2] (   a b^Rational[1, 2] (      a^2 + b)^(-6) ((-315)        a^10 + 3335 a^8 b + 5058 a^6 b^2 + 4158 a^4 b^3 + 1785 a^2 b^4\ + 315 b^5) - 315 ArcTan[a b^Rational[-1, 2]]), 0,   Rational[1, 154828800]    b^Rational[    9, 2] (-a b^Rational[1, 2] (      a^2 + b)^(-7) ((-3465)        a^12 + 48580 a^10 b + 92323 a^8 b^2 + 101376 a^6 b^3 + 65373 a\^4 b^4 + 23100 a^2 b^5 + 3465 b^6) + 3465 ArcTan[     a b^Rational[-1, 2]])}, 0, 13, 1]