Integrating exponential function

Questions
,
1

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?

2

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?

3

Mathematica finds a solution if b is 0

In[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]
4

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] // FullSimplify



Out[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]
5

Do you need a symbolic solution? Or would a numerical approximation be usable as well?