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Analytic Integration of Function Containing the Exponential of an Exponential

  1. Jun 1, 2017 #1
    1. The problem statement, all variables and given/known data
    Can this function be integrated analytically?

    ##f=\exp \left(-\frac{e^{-2 \theta } \left(a \left(b^2 \left(e^{2 \theta }-1\right)^2 L^2+16\right)-32
    \sqrt{a} e^{\theta }+16 e^{2 \theta }\right)}{b L^4}\right),##
    where ##a##, ##b## and ##L## are some real positive constants.
    2. Relevant equations
    This is the integral I am looking at:
    ##I=\int_{-\infty}^{\infty}\exp \left(-\frac{e^{-2 \theta } \left(a \left(b^2 \left(e^{2 \theta }-1\right)^2 L^2+16\right)-32
    \sqrt{a} e^{\theta }+16 e^{2 \theta }\right)}{b L^4}\right) d\theta##


    3. The attempt at a solution

    One can change the coordinates ##u## to ##e^{-\theta}##, but then Jacobian will be inverse in ##x##, as result introduced a pole at ##x=0##. Does anyone know a better solution to it?
     
  2. jcsd
  3. Jun 6, 2017 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
  4. Jun 7, 2017 #3

    SammyS

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    That's very difficult to read. I used \displaystyle in a couple of places each.

    ##\displaystyle f=\exp \left(-\frac{\displaystyle e^{-2 \theta } \left(a \left(b^2 \left(e^{2 \theta }-1\right)^2 L^2+16\right)-32
    \sqrt{a} e^{\theta }+16 e^{2 \theta }\right)}{b L^4}\right) ##

    ##\displaystyle I=\int_{-\infty}^{\infty}\exp \left(-\frac{\displaystyle e^{-2 \theta } \left(a \left(b^2 \left(e^{2 \theta }-1\right)^2 L^2+16\right)-32
    \sqrt{a} e^{\theta }+16 e^{2 \theta }\right)}{b L^4}\right) d\theta##
     
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