Analytic Integration of Function Containing the Exponential of an Exponential

[junt]

Homework Statement

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.

Homework Equations

This is the integral I am looking at:
$I=\int_{-\infty}^{\infty}\exp \left(-\frac{e^{-2 \theta } \left(a \left(b^2 \left(e^{2 \theta }-1\right)^2 L^2+16\right)-32 \sqrt{a} e^{\theta }+16 e^{2 \theta }\right)}{b L^4}\right) d\theta$

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?

 

[SammyS]

Homework Statement

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.

Homework Equations

This is the integral I am looking at:
$I=\int_{-\infty}^{\infty}\exp \left(-\frac{e^{-2 \theta } \left(a \left(b^2 \left(e^{2 \theta }-1\right)^2 L^2+16\right)-32 \sqrt{a} e^{\theta }+16 e^{2 \theta }\right)}{b L^4}\right) d\theta$

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?

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

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

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