MHB Find Analytic Expression for Integral with Approximations

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The discussion focuses on finding a closed-form or analytic expression for a specific integral involving exponential functions and constants. Participants are encouraged to share their insights on how to approach the problem, particularly in terms of techniques or methods that could simplify the integral. The integral includes parameters x, y, and z, which are constants independent of the variable r. The complexity of the integral suggests that advanced mathematical methods may be necessary for a solution. Engaging with the community could lead to collaborative problem-solving and potential breakthroughs in deriving the expression.
venkaiah
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Find the closed form (or) analytic expression form for the following integral

$$
\hspace{0.3cm} \large {\int_{0} ^{\infty} \frac{\frac{1}{x^4} \hspace{0.1cm} e^{- \frac{r}{x^2}}\hspace{0.1cm}e^{- \frac{r}{z^2}} }{ \frac{1}{x^2} \hspace{0.1cm} e^{- \frac{r}{x^2}}+ \frac{1}{y^2} \hspace{0.1cm} e^{- \frac{r}{y^2}}}} dr \hspace{.2cm} ; \hspace{1cm} x>0,y>0,z>0 $$ where $ x $ ,$ y $ and $z $ are constants and independent of $ r $.
 
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Hi venkaiah and welcome to MHB! :D

Any thoughts on how to begin?
 
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