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### Solving the Definite Integral $\int_0^{\infty} \frac{1}{t^{\frac{3}{2}}} e^{-\frac{a}{t}} \, \mathrm{erf}(\sqrt{t})\, \mathrm{d}t$

I would like to solve the following integral $$\int_0^{\infty} \frac{1}{t^{\frac{3}{2}}} e^{-\frac{a}{t}} \, \mathrm{erf}(\sqrt{t})\, \mathrm{d}t$$ with Re$(a)>0$ and erf the error function. Is ...
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### How to compute $\int_0^{\infty}\sqrt x \exp\left(-x-\frac{1}{x}\right) \, dx$?

How to compute this integral? : $$\int_0^{\infty}\sqrt x \exp\left(-x-\frac{1}{x}\right) \, dx$$ Wolframalpha gives the answer $\dfrac{3\sqrt{\pi}}{2e^2}$, but how to compute this?
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### Integral of $1/x^2 \exp(-ax+b-c/x)dx$

I am interested in simplifying the integral $$\int \frac{1}{x^2}\exp(-ax+b-\frac{c}{x})dx$$ with $a, b, c \in \mathbb{R}$. Do you have any idea? With $a=0$ or $c=0$ I know the solutions but what about ...
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### An improper integrals related to probability, $\int_0^\infty\frac1y \exp(\frac{-x_0}y-y)\,dy$

How can I calculate the integral $$\int_0^\infty{\frac1y e^{\frac{-x_0}y-y}}dy$$ in terms of well-known constants and functions? I used some fundamental techniques of integration but got nothing.
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### The closed form of $\int^\infty_{B}e^{-(x+\frac{A}{x})}\,dx$, where $A>0$, $B>0$.

What tools, ways would you propose for getting the closed form of this integral? $$\int^\infty_{B}e^{-\left(x+\frac{A}{x}\right)}\,dx,$$ where $A>0$, $B>0$. When $B=0$, from Table of Integrals,...
Equation no. 3.471.9 of Integral series and products (By Gradeshteyn) is written below \int_0^{\infty}x^{v-1}e^{-\frac{\beta}{x}-\gamma x}dx=2\left(\frac{\beta}{\gamma}\right)^{\frac{v}{2}}K_{v}(2\...
### How to rewrite this integral $I = \int e^{ - \left( {ax + \frac{b}{x}} \right)} dx$ as non-elementary function?
Is it possible to rewrite or evaluate this integral $I = \int\limits_1^p e^{ - \left( {ax + \frac{b}{x}} \right)} dx$ where $a,b,p > 0$ as some known non-elementary function (For example \$\...