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I've been trying for an embarrassingly long time to figure this one out. It looks like it should crack under integration by parts and integration by substitution, but I am having trouble with it. Any pointers?

$$\int {\exp(a \sqrt{x^2 + b} ) \over \sqrt{x^2+b} } dx$$

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You're sure there's no x in the numerator? – user7530 Nov 28 '11 at 11:21
Wolfram Alpha hasn't found result in the terms of standard mathematical functions.. – pedja Nov 28 '11 at 11:24
Wolfram Alpha says it has no expression in elementary terms. – lhf Nov 28 '11 at 11:26
This was part of a larger homework problem in a random signals class where I was trying to prove/disprove independence of two RVs with a joint pdf that looks similar to the integral above. Apparently solving it by directly using integration is not the right approach. – Mark Borgerding Nov 29 '11 at 16:20
up vote 3 down vote accepted

The special case $a=1,b=0$ simplifies to

$$\int \frac{e^x}{x} \, dx$$

Which is known as the Exponential Integral where no closed form is known. Therefore your integral has no general closed form either.

Note: By closed form I mean that it is not expressible in terms of elementary functions.

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...I consider the exponential integral as a closed form expression in its own right. Better to say that the exponential integral is not expressible in terms of the elementary functions... – J. M. Nov 28 '11 at 13:05
I added a note to clarify what I mean, although if you consider $\text{Ei(x)}$ to be a closed form expression you could define every integral as a function and call it closed form :-) – Listing Nov 28 '11 at 13:09
Sure, but another part of my "closed-form" schtick is that there's a whole corpus of identities relating it to a lot of other functions. ;) You don't have that with, say, $\int u^u \mathrm du$. – J. M. Nov 28 '11 at 13:31

The natural substitution is to let $u=a\sqrt{x^2+b}$ so that $du=\frac{ax}{\sqrt{x^2+b}}$ and $x^2=\frac{u^2}{a^2}-b$. Our integral then is

$$\int \frac{e^u}{\sqrt{u^2-ba^2}} du.$$ However, there is no way to deal with something of this form. Are you sure that you are not missing a multiple of $x$? In other words, I think the question should be to integrate $$\int \frac{x\exp(a\sqrt{x^2+b})}{\sqrt{x^2+b}}dx$$ because then it gives $$\frac{1}{a}e^{a\sqrt{x^2+b}}$$ as the anti derivative.

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