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Is there any mathematical condition or some theorem which can state if a given integral has a closed form or not, given that the integral converges?

Or does it depend entirely on the nature of the integrand whether it will have a closed form or not?

I searched on the net but could not find satisfactory results.

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    The "closed form" is somewhat of an arbitrary distinction. Does using $e^x$ in a result count as closed form? What about the Gamma function, Beta function, erf, etc.? – 2017-01-05
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    Please write the reason of downvote. – 2017-01-05
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    @abiessu Yes any standard function, (accepted by all as a standard function) is part of a closed form. Since there are a lot of things that has been derived and hence known about that particular function. I mean I want to express an integral using the known standard functions or closed form integrals, i dont want to define a new function. – 2017-01-05
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    Are you asking about definite integrals, or are you asking about antiderivatives? or maybe both? – 2017-01-05
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    @GerryMyerson Both. Or whichever is possible. – 2017-01-05
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    OK. An answer has been posted that applies to antiderivatives. Definite integrals, it's harder to say; even if $\int f$ can't be given in closed form, there may be values of $a$\int_a^bf$ can, but it's hard to say anything general about this. – 2017-01-05
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    @GerryMyerson Okay. Thanks for your suggestion. :-) – 2017-01-05

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Liouville's theorem states that "elementary antiderivatives, if they exist, must be in the same differential field as the function, plus possibly a finite number of logarithms."

The Risch algorithm is "a method for deciding whether a function has an elementary function as an indefinite integral, and if it does, for determining that indefinite integral."

That said, over the years, people have defined many useful functions that aren't considered elementary, but can be used to solve many other types of integrals. For instance, rational functions of the square root of a cubic or quartic polynomial come up when examining the arc length of ellipses, and in general do not have elementary antiderivatives, but can be solved using Jacobian elliptic functions.