Techniques for noticing expression has no anti-derivative?

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My integration skills are okay. I know most common techniques and currently trying to sharpen up by going over the book Interesting integrals.



However, I do struggle to identify when an expression has no antiderivative in terms of elementary functions and often find myself spending a long time on integrals only for Wolfram Alpha to tell me I’m snookered.



Are there techniques available to be able to identify such things other than by experience?




calculus integration




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  • 3




    This is not an easy problem... From the theoretical standpoint, you can have a look at Liouville's theorem.
    – mathcounterexamples.net
    Jul 25 at 9:29






  • 2




    I only know there is an algorithm, called the Risch algorithm (en.wikipedia.org/wiki/Risch_algorithm), which can verify if a given function has got an anti-derivative or not. But it is a computer algorithm so I am not sure if you can compute it by hand by I think not. I guess to know some of the special functions concerning not integratable function combinations, like the Sinc integral of the integral logarithm, is a first step.
    – mrtaurho
    Jul 25 at 9:30










  • Related: math.stackexchange.com/questions/155/…
    – Hans Lundmark
    Jul 25 at 10:12














up vote
1
down vote

favorite












My integration skills are okay. I know most common techniques and currently trying to sharpen up by going over the book Interesting integrals.



However, I do struggle to identify when an expression has no antiderivative in terms of elementary functions and often find myself spending a long time on integrals only for Wolfram Alpha to tell me I’m snookered.



Are there techniques available to be able to identify such things other than by experience?




calculus integration




share|cite|improve this question















  • 3




    This is not an easy problem... From the theoretical standpoint, you can have a look at Liouville's theorem.
    – mathcounterexamples.net
    Jul 25 at 9:29






  • 2




    I only know there is an algorithm, called the Risch algorithm (en.wikipedia.org/wiki/Risch_algorithm), which can verify if a given function has got an anti-derivative or not. But it is a computer algorithm so I am not sure if you can compute it by hand by I think not. I guess to know some of the special functions concerning not integratable function combinations, like the Sinc integral of the integral logarithm, is a first step.
    – mrtaurho
    Jul 25 at 9:30










  • Related: math.stackexchange.com/questions/155/…
    – Hans Lundmark
    Jul 25 at 10:12












up vote
1
down vote

favorite









up vote
1
down vote

favorite











My integration skills are okay. I know most common techniques and currently trying to sharpen up by going over the book Interesting integrals.



However, I do struggle to identify when an expression has no antiderivative in terms of elementary functions and often find myself spending a long time on integrals only for Wolfram Alpha to tell me I’m snookered.



Are there techniques available to be able to identify such things other than by experience?




calculus integration




share|cite|improve this question











My integration skills are okay. I know most common techniques and currently trying to sharpen up by going over the book Interesting integrals.



However, I do struggle to identify when an expression has no antiderivative in terms of elementary functions and often find myself spending a long time on integrals only for Wolfram Alpha to tell me I’m snookered.



Are there techniques available to be able to identify such things other than by experience?





calculus integration





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked Jul 25 at 9:12









Rumplestillskin

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  • 3




    This is not an easy problem... From the theoretical standpoint, you can have a look at Liouville's theorem.
    – mathcounterexamples.net
    Jul 25 at 9:29






  • 2




    I only know there is an algorithm, called the Risch algorithm (en.wikipedia.org/wiki/Risch_algorithm), which can verify if a given function has got an anti-derivative or not. But it is a computer algorithm so I am not sure if you can compute it by hand by I think not. I guess to know some of the special functions concerning not integratable function combinations, like the Sinc integral of the integral logarithm, is a first step.
    – mrtaurho
    Jul 25 at 9:30










  • Related: math.stackexchange.com/questions/155/…
    – Hans Lundmark
    Jul 25 at 10:12












  • 3




    This is not an easy problem... From the theoretical standpoint, you can have a look at Liouville's theorem.
    – mathcounterexamples.net
    Jul 25 at 9:29






  • 2




    I only know there is an algorithm, called the Risch algorithm (en.wikipedia.org/wiki/Risch_algorithm), which can verify if a given function has got an anti-derivative or not. But it is a computer algorithm so I am not sure if you can compute it by hand by I think not. I guess to know some of the special functions concerning not integratable function combinations, like the Sinc integral of the integral logarithm, is a first step.
    – mrtaurho
    Jul 25 at 9:30










  • Related: math.stackexchange.com/questions/155/…
    – Hans Lundmark
    Jul 25 at 10:12







3




3




This is not an easy problem... From the theoretical standpoint, you can have a look at Liouville's theorem.
– mathcounterexamples.net
Jul 25 at 9:29




This is not an easy problem... From the theoretical standpoint, you can have a look at Liouville's theorem.
– mathcounterexamples.net
Jul 25 at 9:29




2




2




I only know there is an algorithm, called the Risch algorithm (en.wikipedia.org/wiki/Risch_algorithm), which can verify if a given function has got an anti-derivative or not. But it is a computer algorithm so I am not sure if you can compute it by hand by I think not. I guess to know some of the special functions concerning not integratable function combinations, like the Sinc integral of the integral logarithm, is a first step.
– mrtaurho
Jul 25 at 9:30




I only know there is an algorithm, called the Risch algorithm (en.wikipedia.org/wiki/Risch_algorithm), which can verify if a given function has got an anti-derivative or not. But it is a computer algorithm so I am not sure if you can compute it by hand by I think not. I guess to know some of the special functions concerning not integratable function combinations, like the Sinc integral of the integral logarithm, is a first step.
– mrtaurho
Jul 25 at 9:30












Related: math.stackexchange.com/questions/155/…
– Hans Lundmark
Jul 25 at 10:12




Related: math.stackexchange.com/questions/155/…
– Hans Lundmark
Jul 25 at 10:12















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