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Integrate[1/Sqrt[1-x^2+x^4],{x,0,t},Assumptions->t>0]/.t->10.*^5
NIntegrate[1/Sqrt[1-x^2+x^4],{x,0,Infinity}]

Integrate[1/Sqrt[1-x^2+x^4],{x,0,Infinity}]
Out[]//N

enter image description here

The result of Integrate + Limit agrees with that of NIntegrate. But Integrate + N seems give the wrong result, since the integrand is positive.

This integral is a quite standard elliptic integral. Is it a severe bug of Integrate? Any workaround?

see also

Buggy behavior of EllipticE[0,k] with arbitrary precision input

Buggy behavior of EllipticK with arbitrary precision input and $MinPrecision

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    $\begingroup$ Apparently a bug in Limit and not Integrate per se. In[3]:= InputForm[ii = Integrate[1/Sqrt[1-x^2+x^4],{x,0,t},Assumptions->t>0]] Out[3]//InputForm= -((-1)^(1/6)*Sqrt[1 - (-1)^(1/3)*t^2]* Sqrt[(1 + (-1)^(2/3)*t^2)/(1 - t^2 + t^4)]* EllipticF[I*ArcSinh[(-1)^(1/3)*t], (-1)^(2/3)]) In[4]:= InputForm[lii = Limit[ii, t->Infinity]] Out[4]//InputForm= (-1)^(1/6)*(Im[EllipticK[(3 - I*Sqrt[3])/2]] - I*Re[EllipticK[(3 - I*Sqrt[3])/2]]) In[5]:= N[lii] Out[5]= 0. - 1.68575 I $\endgroup$ Aug 6, 2023 at 17:11
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    $\begingroup$ ... which in turn is due to a Series glitch. In[9]:= nss = Normal[Series[ii, {t,Infinity,0}, Assumptions->t>1000]] 2/3 2/3 Out[9]= -((-1) EllipticK[1 - (-1) ]) In[10]:= N[nss] -16 Out[10]= -2.22045 10 - 1.68575 I $\endgroup$ Aug 6, 2023 at 17:14

1 Answer 1

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As shown by @Daniel Lichtblau, this is due to bug of Limit, see also tutorial/SomeNotesOnInternalImplementation#7441:

Definite integrals that involve no singularities are mostly done by taking limits of the indefinite integrals.

Many other definite integrals are done using Marichev[Dash]Adamchik Mellin transform methods. The results are often initially expressed in terms of Meijer G functions, which are converted into hypergeometric functions using Slater's theorem and then simplified.

Hence a simple workaround is to pull Infinity back to finite position, e.g. $x=\tan t$,

Inactive[Integrate][1/Sqrt[1-x^2+x^4],{x,0,Infinity}]//
    IntegrateChangeVariables[#,t,t==ArcTan[x]]&//Activate
(*EllipticK[3/4]*)
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