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Calculate the integral of an elliptic function with assumptions?

Posted 1 month ago
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Mathematica 12 is unable to compute:

Assuming[ m > 0 && m < 1 && k > 0 && EllipticK[m] > 0, Integrate[  
( JacobiDN[k*x, m]^2 - EllipticE[m]/EllipticK[m] )^4 , {x, -EllipticK[m]/k, EllipticK[m]/k}]]

but Mathematica 11.1 can do it !?

Then my question: Is the result from Mathematica 11.1 correct?

Mathematica 12.0 can compute:

sol = Integrate[(JacobiDN[k*x, m]^2 - EllipticE[m]/EllipticK[m])^4, {x, -EllipticK[m]/k, 
      EllipticK[m]/k}, Assumptions -> {0 < m < 1, k > 0, EllipticK[m] > 0}]

(* -(1/(105 k EllipticK[m]^3))(630 EllipticE[m]^4 + 
   840 (-2 + m) EllipticE[m]^3 EllipticK[m] + 
   28 (61 + m (-61 + 16 m)) EllipticE[m]^2 EllipticK[m]^2 + 
   16 (-2 + m) (-5 + 2 m) (-5 + 3 m) EllipticE[m] EllipticK[m]^3 - 
   2 (-1 + m) (71 + m (-71 + 24 m)) EllipticK[m]^4) *)

Check:

  f[k_?NumericQ, m_?NumericQ] := NIntegrate[(JacobiDN[k*x, m]^2 - 
       EllipticE[m]/EllipticK[m])^4, {x, -EllipticK[m]/k, EllipticK[m]/k}]
  f[1, 1/2]
  (*0.00551185 *)

  sol /. k -> 1 /. m -> 1/2 // N
  (*0.00551185 *)

Regards M.I

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