Integrate[InverseJacobiSD[z, m], m] == 2 I Sqrt[m - 1] (EllipticE[I ArcSinh[Sqrt[m - 1] z], m/(m - 1)] - EllipticF[I ArcSinh[Sqrt[m - 1] z], m/(m - 1)]) + (1/(z Sqrt[1 + (m - 1) z^2])) (-2 Sqrt[1 + (m - 1) z^2] + 2 Sqrt[1 + m z^2] + 2 (m - 1) z^2 Sqrt[1 + m z^2] - Sqrt[1 + (m - 1) z^2] Log[(1/4) (2 + (2 m - 1) z^2 + 2 Sqrt[1 + (m - 1) z^2] Sqrt[1 + m z^2])]) /; z > 0 && m > 0

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Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List["InverseJacobiSD", "[", RowBox[List["z_", ",", "m_"]], "]"]], RowBox[List["\[DifferentialD]", "m_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", SqrtBox[RowBox[List["m", "-", "1"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["EllipticE", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSinh", "[", RowBox[List[SqrtBox[RowBox[List["m", "-", "1"]]], " ", "z"]], "]"]]]], ",", FractionBox["m", RowBox[List["m", "-", "1"]]]]], "]"]], "-", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSinh", "[", RowBox[List[SqrtBox[RowBox[List["m", "-", "1"]]], " ", "z"]], "]"]]]], ",", FractionBox["m", RowBox[List["m", "-", "1"]]]]], "]"]]]], ")"]]]], "+", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", SqrtBox[RowBox[List["1", "+", RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], " ", SuperscriptBox["z", "2"]]]]]]]], "+", RowBox[List["2", " ", SqrtBox[RowBox[List["1", "+", RowBox[List["m", " ", SuperscriptBox["z", "2"]]]]]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], " ", SuperscriptBox["z", "2"], " ", SqrtBox[RowBox[List["1", "+", RowBox[List["m", " ", SuperscriptBox["z", "2"]]]]]]]], "-", RowBox[List[SqrtBox[RowBox[List["1", "+", RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], " ", SuperscriptBox["z", "2"]]]]]], " ", RowBox[List["Log", "[", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["2", "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "m"]], "-", "1"]], ")"]], " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["2", " ", SqrtBox[RowBox[List["1", "+", RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], " ", SuperscriptBox["z", "2"]]]]]], " ", SqrtBox[RowBox[List["1", "+", RowBox[List["m", " ", SuperscriptBox["z", "2"]]]]]]]]]], ")"]]]], "]"]]]]]], RowBox[List["z", " ", SqrtBox[RowBox[List["1", "+", RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], " ", SuperscriptBox["z", "2"]]]]]]]]]]], "/;", RowBox[List[RowBox[List["z", ">", "0"]], "&&", RowBox[List["m", ">", "0"]]]]]]]]]]

2001-10-29