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Csch






Mathematica Notation

Traditional Notation









Elementary Functions > Csch[z] > Integration > Indefinite integration > Involving functions of the direct function and hyperbolic functions > Involving algebraic functions of the direct function and hyperbolic functions > Involving tanh > Involving tanh(c z)(a+b csch2(c z))beta





http://functions.wolfram.com/01.23.21.0453.01









  


  










Input Form





Integrate[Tanh[c z] Sqrt[a + b Csch[c z]^2], z] == ((Sqrt[-2 a + 2 b] ArcTan[(Sqrt[-2 a + 2 b] Sinh[c z])/ Sqrt[-a + 2 b + a Cosh[2 c z]]] Sqrt[-a + 2 b + a Cosh[2 c z]] + Sqrt[2] Sqrt[a] Sqrt[b] ArcSinh[(Sqrt[a] Sinh[c z])/Sqrt[b]] Sqrt[(-a + 2 b + a Cosh[2 c z])/b]) Sqrt[a + b Csch[c z]^2] Sinh[c z])/ (c (-a + 2 b + a Cosh[2 c z]))










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["Tanh", "[", RowBox[List["c", " ", "z"]], "]"]], SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Csch", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "a"]], "+", RowBox[List["2", " ", "b"]]]]], " ", RowBox[List["ArcTan", "[", FractionBox[RowBox[List[SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "a"]], "+", RowBox[List["2", " ", "b"]]]]], " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]], SqrtBox[RowBox[List[RowBox[List["-", "a"]], "+", RowBox[List["2", " ", "b"]], "+", RowBox[List["a", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]]], "]"]], " ", SqrtBox[RowBox[List[RowBox[List["-", "a"]], "+", RowBox[List["2", " ", "b"]], "+", RowBox[List["a", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]]]], "+", RowBox[List[SqrtBox["2"], " ", SqrtBox["a"], " ", SqrtBox["b"], " ", RowBox[List["ArcSinh", "[", FractionBox[RowBox[List[SqrtBox["a"], " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]], SqrtBox["b"]], "]"]], " ", SqrtBox[FractionBox[RowBox[List[RowBox[List["-", "a"]], "+", RowBox[List["2", " ", "b"]], "+", RowBox[List["a", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], "b"]]]]]], ")"]], " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Csch", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]]], " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", RowBox[List["2", " ", "b"]], "+", RowBox[List["a", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], ")"]]]], ")"]]]]]]]]










MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[RowBox[List["Tanh", "[", RowBox[List["c_", " ", "z_"]], "]"]], " ", SqrtBox[RowBox[List["a_", "+", RowBox[List["b_", " ", SuperscriptBox[RowBox[List["Csch", "[", RowBox[List["c_", " ", "z_"]], "]"]], "2"]]]]]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "a"]], "+", RowBox[List["2", " ", "b"]]]]], " ", RowBox[List["ArcTan", "[", FractionBox[RowBox[List[SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "a"]], "+", RowBox[List["2", " ", "b"]]]]], " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]], SqrtBox[RowBox[List[RowBox[List["-", "a"]], "+", RowBox[List["2", " ", "b"]], "+", RowBox[List["a", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]]], "]"]], " ", SqrtBox[RowBox[List[RowBox[List["-", "a"]], "+", RowBox[List["2", " ", "b"]], "+", RowBox[List["a", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]]]], "+", RowBox[List[SqrtBox["2"], " ", SqrtBox["a"], " ", SqrtBox["b"], " ", RowBox[List["ArcSinh", "[", FractionBox[RowBox[List[SqrtBox["a"], " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]], SqrtBox["b"]], "]"]], " ", SqrtBox[FractionBox[RowBox[List[RowBox[List["-", "a"]], "+", RowBox[List["2", " ", "b"]], "+", RowBox[List["a", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], "b"]]]]]], ")"]], " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Csch", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]]], " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]], RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", RowBox[List["2", " ", "b"]], "+", RowBox[List["a", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], ")"]]]]]]]]]










Date Added to functions.wolfram.com (modification date)





2002-12-18





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