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 Tanh

 http://functions.wolfram.com/01.21.21.0133.01

 Input Form

 Integrate[(Cosh[2 c z] Tanh[c z])/Sqrt[a + b Cosh[2 c z]], z] == (Cosh[(c z)/2]^2 Sqrt[(a + b Cosh[2 c z])/(1 + Cosh[c z])^2] (2 b Log[(2 (a - b + Sqrt[a - b] Sqrt[(a + b Cosh[2 c z]) Sech[(c z)/2]^4] + (-a + b) Tanh[(c z)/2]^2))/ (Sqrt[a - b] (1 + Tanh[(c z)/2]^2))] + Sqrt[a - b] Sqrt[(a + b Cosh[2 c z]) Sech[(c z)/2]^4] + Sqrt[a - b] Cosh[c z] Sqrt[(a + b Cosh[2 c z]) Sech[(c z)/2]^4]))/ (Sqrt[a - b] b c Sqrt[a + b Cosh[2 c z]])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List[RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]], " ", RowBox[List["Tanh", "[", RowBox[List["c", " ", "z"]], "]"]]]], SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["Cosh", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "2"], " ", SqrtBox[FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]]]], ")"]], "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b", " ", RowBox[List["Log", "[", RowBox[List[RowBox[List["(", RowBox[List["2", " ", RowBox[List["(", RowBox[List["a", "-", "b", "+", RowBox[List[SqrtBox[RowBox[List["a", "-", "b"]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Sech", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "4"]]]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Tanh", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "2"]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["a", "-", "b"]]], " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["Tanh", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "2"]]], ")"]]]], ")"]]]], "]"]]]], "+", RowBox[List[SqrtBox[RowBox[List["a", "-", "b"]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Sech", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "4"]]]]]], "+", RowBox[List[SqrtBox[RowBox[List["a", "-", "b"]]], " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Sech", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "4"]]]]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["a", "-", "b"]]], " ", "b", " ", "c", " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]]]], ")"]]]]]]]]

 MathML Form

 cosh ( 2 c z ) tanh ( c z ) a + b cosh ( 2 c z ) z cosh 2 ( c z 2 ) a + b cosh ( 2 c z ) ( cosh ( c z ) + 1 ) 2 ( a - b ( a + b cosh ( 2 c z ) ) sech 4 ( c z 2 ) cosh ( c z ) + 2 b log ( 2 ( ( b - a ) tanh 2 ( c z 2 ) + a - b + a - b ( a + b cosh ( 2 c z ) ) sech 4 ( c z 2 ) ) a - b ( tanh 2 ( c z 2 ) + 1 ) ) + a - b ( a + b cosh ( 2 c z ) ) sech 4 ( c z 2 ) ) / ( a - b b c a + b cosh ( 2 c z ) ) z 2 c z c z a b 2 c z 1 2 -1 c z 2 -1 2 a b 2 c z c z 1 2 -1 1 2 a -1 b 1 2 a b 2 c z c z 2 -1 4 1 2 c z 2 b 2 b -1 a c z 2 -1 2 a -1 b a -1 b 1 2 a b 2 c z c z 2 -1 4 1 2 a -1 b 1 2 c z 2 -1 2 1 -1 a -1 b 1 2 a b 2 c z c z 2 -1 4 1 2 a -1 b 1 2 b c a b 2 c z 1 2 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List[RowBox[List["Cosh", "[", RowBox[List["2", " ", "c_", " ", "z_"]], "]"]], " ", RowBox[List["Tanh", "[", RowBox[List["c_", " ", "z_"]], "]"]]]], SqrtBox[RowBox[List["a_", "+", RowBox[List["b_", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c_", " ", "z_"]], "]"]]]]]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["Cosh", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "2"], " ", SqrtBox[FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]]]], ")"]], "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b", " ", RowBox[List["Log", "[", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["a", "-", "b", "+", RowBox[List[SqrtBox[RowBox[List["a", "-", "b"]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Sech", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "4"]]]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Tanh", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "2"]]]]], ")"]]]], RowBox[List[SqrtBox[RowBox[List["a", "-", "b"]]], " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["Tanh", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "2"]]], ")"]]]]], "]"]]]], "+", RowBox[List[SqrtBox[RowBox[List["a", "-", "b"]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Sech", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "4"]]]]]], "+", RowBox[List[SqrtBox[RowBox[List["a", "-", "b"]]], " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Sech", "[", FractionBox[RowBox[List["c", " ", "z"]], "2"], "]"]], "4"]]]]]]]], ")"]]]], RowBox[List[SqrtBox[RowBox[List["a", "-", "b"]]], " ", "b", " ", "c", " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]]]]]]]]]

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

 2002-12-18