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 Tanh

 http://functions.wolfram.com/01.21.21.0342.01

 Input Form

 Integrate[Cosh[2 c z] (a + b Cosh[2 c z])^\[Beta] Tanh[c z]^\[Nu], z] == (1/(c (1 + \[Nu]) (3 + \[Nu]))) (((a + b Cosh[2 c z])^\[Beta] (Sech[c z]^2)^\[Beta] Tanh[c z]^(1 + \[Nu]) ((3 + \[Nu]) AppellF1[(1 + \[Nu])/2, 2 + \[Beta], -\[Beta], (3 + \[Nu])/2, Tanh[c z]^2, ((a - b) Tanh[c z]^2)/(a + b)] + (1 + \[Nu]) AppellF1[(3 + \[Nu])/2, 2 + \[Beta], -\[Beta], (5 + \[Nu])/2, Tanh[c z]^2, ((a - b) Tanh[c z]^2)/(a + b)] Tanh[c z]^2))/(1 + ((-a + b) Tanh[c z]^2)/(a + b))^\[Beta])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["Cosh", "[", RowBox[List["2", "c", " ", "z"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", "c", " ", "z"]], "]"]]]]]], ")"]], "\[Beta]"], " ", SuperscriptBox[RowBox[List["Tanh", "[", RowBox[List["c", " ", "z"]], "]"]], "\[Nu]"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["c", " ", RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", "\[Nu]"]], ")"]]]]], RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], ")"]], "\[Beta]"], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["Sech", "[", RowBox[List["c", " ", "z"]], "]"]], "2"], ")"]], "\[Beta]"], " ", SuperscriptBox[RowBox[List["Tanh", "[", RowBox[List["c", " ", "z"]], "]"]], RowBox[List["1", "+", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Tanh", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]], RowBox[List["a", "+", "b"]]]]], ")"]], RowBox[List["-", "\[Beta]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["3", "+", "\[Nu]"]], ")"]], " ", RowBox[List["AppellF1", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], ",", RowBox[List["2", "+", "\[Beta]"]], ",", RowBox[List["-", "\[Beta]"]], ",", FractionBox[RowBox[List["3", "+", "\[Nu]"]], "2"], ",", SuperscriptBox[RowBox[List["Tanh", "[", RowBox[List["c", " ", "z"]], "]"]], "2"], ",", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Tanh", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]], RowBox[List["a", "+", "b"]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]], " ", RowBox[List["AppellF1", "[", RowBox[List[FractionBox[RowBox[List["3", "+", "\[Nu]"]], "2"], ",", RowBox[List["2", "+", "\[Beta]"]], ",", RowBox[List["-", "\[Beta]"]], ",", FractionBox[RowBox[List["5", "+", "\[Nu]"]], "2"], ",", SuperscriptBox[RowBox[List["Tanh", "[", RowBox[List["c", " ", "z"]], "]"]], "2"], ",", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Tanh", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]], RowBox[List["a", "+", "b"]]]]], "]"]], " ", SuperscriptBox[RowBox[List["Tanh", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]], ")"]]]], ")"]]]]]]]]

 MathML Form

 cosh ( 2 c z ) ( a + b cosh ( 2 c z ) ) β tanh ν ( c z ) z 1 c ( ν + 1 ) ( ν + 3 ) ( ( a + b cosh ( 2 c z ) ) β sech 2 ( c z ) β tanh ν + 1 ( c z ) ( ( b - a ) tanh 2 ( c z ) a + b + 1 ) - β ( ( ν + 1 ) F 1 AppellF1 ( ν + 3 2 ; β + 2 , - β ; ν + 5 2 ; tanh 2 ( c z ) , ( a - b ) tanh 2 ( c z ) a + b ) tanh 2 ( c z ) + ( ν + 3 ) F 1 AppellF1 ( ν + 1 2 ; β + 2 , - β ; ν + 3 2 ; tanh 2 ( c z ) , ( a - b ) tanh 2 ( c z ) a + b ) ) ) z 2 c z a b 2 c z β c z ν 1 c ν 1 ν 3 -1 a b 2 c z β c z 2 β c z ν 1 b -1 a c z 2 a b -1 1 -1 β ν 1 AppellF1 ν 3 2 -1 β 2 -1 β ν 5 2 -1 c z 2 a -1 b c z 2 a b -1 c z 2 ν 3 AppellF1 ν 1 2 -1 β 2 -1 β ν 3 2 -1 c z 2 a -1 b c z 2 a b -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[RowBox[List["Cosh", "[", RowBox[List["2", " ", "c_", " ", "z_"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a_", "+", RowBox[List["b_", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c_", " ", "z_"]], "]"]]]]]], ")"]], "\[Beta]_"], " ", SuperscriptBox[RowBox[List["Tanh", "[", RowBox[List["c_", " ", "z_"]], "]"]], "\[Nu]_"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], ")"]], "\[Beta]"], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["Sech", "[", RowBox[List["c", " ", "z"]], "]"]], "2"], ")"]], "\[Beta]"], " ", SuperscriptBox[RowBox[List["Tanh", "[", RowBox[List["c", " ", "z"]], "]"]], RowBox[List["1", "+", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Tanh", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]], RowBox[List["a", "+", "b"]]]]], ")"]], RowBox[List["-", "\[Beta]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["3", "+", "\[Nu]"]], ")"]], " ", RowBox[List["AppellF1", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], ",", RowBox[List["2", "+", "\[Beta]"]], ",", RowBox[List["-", "\[Beta]"]], ",", FractionBox[RowBox[List["3", "+", "\[Nu]"]], "2"], ",", SuperscriptBox[RowBox[List["Tanh", "[", RowBox[List["c", " ", "z"]], "]"]], "2"], ",", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Tanh", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]], RowBox[List["a", "+", "b"]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]], " ", RowBox[List["AppellF1", "[", RowBox[List[FractionBox[RowBox[List["3", "+", "\[Nu]"]], "2"], ",", RowBox[List["2", "+", "\[Beta]"]], ",", RowBox[List["-", "\[Beta]"]], ",", FractionBox[RowBox[List["5", "+", "\[Nu]"]], "2"], ",", SuperscriptBox[RowBox[List["Tanh", "[", RowBox[List["c", " ", "z"]], "]"]], "2"], ",", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["Tanh", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]], RowBox[List["a", "+", "b"]]]]], "]"]], " ", SuperscriptBox[RowBox[List["Tanh", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]], ")"]]]], RowBox[List["c", " ", RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", "\[Nu]"]], ")"]]]]]]]]]

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

 2002-12-18