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 Cosh

 http://functions.wolfram.com/01.20.21.4602.01

 Input Form

 Integrate[(z Sinh[2 c z])/(a Sinh[c z] + b Cosh[c z]), z] == (1/((a + b) c^2)) (2 ((2 a b ArcTan[(a + b Tanh[(c z)/2])/Sqrt[-a^2 + b^2]] ArcTanh[b/a])/((a - b) Sqrt[-a^2 + b^2]) + ((a + b c z) Cosh[c z])/ (-a + b) - (b ((c z + ArcTanh[b/a]) (Log[1 - E^((-c) z - ArcTanh[b/a])] - Log[1 + E^((-c) z - ArcTanh[b/a])]) + PolyLog[2, -E^((-c) z - ArcTanh[b/a])] - PolyLog[2, E^((-c) z - ArcTanh[b/a])]))/((a - b) Sqrt[1 - b^2/a^2]) + ((b + a c z) Sinh[c z])/(a - b)))

 Standard Form

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

 MathML Form

 z sinh ( 2 c z ) a sinh ( c z ) + b cosh ( c z ) z 1 ( a + b ) c 2 ( 2 ( 2 a b tan - 1 ( a + b tanh ( c z 2 ) b 2 - a 2 ) tanh - 1 ( b a ) ( a - b ) b 2 - a 2 + ( a + b c z ) cosh ( c z ) b - a + ( b + a c z ) sinh ( c z ) a - b - 1 ( a - b ) 1 - b 2 a 2 ( b ( ( c z + tanh - 1 ( b a ) ) ( log ( 1 - - c z - tanh - 1 ( b a ) ) - log ( 1 + - c z - tanh - 1 ( b a ) ) ) + Li PolyLog 2 ( - - c z - tanh - 1 ( b a ) ) - Li PolyLog 2 ( - c z - tanh - 1 ( b a ) ) ) ) ) ) z z 2 c z a c z b c z -1 1 a b c 2 -1 2 2 a b a b c z 2 -1 b 2 -1 a 2 1 2 -1 b a -1 a -1 b b 2 -1 a 2 1 2 -1 a b c z c z b -1 a -1 b a c z c z a -1 b -1 -1 1 a -1 b 1 -1 b 2 a 2 -1 1 2 -1 b c z b a -1 1 -1 -1 c z -1 b a -1 -1 1 -1 c z -1 b a -1 PolyLog 2 -1 -1 c z -1 b a -1 -1 PolyLog 2 -1 c z -1 b a -1 [/itex]

 Rule Form

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

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

 2002-12-18