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 Cosh

 http://functions.wolfram.com/01.20.21.1548.01

 Input Form

 Integrate[Sinh[c z]^\[Mu] Cosh[b + a z], z] == ((1/2) (-((1/(-a + c \[Mu])) (E^(2 b + a z) Hypergeometric2F1[ -((-a + c \[Mu])/(2 c)), -\[Mu], (1/2) (2 + a/c - \[Mu]), E^(2 c z)])) + Hypergeometric2F1[-((a + c \[Mu])/(2 c)), -\[Mu], -((a + c (-2 + \[Mu]))/(2 c)), E^(2 c z)]/(E^(a z) (-a - c \[Mu]))) Sinh[c z]^\[Mu])/(E^b (1 - E^(2 c z))^\[Mu])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]], "\[Mu]"], RowBox[List["Cosh", "[", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", "b"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], ")"]], RowBox[List["-", "\[Mu]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["1", RowBox[List[RowBox[List["-", "a"]], "+", RowBox[List["c", " ", "\[Mu]"]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["2", " ", "b"]], "+", RowBox[List["a", " ", "z"]]]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["-", "a"]], "+", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]]]], ",", RowBox[List["-", "\[Mu]"]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", FractionBox["a", "c"], "-", "\[Mu]"]], ")"]]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "]"]]]], ")"]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "a"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["a", "+", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]]]], ",", RowBox[List["-", "\[Mu]"]], ",", RowBox[List["-", FractionBox[RowBox[List["a", "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "\[Mu]"]], ")"]]]]]], RowBox[List["2", " ", "c"]]]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "]"]]]], RowBox[List[RowBox[List["-", "a"]], "-", RowBox[List["c", " ", "\[Mu]"]]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]], "\[Mu]"]]]]]]]

 MathML Form

 sinh μ ( c z ) cosh ( b + a z ) z 1 2 - b ( 1 - 2 c z ) - μ ( - a z - a - c μ 2 F 1 ( - a + c μ 2 c , - μ ; - a + c ( μ - 2 ) 2 c ; 2 c z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", FractionBox[RowBox[List["a", "+", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]]]], Hypergeometric2F1], ",", TagBox[RowBox[List["-", "\[Mu]"]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List["-", FractionBox[RowBox[List["a", "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List["\[Mu]", "-", "2"]], ")"]]]]]], RowBox[List["2", " ", "c"]]]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] - 2 b + a z c μ - a 2 F 1 ( - c μ - a 2 c , - μ ; 1 2 ( a c - μ + 2 ) ; 2 c z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["c", " ", "\[Mu]"]], "-", "a"]], RowBox[List["2", " ", "c"]]]]], Hypergeometric2F1], ",", TagBox[RowBox[List["-", "\[Mu]"]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[FractionBox["a", "c"], "-", "\[Mu]", "+", "2"]], ")"]]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] ) sinh μ ( c z ) z c z μ b a z 1 2 -1 b 1 -1 2 c z -1 μ -1 a z -1 a -1 c μ -1 Hypergeometric2F1 -1 a c μ 2 c -1 -1 μ -1 a c μ -2 2 c -1 2 c z -1 2 b a z c μ -1 a -1 Hypergeometric2F1 -1 c μ -1 a 2 c -1 -1 μ 1 2 a c -1 -1 μ 2 2 c z c z μ [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["c_", " ", "z_"]], "]"]], "\[Mu]_"], " ", RowBox[List["Cosh", "[", RowBox[List["b_", "+", RowBox[List["a_", " ", "z_"]]]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", "b"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], ")"]], RowBox[List["-", "\[Mu]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["2", " ", "b"]], "+", RowBox[List["a", " ", "z"]]]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["-", "a"]], "+", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]]]], ",", RowBox[List["-", "\[Mu]"]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", FractionBox["a", "c"], "-", "\[Mu]"]], ")"]]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "]"]]]], RowBox[List[RowBox[List["-", "a"]], "+", RowBox[List["c", " ", "\[Mu]"]]]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "a"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["a", "+", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]]]], ",", RowBox[List["-", "\[Mu]"]], ",", RowBox[List["-", FractionBox[RowBox[List["a", "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "\[Mu]"]], ")"]]]]]], RowBox[List["2", " ", "c"]]]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "]"]]]], RowBox[List[RowBox[List["-", "a"]], "-", RowBox[List["c", " ", "\[Mu]"]]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]], "\[Mu]"]]]]]]]

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

 2002-12-18