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 Coth

 http://functions.wolfram.com/01.22.21.0180.01

 Input Form

 Integrate[Sin[a z] Cosh[b z] Coth[c z], z] == (1/4) I (-((E^(((-I) a - b) z) Hypergeometric2F1[((-I) a - b)/(2 c), 1, ((-I) a - b + 2 c)/(2 c), E^(2 c z)])/((-I) a - b)) + (E^((I a - b) z) Hypergeometric2F1[(I a - b)/(2 c), 1, (I a - b + 2 c)/(2 c), E^(2 c z)])/(I a - b) - (E^(((-I) a + b) z) Hypergeometric2F1[((-I) a + b)/(2 c), 1, ((-I) a + b + 2 c)/(2 c), E^(2 c z)])/((-I) a + b) + (E^((I a + b) z) Hypergeometric2F1[(I a + b)/(2 c), 1, (I a + b + 2 c)/(2 c), E^(2 c z)])/(I a + b) - (E^(((-I) a - b + 2 c) z) Hypergeometric2F1[((-I) a - b + 2 c)/(2 c), 1, ((-I) a - b + 4 c)/(2 c), E^(2 c z)])/((-I) a - b + 2 c) + (E^((I a - b + 2 c) z) Hypergeometric2F1[(I a - b + 2 c)/(2 c), 1, (I a - b + 4 c)/(2 c), E^(2 c z)])/(I a - b + 2 c) - (E^(((-I) a + b + 2 c) z) Hypergeometric2F1[((-I) a + b + 2 c)/(2 c), 1, ((-I) a + b + 4 c)/(2 c), E^(2 c z)])/((-I) a + b + 2 c) + (E^((I a + b + 2 c) z) Hypergeometric2F1[(I a + b + 2 c)/(2 c), 1, (I a + b + 4 c)/(2 c), E^(2 c z)])/(I a + b + 2 c))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["Sin", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["Cosh", "[", RowBox[List["b", " ", "z"]], "]"]], RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]], " ", RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "4"], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "-", "b"]], ")"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "-", "b"]], RowBox[List["2", " ", "c"]]], ",", "1", ",", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "-", "b", "+", RowBox[List["2", " ", "c"]]]], RowBox[List["2", " ", "c"]]], ",", SuperscriptBox["\[ExponentialE]", 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 MathML Form

 sin ( a z ) cosh ( b z ) coth ( c z ) z 1 4 ( - ( - b - a ) z 2 F 1 ( - b - a 2 c , 1 ; - b + 2 c - a 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[FractionBox[RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List["\[ImaginaryI]", " ", "a", " "]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], ",", TagBox["1", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["2", " ", "c"]], "-", RowBox[List["\[ImaginaryI]", " ", "a", " "]]]], 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] - b - a + ( a - b ) z 2 F 1 ( a - b 2 c , 1 ; 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b + 2 c + a 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[FractionBox[RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "a", " "]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], ",", TagBox["1", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List["b", "+", RowBox[List["2", " ", "c"]], "+", RowBox[List["\[ImaginaryI]", " ", "a", " "]]]], 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] b + a + ( - b + 2 c + a ) z 2 F 1 ( - b + 2 c + a 2 c , 1 ; 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b + 4 c + a 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[FractionBox[RowBox[List["b", "+", RowBox[List["2", " ", "c"]], "+", RowBox[List["\[ImaginaryI]", " ", "a", " "]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], ",", TagBox["1", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List["b", "+", RowBox[List["4", " ", "c"]], "+", RowBox[List["\[ImaginaryI]", " ", "a", " "]]]], 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] b + 2 c + a - ( b - a ) z 2 F 1 ( b - a 2 c , 1 ; 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b + 4 c - a 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[FractionBox[RowBox[List["b", "+", RowBox[List["2", " ", "c"]], "-", RowBox[List["\[ImaginaryI]", " ", "a", " "]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], ",", TagBox["1", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List["b", "+", RowBox[List["4", " ", "c"]], "-", RowBox[List["\[ImaginaryI]", " ", "a", " "]]]], 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] b + 2 c - a ) z a z b z c z 1 4 -1 -1 b -1 a z Hypergeometric2F1 -1 b -1 a 2 c -1 1 -1 b 2 c -1 a 2 c -1 2 c z -1 b -1 a -1 a -1 b z Hypergeometric2F1 a -1 b 2 c -1 1 -1 b 2 c a 2 c -1 2 c z a -1 b -1 b a z Hypergeometric2F1 b a 2 c -1 1 b 2 c a 2 c -1 2 c z b a -1 -1 b 2 c a z Hypergeometric2F1 -1 b 2 c a 2 c -1 1 -1 b 4 c a 2 c -1 2 c z -1 b 2 c a -1 b 2 c a z Hypergeometric2F1 b 2 c a 2 c -1 1 b 4 c a 2 c -1 2 c z b 2 c a -1 -1 b -1 a z Hypergeometric2F1 b -1 a 2 c -1 1 b 2 c -1 a 2 c -1 2 c z b -1 a -1 -1 -1 b 2 c -1 a z Hypergeometric2F1 -1 b 2 c -1 a 2 c -1 1 -1 b 4 c -1 a 2 c -1 2 c z -1 b 2 c -1 a -1 -1 b 2 c -1 a z Hypergeometric2F1 b 2 c -1 a 2 c -1 1 b 4 c -1 a 2 c -1 2 c z b 2 c -1 a -1 [/itex]

 Rule Form

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

 2002-12-18