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 Coth

 http://functions.wolfram.com/01.22.21.0167.01

 Input Form

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

 Standard Form

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

 MathML Form

 p z cosh ( b z ) coth ( c z ) z - 1 4 ( 2 ( p - b ) z 2 F 1 ( p - b 2 c , 1 ; - b + 2 c + p 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["p", "-", "b"]], 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"]], "+", "p"]], 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] p - b + 2 ( b + p ) z 2 F 1 ( b + p 2 c , 1 ; b + 2 c + p 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", "+", "p"]], 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"]], "+", "p"]], 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 + p + 2 ( - b + 2 c + p ) z 2 F 1 ( - b + 2 c + p 2 c , 1 ; - b + 4 c + p 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["2", " ", "c"]], "+", "p"]], 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["4", " ", "c"]], "+", "p"]], 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 + p + 2 ( b + 2 c + p ) z 2 F 1 ( b + 2 c + p 2 c , 1 ; b + 4 c + p 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"]], "+", "p"]], 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"]], "+", "p"]], 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 + p ) z p z b z c z -1 1 4 2 p -1 b z Hypergeometric2F1 p -1 b 2 c -1 1 -1 b 2 c p 2 c -1 2 c z p -1 b -1 2 b p z Hypergeometric2F1 b p 2 c -1 1 b 2 c p 2 c -1 2 c z b p -1 2 -1 b 2 c p z Hypergeometric2F1 -1 b 2 c p 2 c -1 1 -1 b 4 c p 2 c -1 2 c z -1 b 2 c p -1 2 b 2 c p z Hypergeometric2F1 b 2 c p 2 c -1 1 b 4 c p 2 c -1 2 c z b 2 c p -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["p_", " ", "z_"]]], " ", RowBox[List["Cosh", "[", RowBox[List["b_", " ", "z_"]], "]"]], " ", RowBox[List["Coth", "[", RowBox[List["c_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", "p"]], ")"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", "p"]], RowBox[List["2", " ", "c"]]], ",", "1", ",", FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["2", " ", "c"]], "+", "p"]], RowBox[List["2", " ", "c"]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "]"]]]], RowBox[List[RowBox[List["-", "b"]], "+", "p"]]], "+", FractionBox[RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["b", "+", "p"]], ")"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["b", "+", "p"]], RowBox[List["2", " ", "c"]]], ",", "1", ",", FractionBox[RowBox[List["b", "+", RowBox[List["2", " ", "c"]], "+", "p"]], RowBox[List["2", " ", "c"]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "]"]]]], RowBox[List["b", "+", "p"]]], "+", FractionBox[RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["2", " ", "c"]], "+", "p"]], ")"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["2", " ", "c"]], "+", "p"]], RowBox[List["2", " ", "c"]]], ",", "1", ",", FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["4", " ", "c"]], "+", "p"]], RowBox[List["2", " ", "c"]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "]"]]]], RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["2", " ", "c"]], "+", "p"]]], "+", FractionBox[RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["b", "+", RowBox[List["2", " ", "c"]], "+", "p"]], ")"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["b", "+", RowBox[List["2", " ", "c"]], "+", "p"]], RowBox[List["2", " ", "c"]]], ",", "1", ",", FractionBox[RowBox[List["b", "+", RowBox[List["4", " ", "c"]], "+", "p"]], RowBox[List["2", " ", "c"]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], "]"]]]], RowBox[List["b", "+", RowBox[List["2", " ", "c"]], "+", "p"]]]]], ")"]]]]]]]]

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

 2002-12-18