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 Cosh

 http://functions.wolfram.com/01.20.21.1345.01

 Input Form

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

 Standard Form

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 MathML Form

 b z cot ( c z ) cosh ( c z ) z - 1 2 ( ( b - c ) z 2 F 1 ( ( c - b ) 2 c , 1 ; ( 1 + 2 ) - b 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["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["c", "-", "b"]], ")"]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], ",", TagBox["1", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", FractionBox["\[ImaginaryI]", "2"]]], ")"]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "b"]], RowBox[List["2", " ", "c"]]]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] b - c + ( b + c ) z 2 F 1 ( - ( b + c ) 2 c , 1 ; 1 - ( b + c ) 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["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]]]], RowBox[List["2", " ", "c"]]]]], Hypergeometric2F1], ",", TagBox["1", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List["1", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]]]], RowBox[List["2", " ", "c"]]]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] b + c + ( b + c ( 1 + 2 ) ) z 2 F 1 ( - ( b + c ( 1 + 2 ) ) 2 c , 1 ; - ( b + c ( 1 + 4 ) ) 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["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[ImaginaryI]"]]]], " ", ")"]]]]]], ")"]]]], RowBox[List["2", " ", "c"]]]]], Hypergeometric2F1], ",", TagBox["1", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["4", " ", "\[ImaginaryI]"]]]], " ", ")"]]]]]], ")"]]]], RowBox[List["2", " ", "c"]]]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] b + c ( 1 + 2 ) + ( b - ( 1 - 2 ) c ) z 2 F 1 ( ( 1 + 2 ) - b 2 c , 1 ; ( 2 + 2 ) - b 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[RowBox[List["(", RowBox[List["1", "+", FractionBox["\[ImaginaryI]", "2"]]], ")"]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "b"]], RowBox[List["2", " ", "c"]]]]], Hypergeometric2F1], ",", TagBox["1", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List[RowBox[List["(", RowBox[List["2", "+", FractionBox["\[ImaginaryI]", "2"]]], ")"]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "b"]], RowBox[List["2", " ", "c"]]]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] b - ( 1 - 2 ) c ) z b z c z c z -1 1 2 b -1 c z Hypergeometric2F1 c -1 b 2 c -1 1 1 2 -1 -1 b 2 c -1 2 c z b -1 c -1 b c z Hypergeometric2F1 -1 b c 2 c -1 1 1 -1 b c 2 c -1 2 c z b c -1 b c 1 2 z Hypergeometric2F1 -1 b c 1 2 2 c -1 1 -1 b c 1 4 2 c -1 2 c z b c 1 2 -1 b -1 1 -2 c z Hypergeometric2F1 1 2 -1 -1 b 2 c -1 1 2 2 -1 -1 b 2 c -1 2 c z b -1 1 -2 c -1 [/itex]

 Rule Form

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

 2002-12-18