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 Tanh

 http://functions.wolfram.com/01.21.21.0177.01

 Input Form

 Integrate[z^n E^(p z) Cosh[b z] Tanh[c z], z] == (1/2) n! ((-E^((-b + p) z)) Sum[(1/(-j + n)!) ((-1)^j (-b + p)^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[a, 1], \[Ellipsis], Subscript[a, 1 + j], 1}, {1 + Subscript[a, 1], \[Ellipsis], 1 + Subscript[a, 1 + j]}, -E^(2 c z)]), {j, 0, n}] + E^((-b + 2 c + p) z) Sum[(1/(-j + n)!) ((-1)^j (-b + 2 c + p)^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[b, 1], \[Ellipsis], Subscript[b, 1 + j], 1}, {1 + Subscript[b, 1], \[Ellipsis], 1 + Subscript[b, 1 + j]}, -E^(2 c z)]), {j, 0, n}] - E^((b + p) z) Sum[(1/(-j + n)!) ((-1)^j (b + p)^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[c, 1], \[Ellipsis], Subscript[c, 1 + j], 1}, {1 + Subscript[c, 1], \[Ellipsis], 1 + Subscript[c, 1 + j]}, -E^(2 c z)]), {j, 0, n}] + E^((b + 2 c + p) z) Sum[(1/(-j + n)!) ((-1)^j (b + 2 c + p)^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[d, 1], \[Ellipsis], Subscript[d, 1 + j], 1}, {1 + Subscript[d, 1], \[Ellipsis], 1 + Subscript[d, 1 + j]}, -E^(2 c z)]), {j, 0, n}]) /; Subscript[a, 1] == Subscript[a, 2] == \[Ellipsis] == Subscript[a, n + 1] == (-b + p)/(2 c) && Subscript[b, 1] == Subscript[b, 2] == \[Ellipsis] == Subscript[b, n + 1] == (-b + 2 c + p)/(2 c) && Subscript[c, 1] == Subscript[c, 2] == \[Ellipsis] == Subscript[c, n + 1] == (b + p)/(2 c) && Subscript[d, 1] == Subscript[d, 2] == \[Ellipsis] == Subscript[d, n + 1] == (b + 2 c + p)/(2 c) && Element[n, Integers] && n >= 0

 Standard Form

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 MathML Form

 z n p z cosh ( b z ) tanh ( c z ) z 1 2 n ! ( ( b + 2 c + p ) z j = 0 n ( - 1 ) j ( b + 2 c + p ) - j - 1 z n - j ( n - j ) ! j + 2 F j + 1 ( b + 2 c + p 2 c , , b + 2 c + p 2 c , 1 ; b + 2 c + p 2 c + 1 , , b + 2 c + p 2 c + 1 ; - 2 c z ) TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["j", "+", "2"]], TraditionalForm]], SubscriptBox["F", FormBox[RowBox[List["j", "+", "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"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List["b", "+", RowBox[List["2", " ", "c"]], "+", "p"]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["1", HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox[RowBox[List["b", "+", RowBox[List["2", " ", "c"]], "+", "p"]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox[RowBox[List["b", "+", RowBox[List["2", " ", "c"]], "+", "p"]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]] - ( b + p ) z j = 0 n ( - 1 ) j ( b + p ) - j - 1 z n - j ( n - j ) ! j + 2 F j + 1 ( b + p 2 c , , b + p 2 c , 1 ; b + p 2 c + 1 , , b + p 2 c + 1 ; - 2 c z ) TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["j", "+", "2"]], TraditionalForm]], SubscriptBox["F", FormBox[RowBox[List["j", "+", "1"]], TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["b", "+", "p"]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List["b", "+", "p"]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["1", HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox[RowBox[List["b", "+", "p"]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox[RowBox[List["b", "+", "p"]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]] + ( - b + 2 c + p ) z j = 0 n ( - 1 ) j ( - b + 2 c + p ) - j - 1 z n - j ( n - j ) ! j + 2 F j + 1 ( - b + 2 c + p 2 c , , - b + 2 c + p 2 c , 1 ; - b + 2 c + p 2 c + 1 , , - b + 2 c + p 2 c + 1 ; - 2 c z ) TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["j", "+", "2"]], TraditionalForm]], SubscriptBox["F", FormBox[RowBox[List["j", "+", "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"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["2", " ", "c"]], "+", "p"]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["1", HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["2", " ", "c"]], "+", "p"]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["2", " ", "c"]], "+", "p"]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]] - ( p - b ) z j = 0 n ( - 1 ) j ( p - b ) - j - 1 z n - j ( n - j ) ! j + 2 F j + 1 ( p - b 2 c , , p - b 2 c , 1 ; p - b 2 c + 1 , , p - b 2 c + 1 ; - 2 c z ) TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["j", "+", "2"]], TraditionalForm]], SubscriptBox["F", FormBox[RowBox[List["j", "+", "1"]], TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["p", "-", "b"]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List["p", "-", "b"]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["1", HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox[RowBox[List["p", "-", "b"]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox[RowBox[List["p", "-", "b"]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]] ) /; n Condition z z n p z b z c z 1 2 n b 2 c p z j 0 n -1 j b 2 c p -1 j -1 z n -1 j n -1 j -1 HypergeometricPFQ b 2 c p 2 c -1 b 2 c p 2 c -1 1 b 2 c p 2 c -1 1 b 2 c p 2 c -1 1 -1 2 c z -1 b p z j 0 n -1 j b p -1 j -1 z n -1 j n -1 j -1 HypergeometricPFQ b p 2 c -1 b p 2 c -1 1 b p 2 c -1 1 b p 2 c -1 1 -1 2 c z -1 b 2 c p z j 0 n -1 j -1 b 2 c p -1 j -1 z n -1 j n -1 j -1 HypergeometricPFQ -1 b 2 c p 2 c -1 -1 b 2 c p 2 c -1 1 -1 b 2 c p 2 c -1 1 -1 b 2 c p 2 c -1 1 -1 2 c z -1 p -1 b z j 0 n -1 j p -1 b -1 j -1 z n -1 j n -1 j -1 HypergeometricPFQ p -1 b 2 c -1 p -1 b 2 c -1 1 p -1 b 2 c -1 1 p -1 b 2 c -1 1 -1 2 c z n [/itex]

 Rule Form

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

 2002-12-18