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 Tanh

 http://functions.wolfram.com/01.21.21.0173.01

 Input Form

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

 Standard Form

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 MathML Form

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