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 Sech

 http://functions.wolfram.com/01.24.21.0225.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "n"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]]]], " ", "z"]]], RowBox[List["Sinh", "[", RowBox[List["a", "+", RowBox[List["b", " ", "z"]]]], "]"]], RowBox[List["Sech", "[", RowBox[List["c", " ", "z"]], "]"]], " ", RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", "a"], SuperscriptBox["z", RowBox[List["1", "+", "n"]]], " "]], RowBox[List["1", "+", "n"]]], "-", "\[IndentingNewLine]", RowBox[List[RowBox[List["n", "!"]], SuperscriptBox["\[ExponentialE]", RowBox[List["a", "+", RowBox[List["2", "c", " ", "z"]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", "c"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "j"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "j"]], "+", "n"]]], " "]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "j"]], "+", "n"]], ")"]], "!"]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["a", RowBox[List["j", "+", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["a", RowBox[List["j", "+", "1"]]]]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]]]]]], "+", RowBox[List[RowBox[List["n", "!"]], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "a"]], "-", RowBox[List["2", "b", " ", "z"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[FractionBox[SuperscriptBox["z", RowBox[List[RowBox[List["-", "j"]], "+", "n"]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n", "-", "j"]], ")"]], "!"]], SuperscriptBox[RowBox[List["(", RowBox[List["2", "b"]], ")"]], RowBox[List["j", "+", "1"]]]]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["b", RowBox[List["j", "+", "1"]]], ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["b", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["b", RowBox[List["j", "+", "1"]]]]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["a", "1"], "\[Equal]", SubscriptBox["a", "2"], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["a", RowBox[List["n", "+", "2"]]], "\[Equal]", "1"]], "\[And]", RowBox[List[SubscriptBox["b", "1"], "\[Equal]", SubscriptBox["b", "2"], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["b", RowBox[List["n", "+", "1"]]], "\[Equal]", RowBox[List["-", FractionBox["b", "c"]]]]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 z n - ( b + c ) z sinh ( a + b z ) sech ( c z ) z a z n + 1 n + 1 + n ! - a - 2 b z j = 0 n z n - j ( n - j ) ! ( 2 b ) j + 1 j + 2 F j + 1 ( - b c , , - b c , 1 ; - b c + 1 , , - b 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[RowBox[List["-", FractionBox["b", "c"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List["-", FractionBox["b", "c"]]], HypergeometricPFQ], ",", TagBox["1", HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[RowBox[List["-", FractionBox["b", "c"]]], "+", "1"]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List[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]]]]] - n ! a + 2 c z j = 0 n ( - 1 ) j ( 2 c ) - j - 1 z n - j ( n - j ) ! j + 2 F j + 1 ( 1 , , 1 , 1 ; 2 , , 2 ; - 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["1", HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox["1", HypergeometricPFQ], ",", TagBox["1", 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]]]]] /; n Condition z z n -1 b c z a b z c z a z n 1 n 1 -1 n -1 a -1 2 b z j 0 n z n -1 j n -1 j 2 b j 1 -1 HypergeometricPFQ -1 b c -1 -1 b c -1 1 -1 b c -1 1 -1 b c -1 1 -1 2 c z -1 n a 2 c z j 0 n -1 j 2 c -1 j -1 z n -1 j n -1 j -1 HypergeometricPFQ 1 1 1 2 2 -1 2 c z n [/itex]

 Rule Form

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

 2002-12-18