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 Sech

 http://functions.wolfram.com/01.24.21.0265.01

 Input Form

 Integrate[E^(p z) Cos[a z] Csch[c z] Sech[c z], z] == (2 ((1/(I a - 2 c + p)) (E^((I a + p) z) HypergeometricPFQ[ {1, 1/2 - (I a)/(4 c) - p/(4 c)}, {3/2 - (I a)/(4 c) - p/(4 c)}, E^(-4 c z)]) + (1/((-I) a - 2 c + p)) (E^(((-I) a + p) z) HypergeometricPFQ[{1, 1/2 + (I a)/(4 c) - p/(4 c)}, {3/2 + (I a)/(4 c) - p/(4 c)}, E^(-4 c z)])))/E^(2 c z)

 Standard Form

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 MathML Form

 p z cos ( a z ) csch ( c z ) sech ( c z ) z 2 - 2 c z ( ( - a + p ) z 2 F 1 ( 1 , 1 2 + a 4 c - p 4 c ; 3 2 + a 4 c - p 4 c ; - 4 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["1", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox["1", "2"], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a"]], RowBox[List["4", " ", "c"]]], "-", FractionBox["p", RowBox[List["4", " ", "c"]]]]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["3", "2"], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a"]], RowBox[List["4", " ", "c"]]], "-", FractionBox["p", RowBox[List["4", " ", "c"]]]]], HypergeometricPFQ], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "4"]], " ", "c", " ", "z"]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] - 2 c - a + p + ( a + p ) z 2 F 1 ( 1 , 1 2 - a 4 c - p 4 c ; 3 2 - a 4 c - p 4 c ; - 4 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["1", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a"]], RowBox[List["4", " ", "c"]]], "-", FractionBox["p", RowBox[List["4", " ", "c"]]]]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["3", "2"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a"]], RowBox[List["4", " ", "c"]]], "-", FractionBox["p", RowBox[List["4", " ", "c"]]]]], HypergeometricPFQ], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "4"]], " ", "c", " ", "z"]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] - 2 c + a + p ) z p z a z c z c z 2 -2 c z -1 a p z HypergeometricPFQ 1 1 2 a 4 c -1 -1 p 4 c -1 3 2 a 4 c -1 -1 p 4 c -1 -4 c z -2 c -1 a p -1 a p z HypergeometricPFQ 1 1 2 -1 a 4 c -1 -1 p 4 c -1 3 2 -1 a 4 c -1 -1 p 4 c -1 -4 c z -2 c a 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["Cos", "[", RowBox[List["a_", " ", "z_"]], "]"]], " ", RowBox[List["Csch", "[", RowBox[List["c_", " ", "z_"]], "]"]], " ", RowBox[List["Sech", "[", RowBox[List["c_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "c", " ", "z"]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "+", "p"]], ")"]], " ", "z"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a"]], RowBox[List["4", " ", "c"]]], "-", FractionBox["p", RowBox[List["4", " ", "c"]]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a"]], RowBox[List["4", " ", "c"]]], "-", FractionBox["p", RowBox[List["4", " ", "c"]]]]], "}"]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "4"]], " ", "c", " ", "z"]]]]], "]"]]]], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "-", RowBox[List["2", " ", "c"]], "+", "p"]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "+", "p"]], ")"]], " ", "z"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List[FractionBox["1", "2"], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a"]], RowBox[List["4", " ", "c"]]], "-", FractionBox["p", RowBox[List["4", " ", "c"]]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a"]], RowBox[List["4", " ", "c"]]], "-", FractionBox["p", RowBox[List["4", " ", "c"]]]]], "}"]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "4"]], " ", "c", " ", "z"]]]]], "]"]]]], RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "-", RowBox[List["2", " ", "c"]], "+", "p"]]]]], ")"]]]]]]]]

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

 2002-12-18