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 Sech

 http://functions.wolfram.com/01.24.21.0249.01

 Input Form

 Integrate[E^(p z) Sin[a z] Cosh[b z] Sech[c z], z] == (-(1/2)) I ((1/((-I) a - b + c + p)) (E^(I Pi + ((-I) a - b + c + p) z) Hypergeometric2F1[1, ((-I) a - b + c + p)/(2 c), ((-I) a - b + 3 c + p)/(2 c), -E^(2 c z)]) + (1/(I a - b + c + p)) (E^((I a - b + c + p) z) Hypergeometric2F1[1, (I a - b + c + p)/(2 c), (I a - b + 3 c + p)/(2 c), -E^(2 c z)]) + (1/((-I) a + b + c + p)) (E^(I Pi + ((-I) a + b + c + p) z) Hypergeometric2F1[1, ((-I) a + b + c + p)/(2 c), ((-I) a + b + 3 c + p)/(2 c), -E^(2 c z)]) + (1/(I a + b + c + p)) (E^((I a + b + c + p) z) Hypergeometric2F1[1, (I a + b + c + p)/(2 c), (I a + b + 3 c + p)/(2 c), -E^(2 c z)]))

 Standard Form

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 MathML Form

 p z sin ( a z ) cosh ( b z ) sech ( c z ) z - 1 2 ( π + ( - b + c - a + p ) z 2 F 1 ( 1 , - b + c - a + p 2 c ; - b + 3 c - a + p 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["1", Hypergeometric2F1], ",", TagBox[FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", "c", "-", RowBox[List["\[ImaginaryI]", " ", "a"]], " ", "+", "p"]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["3", " ", "c"]], "-", RowBox[List["\[ImaginaryI]", " ", "a"]], " ", "+", "p"]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] - b + c - a + p + ( - b + c + a + p ) z 2 F 1 ( 1 , - b + c + a + p 2 c ; - b + 3 c + a + p 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["1", Hypergeometric2F1], ",", TagBox[FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", "c", "+", RowBox[List["\[ImaginaryI]", " ", "a"]], " ", "+", "p"]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["3", " ", "c"]], "+", RowBox[List["\[ImaginaryI]", " ", "a"]], " ", "+", "p"]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] - b + c + a + p + π + ( b + c - a + p ) z 2 F 1 ( 1 , b + c - a + p 2 c ; b + 3 c - a + p 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["1", Hypergeometric2F1], ",", TagBox[FractionBox[RowBox[List["b", "+", "c", "-", RowBox[List["\[ImaginaryI]", " ", "a"]], " ", "+", "p"]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List["b", "+", RowBox[List["3", " ", "c"]], "-", RowBox[List["\[ImaginaryI]", " ", "a"]], " ", "+", "p"]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] b + c - a + p + ( b + c + a + p ) z 2 F 1 ( 1 , b + c + a + p 2 c ; b + 3 c + a + p 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["1", Hypergeometric2F1], ",", TagBox[FractionBox[RowBox[List["b", "+", "c", "+", RowBox[List["\[ImaginaryI]", " ", "a"]], " ", "+", "p"]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List["b", "+", RowBox[List["3", " ", "c"]], "+", RowBox[List["\[ImaginaryI]", " ", "a"]], " ", "+", "p"]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] b + c + a + p ) z p z a z b z c z -1 1 2 -1 b c -1 a p z Hypergeometric2F1 1 -1 b c -1 a p 2 c -1 -1 b 3 c -1 a p 2 c -1 -1 2 c z -1 b c -1 a p -1 -1 b c a p z Hypergeometric2F1 1 -1 b c a p 2 c -1 -1 b 3 c a p 2 c -1 -1 2 c z -1 b c a p -1 b c -1 a p z Hypergeometric2F1 1 b c -1 a p 2 c -1 b 3 c -1 a p 2 c -1 -1 2 c z b c -1 a p -1 b c a p z Hypergeometric2F1 1 b c a p 2 c -1 b 3 c a p 2 c -1 -1 2 c z b 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["Sin", "[", RowBox[List["a_", " ", "z_"]], "]"]], " ", RowBox[List["Cosh", "[", RowBox[List["b_", " ", "z_"]], "]"]], " ", RowBox[List["Sech", "[", RowBox[List["c_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "-", "b", "+", "c", "+", "p"]], ")"]], " ", "z"]]]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["1", ",", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "-", "b", "+", "c", "+", "p"]], RowBox[List["2", " ", "c"]]], ",", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "-", "b", "+", RowBox[List["3", " ", "c"]], "+", "p"]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "-", "b", "+", "c", "+", "p"]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "-", "b", "+", "c", "+", "p"]], ")"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["1", ",", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "-", "b", "+", "c", "+", "p"]], RowBox[List["2", " ", "c"]]], ",", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "-", "b", "+", RowBox[List["3", " ", "c"]], "+", "p"]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "-", "b", "+", "c", "+", "p"]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "+", "b", "+", "c", "+", "p"]], ")"]], " ", "z"]]]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["1", ",", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "+", "b", "+", "c", "+", "p"]], RowBox[List["2", " ", "c"]]], ",", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "+", "b", "+", RowBox[List["3", " ", "c"]], "+", "p"]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "+", "b", "+", "c", "+", "p"]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "+", "b", "+", "c", "+", "p"]], ")"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["1", ",", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "+", "b", "+", "c", "+", "p"]], RowBox[List["2", " ", "c"]]], ",", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "+", "b", "+", RowBox[List["3", " ", "c"]], "+", "p"]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "+", "b", "+", "c", "+", "p"]]]]], ")"]]]]]]]]

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

 2002-12-18