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 Cosh

 http://functions.wolfram.com/01.20.21.1203.01

 Input Form

 Integrate[(Sin[e z] Cosh[c z])/(a + b E^(d z))^n, z] == ((1/4) I ((1/(c - I e)) (E^((-c) z + I e z) (Hypergeometric2F1[-((c - I e)/d), n, (-c + d + I e)/d, -((b E^(d z))/a)] + E^(2 (c - I e) z) Hypergeometric2F1[(c - I e)/d, n, (c + d - I e)/d, -((b E^(d z))/a)])) - (1/(c + I e)) ((Hypergeometric2F1[-((c + I e)/d), n, (-c + d - I e)/d, -((b E^(d z))/a)] + E^(2 (c + I e) z) Hypergeometric2F1[(c + I e)/d, n, (c + d + I e)/d, -((b E^(d z))/a)])/E^((c + I e) z))))/a^n /; Element[n, Integers] && n > 0

 Standard Form

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 MathML Form

 sin ( e z ) cosh ( c z ) ( a + b d z ) n z 1 4 a - n ( e z - c z c - e ( 2 ( c - e ) z 2 F 1 ( c - e d , n ; c + d - e d ; - b d z a ) 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[FractionBox[RowBox[List["c", "-", RowBox[List["\[ImaginaryI]", " ", "e"]]]], "d"], Hypergeometric2F1], ",", TagBox["n", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List["c", "+", "d", "-", RowBox[List["\[ImaginaryI]", " ", "e"]]]], "d"], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["d", " ", "z"]]]]], "a"]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] + 2 F 1 ( - c - e d , n ; - c + d + e d ; - b d z a ) 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[RowBox[List["-", FractionBox[RowBox[List["c", "-", RowBox[List["\[ImaginaryI]", " ", "e"]]]], "d"]]], Hypergeometric2F1], ",", TagBox["n", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List[RowBox[List["-", "c"]], "+", "d", "+", RowBox[List["\[ImaginaryI]", " ", "e", " "]]]], "d"], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["d", " ", "z"]]]]], "a"]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] ) - - ( c + e ) z c + e ( 2 ( c + e ) z 2 F 1 ( c + e d , n ; c + d + e d ; - b d z a ) 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[FractionBox[RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "e", " "]]]], "d"], Hypergeometric2F1], ",", TagBox["n", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List["c", "+", "d", "+", RowBox[List["\[ImaginaryI]", " ", "e", " "]]]], "d"], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["d", " ", "z"]]]]], "a"]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] + 2 F 1 ( - c + e d , n ; - c + d - e d ; - b d z a ) 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[RowBox[List["-", FractionBox[RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "e", " "]]]], "d"]]], Hypergeometric2F1], ",", TagBox["n", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List[RowBox[List["-", "c"]], "+", "d", "-", RowBox[List["\[ImaginaryI]", " ", "e"]]]], "d"], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["d", " ", "z"]]]]], "a"]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] ) ) /; n + Condition z e z c z a b d z n -1 1 4 a -1 n e z -1 c z c -1 e -1 2 c -1 e z Hypergeometric2F1 c -1 e d -1 n c d -1 e d -1 -1 b d z a -1 Hypergeometric2F1 -1 c -1 e d -1 n -1 c d e d -1 -1 b d z a -1 -1 -1 c e z c e -1 2 c e z Hypergeometric2F1 c e d -1 n c d e d -1 -1 b d z a -1 Hypergeometric2F1 -1 c e d -1 n -1 c d -1 e d -1 -1 b d z a -1 n SuperPlus [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List[RowBox[List["Sin", "[", RowBox[List["e_", " ", "z_"]], "]"]], " ", RowBox[List["Cosh", "[", RowBox[List["c_", " ", "z_"]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["a_", "+", RowBox[List["b_", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["d_", " ", "z_"]]]]]]], ")"]], "n_"]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", "\[ImaginaryI]", " ", SuperscriptBox["a", RowBox[List["-", "n"]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", "c"]], " ", "z"]], "+", RowBox[List["\[ImaginaryI]", " ", "e", " ", "z"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["c", "-", RowBox[List["\[ImaginaryI]", " ", "e"]]]], "d"]]], ",", "n", ",", FractionBox[RowBox[List[RowBox[List["-", "c"]], "+", "d", "+", RowBox[List["\[ImaginaryI]", " ", "e"]]]], "d"], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["d", " ", "z"]]]]], "a"]]]]], "]"]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", RowBox[List["(", RowBox[List["c", "-", RowBox[List["\[ImaginaryI]", " ", "e"]]]], ")"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["c", "-", RowBox[List["\[ImaginaryI]", " ", "e"]]]], "d"], ",", "n", ",", FractionBox[RowBox[List["c", "+", "d", "-", RowBox[List["\[ImaginaryI]", " ", "e"]]]], "d"], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["d", " ", "z"]]]]], "a"]]]]], "]"]]]]]], ")"]]]], RowBox[List["c", "-", RowBox[List["\[ImaginaryI]", " ", "e"]]]]], "-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "e"]]]], ")"]]]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "e"]]]], "d"]]], ",", "n", ",", FractionBox[RowBox[List[RowBox[List["-", "c"]], "+", "d", "-", RowBox[List["\[ImaginaryI]", " ", "e"]]]], "d"], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["d", " ", "z"]]]]], "a"]]]]], "]"]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", RowBox[List["(", RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "e"]]]], ")"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "e"]]]], "d"], ",", "n", ",", FractionBox[RowBox[List["c", "+", "d", "+", RowBox[List["\[ImaginaryI]", " ", "e"]]]], "d"], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["d", " ", "z"]]]]], "a"]]]]], "]"]]]]]], ")"]]]], RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "e"]]]]]]], ")"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]

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

 2002-12-18