html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 Cosh

 http://functions.wolfram.com/01.20.21.1204.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["p", " ", "z"]]], 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"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "4"], " ", "\[ImaginaryI]", " ", SuperscriptBox["a", RowBox[List["-", "n"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["c", "-", RowBox[List["\[ImaginaryI]", " ", "e"]], "+", "p"]], ")"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List["c", "-", RowBox[List["\[ImaginaryI]", " ", "e"]], "-", "p"]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["c", "-", RowBox[List["\[ImaginaryI]", " ", "e"]], "+", "p"]], "d"], ",", "n", ",", FractionBox[RowBox[List["c", "+", "d", "-", RowBox[List["\[ImaginaryI]", " ", "e"]], "+", "p"]], "d"], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["d", " ", "z"]]]]], "a"]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", RowBox[List["\[ImaginaryI]", " ", "e"]], "+", "p"]], ")"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List["c", "-", RowBox[List["\[ImaginaryI]", " ", "e"]], "+", "p"]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["-", "c"]], "+", RowBox[List["\[ImaginaryI]", " ", "e"]], "+", "p"]], "d"], ",", "n", ",", FractionBox[RowBox[List[RowBox[List["-", "c"]], "+", "d", "+", RowBox[List["\[ImaginaryI]", " ", "e"]], "+", "p"]], "d"], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["d", " ", "z"]]]]], "a"]]]]], "]"]]]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["c", "-", RowBox[List["\[ImaginaryI]", " ", "e"]], "-", "p"]], ")"]], " ", RowBox[List["(", RowBox[List["c", "-", RowBox[List["\[ImaginaryI]", " ", "e"]], "+", "p"]], ")"]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "-", RowBox[List["\[ImaginaryI]", " ", "e"]], "+", "p"]], ")"]], " ", "z"]]]]], " ", RowBox[List["(", RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "e"]], "+", "p"]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["-", "c"]], "-", RowBox[List["\[ImaginaryI]", " ", "e"]], "+", "p"]], "d"], ",", "n", ",", FractionBox[RowBox[List[RowBox[List["-", "c"]], "+", "d", "-", RowBox[List["\[ImaginaryI]", " ", "e"]], "+", "p"]], "d"], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["d", " ", "z"]]]]], "a"]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "e"]], "+", "p"]], ")"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "-", RowBox[List["\[ImaginaryI]", " ", "e"]], "+", "p"]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "e"]], "+", "p"]], "d"], ",", "n", ",", FractionBox[RowBox[List["c", "+", "d", "+", RowBox[List["\[ImaginaryI]", " ", "e"]], "+", "p"]], "d"], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["d", " ", "z"]]]]], "a"]]]]], "]"]]]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "e"]], "-", "p"]], ")"]], " ", RowBox[List["(", RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "e"]], "+", "p"]], ")"]]]], ")"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]]]]]]]]

 MathML Form

 p z sin ( e z ) cosh ( c z ) ( a + b d z ) n z 1 4 a - n ( 1 ( c - e - p ) ( c - e + p ) ( ( - c + e + p ) z ( c - e + p ) 2 F 1 ( - c + e + p d , n ; - c + d + e + p 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[RowBox[List["-", "c"]], "+", RowBox[List["\[ImaginaryI]", " ", "e"]], " ", "+", "p"]], "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"]], " ", "+", "p"]], "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 + p ) z ( c - e - p ) 2 F 1 ( c - e + p d , n ; c + d - e + p 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"]], "+", "p"]], "d"], Hypergeometric2F1], ",", TagBox["n", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List["c", "+", "d", "-", RowBox[List["\[ImaginaryI]", " ", "e"]], "+", "p"]], "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] ) + 1 ( c + e - p ) ( c + e + p ) ( ( c + e + p ) z ( - c - e + p ) 2 F 1 ( c + e + p d , n ; c + d + e + p 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"]], " ", "+", "p"]], "d"], Hypergeometric2F1], ",", TagBox["n", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List["c", "+", "d", "+", RowBox[List["\[ImaginaryI]", " ", "e"]], " ", "+", "p"]], "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 + p ) z ( c + e + p ) 2 F 1 ( - c - e + p d , n ; - c + d - e + p 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[RowBox[List["-", "c"]], "-", RowBox[List["\[ImaginaryI]", " ", "e"]], "+", "p"]], "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"]], "+", "p"]], "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 p z e z c z a b d z n -1 1 4 a -1 n 1 c -1 e -1 p c -1 e p -1 -1 c e p z c -1 e p Hypergeometric2F1 -1 c e p d -1 n -1 c d e p d -1 -1 b d z a -1 c -1 e p z c -1 e -1 p Hypergeometric2F1 c -1 e p d -1 n c d -1 e p d -1 -1 b d z a -1 1 c e -1 p c e p -1 c e p z -1 c -1 e p Hypergeometric2F1 c e p d -1 n c d e p d -1 -1 b d z a -1 -1 -1 c -1 e p z c e p Hypergeometric2F1 -1 c -1 e p d -1 n -1 c d -1 e p 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[SuperscriptBox["\[ExponentialE]", RowBox[List["p_", " ", "z_"]]], " ", 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[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["c", "-", RowBox[List["\[ImaginaryI]", " ", "e"]], "+", "p"]], ")"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List["c", "-", RowBox[List["\[ImaginaryI]", " ", "e"]], "-", "p"]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["c", "-", RowBox[List["\[ImaginaryI]", " ", "e"]], "+", "p"]], "d"], ",", "n", ",", FractionBox[RowBox[List["c", "+", "d", "-", RowBox[List["\[ImaginaryI]", " ", "e"]], "+", "p"]], "d"], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["d", " ", "z"]]]]], "a"]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", RowBox[List["\[ImaginaryI]", " ", "e"]], "+", "p"]], ")"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List["c", "-", RowBox[List["\[ImaginaryI]", " ", "e"]], "+", "p"]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["-", "c"]], "+", RowBox[List["\[ImaginaryI]", " ", "e"]], "+", "p"]], "d"], ",", "n", ",", FractionBox[RowBox[List[RowBox[List["-", "c"]], "+", "d", "+", RowBox[List["\[ImaginaryI]", " ", "e"]], "+", "p"]], "d"], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["d", " ", "z"]]]]], "a"]]]]], "]"]]]]]], RowBox[List[RowBox[List["(", RowBox[List["c", "-", RowBox[List["\[ImaginaryI]", " ", "e"]], "-", "p"]], ")"]], " ", RowBox[List["(", RowBox[List["c", "-", RowBox[List["\[ImaginaryI]", " ", "e"]], "+", "p"]], ")"]]]]], "+", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "-", RowBox[List["\[ImaginaryI]", " ", "e"]], "+", "p"]], ")"]], " ", "z"]]]]], " ", RowBox[List["(", RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "e"]], "+", "p"]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["-", "c"]], "-", RowBox[List["\[ImaginaryI]", " ", "e"]], "+", "p"]], "d"], ",", "n", ",", FractionBox[RowBox[List[RowBox[List["-", "c"]], "+", "d", "-", RowBox[List["\[ImaginaryI]", " ", "e"]], "+", "p"]], "d"], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["d", " ", "z"]]]]], "a"]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "e"]], "+", "p"]], ")"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "-", RowBox[List["\[ImaginaryI]", " ", "e"]], "+", "p"]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "e"]], "+", "p"]], "d"], ",", "n", ",", FractionBox[RowBox[List["c", "+", "d", "+", RowBox[List["\[ImaginaryI]", " ", "e"]], "+", "p"]], "d"], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["d", " ", "z"]]]]], "a"]]]]], "]"]]]]]], RowBox[List[RowBox[List["(", RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "e"]], "-", "p"]], ")"]], " ", RowBox[List["(", RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "e"]], "+", "p"]], ")"]]]]]]], ")"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]

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

 2002-12-18