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 Cosh

 http://functions.wolfram.com/01.20.21.1248.01

 Input Form

 Integrate[E^(p z) (a + b Sin[d z])^\[Beta] Cosh[c z], z] == (1/2) ((1/(-c + p - I d \[Beta])) ((E^((-c + p) z) AppellF1[-((I (-c + p))/d) - \[Beta], -\[Beta], -\[Beta], 1 - (I (-c + p))/d - \[Beta], (I b E^(I d z))/ (a + Sqrt[a^2 - b^2]), -((I b E^(I d z))/(-a + Sqrt[a^2 - b^2]))] (a + b Sin[d z])^\[Beta])/((1 - (I b E^(I d z))/(a - Sqrt[a^2 - b^2]))^ \[Beta] (1 - (I b E^(I d z))/(a + Sqrt[a^2 - b^2]))^\[Beta])) + (1/(c + p - I d \[Beta])) ((E^((c + p) z) AppellF1[-((I (c + p))/d) - \[Beta], -\[Beta], -\[Beta], 1 - (I (c + p))/d - \[Beta], (I b E^(I d z))/(a + Sqrt[a^2 - b^2]), -((I b E^(I d z))/(-a + Sqrt[a^2 - b^2]))] (a + b Sin[d z])^\[Beta])/ ((1 - (I b E^(I d z))/(a - Sqrt[a^2 - b^2]))^\[Beta] (1 - (I b E^(I d z))/(a + Sqrt[a^2 - b^2]))^\[Beta])))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["p", " ", "z"]]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sin", "[", RowBox[List["d", " ", "z"]], "]"]]]]]], ")"]], "\[Beta]"], RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", RowBox[List[RowBox[List["-", "c"]], "+", "p", "-", RowBox[List["\[ImaginaryI]", " ", "d", " ", "\[Beta]"]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", "p"]], ")"]], " ", "z"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "d", " ", "z"]]]]], RowBox[List["a", "-", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]]]]], ")"]], RowBox[List["-", "\[Beta]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "d", " ", "z"]]]]], RowBox[List["a", "+", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]]]]], ")"]], RowBox[List["-", "\[Beta]"]]], " ", RowBox[List["AppellF1", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", "p"]], ")"]]]], "d"]]], "-", "\[Beta]"]], ",", RowBox[List["-", "\[Beta]"]], ",", RowBox[List["-", "\[Beta]"]], ",", RowBox[List["1", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", "p"]], ")"]]]], "d"], "-", "\[Beta]"]], ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", "b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "d", " ", "z"]]]]], RowBox[List["a", "+", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]]], ",", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "d", " ", "z"]]]]], RowBox[List[RowBox[List["-", "a"]], "+", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]]]]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sin", "[", RowBox[List["d", " ", "z"]], "]"]]]]]], ")"]], "\[Beta]"]]], ")"]]]], "+", RowBox[List[FractionBox["1", RowBox[List["c", "+", "p", "-", RowBox[List["\[ImaginaryI]", " ", "d", " ", "\[Beta]"]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["c", "+", "p"]], ")"]], " ", "z"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "d", " ", "z"]]]]], RowBox[List["a", "-", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]]]]], ")"]], RowBox[List["-", "\[Beta]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "d", " ", "z"]]]]], RowBox[List["a", "+", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]]]]], ")"]], RowBox[List["-", "\[Beta]"]]], " ", RowBox[List["AppellF1", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["c", "+", "p"]], ")"]]]], "d"]]], "-", "\[Beta]"]], ",", RowBox[List["-", "\[Beta]"]], ",", RowBox[List["-", "\[Beta]"]], ",", RowBox[List["1", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["c", "+", "p"]], ")"]]]], "d"], "-", "\[Beta]"]], ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", "b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "d", " ", "z"]]]]], RowBox[List["a", "+", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]]], ",", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "d", " ", "z"]]]]], RowBox[List[RowBox[List["-", "a"]], "+", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]]]]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sin", "[", RowBox[List["d", " ", "z"]], "]"]]]]]], ")"]], "\[Beta]"]]], ")"]]]]]], ")"]]]]]]]]

 MathML Form

 p z ( a + b sin ( d z ) ) β cosh ( c z ) z 1 2 ( 1 - c + p - d β ( ( p - c ) z ( 1 - b d z a - a 2 - b 2 ) - β ( 1 - b d z a + a 2 - b 2 ) - β F 1 AppellF1 ( - ( p - c ) d - β ; - β , - β ; - ( p - c ) d - β + 1 ; b d z a + a 2 - b 2 , - b d z a 2 - b 2 - a ) ( a + b sin ( d z ) ) β ) + 1 c + p - d β ( ( c + p ) z ( 1 - b d z a - a 2 - b 2 ) - β ( 1 - b d z a + a 2 - b 2 ) - β F 1 AppellF1 ( - ( c + p ) d - β ; - β , - β ; - ( c + p ) d - β + 1 ; b d z a + a 2 - b 2 , - b d z a 2 - b 2 - a ) ( a + b sin ( d z ) ) β ) ) z p z a b d z β c z 1 2 1 -1 c p -1 d β -1 p -1 c z 1 -1 b d z a -1 a 2 -1 b 2 1 2 -1 -1 β 1 -1 b d z a a 2 -1 b 2 1 2 -1 -1 β AppellF1 -1 p -1 c d -1 -1 β -1 β -1 β -1 p -1 c d -1 -1 β 1 b d z a a 2 -1 b 2 1 2 -1 -1 b d z a 2 -1 b 2 1 2 -1 a -1 a b d z β 1 c p -1 d β -1 c p z 1 -1 b d z a -1 a 2 -1 b 2 1 2 -1 -1 β 1 -1 b d z a a 2 -1 b 2 1 2 -1 -1 β AppellF1 -1 c p d -1 -1 β -1 β -1 β -1 c p d -1 -1 β 1 b d z a a 2 -1 b 2 1 2 -1 -1 b d z a 2 -1 b 2 1 2 -1 a -1 a b d z β [/itex]

 Rule Form

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

 2002-12-18