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 Cosh

 http://functions.wolfram.com/01.20.21.0884.01

 Input Form

 Integrate[z^(\[Alpha] - 1) Sin[c z] Cosh[b + a z], z] == ((1/4) z^\[Alpha] ((-I) ((a - I c) z)^\[Alpha] ((-(a + I c)^2) z^2)^\[Alpha] Gamma[\[Alpha], (-(a - I c)) z] (Cosh[b] + Sinh[b]) + I (((-(a - I c)) z)^\[Alpha] ((-(a + I c)^2) z^2)^\[Alpha] Gamma[\[Alpha], (a - I c) z] (Cosh[b] - Sinh[b]) + ((-(a - I c)^2) z^2)^\[Alpha] ((-((-(a + I c)) z)^\[Alpha]) Gamma[\[Alpha], (a + I c) z] (Cosh[b] - Sinh[b]) + ((a + I c) z)^\[Alpha] Gamma[\[Alpha], (-(a + I c)) z] (Cosh[b] + Sinh[b])))))/(((-(a - I c)^2) z^2)^\[Alpha] ((-(a + I c)^2) z^2)^\[Alpha])

 Standard Form

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 MathML Form

 z α - 1 sin ( c z ) cosh ( b + a z ) z 1 4 z α ( - ( a - c ) 2 z 2 ) - α ( - ( a + c ) 2 z 2 ) - α ( ( ( - ( a - c ) z ) α Γ ( α , ( a - c ) z ) ( cosh ( b ) - sinh ( b ) ) ( - ( a + c ) 2 z 2 ) α + ( - ( a - c ) 2 z 2 ) α ( ( ( a + c ) z ) α Γ ( α , - ( a + c ) z ) ( cosh ( b ) + sinh ( b ) ) - ( - ( a + c ) z ) α Γ ( α , ( a + c ) z ) ( cosh ( b ) - sinh ( b ) ) ) ) - ( ( a - c ) z ) α ( - ( a + c ) 2 z 2 ) α Γ ( α , - ( a - c ) z ) ( cosh ( b ) + sinh ( b ) ) ) z z α -1 c z b a z 1 4 z α -1 a -1 c 2 z 2 -1 α -1 a c 2 z 2 -1 α -1 a -1 c z α Gamma α a -1 c z b -1 b -1 a c 2 z 2 α -1 a -1 c 2 z 2 α a c z α Gamma α -1 a c z b b -1 -1 a c z α Gamma α a c z b -1 b -1 a -1 c z α -1 a c 2 z 2 α Gamma α -1 a -1 c z b b [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", RowBox[List["Sin", "[", RowBox[List["c_", " ", "z_"]], "]"]], " ", RowBox[List["Cosh", "[", RowBox[List["b_", "+", RowBox[List["a_", " ", "z_"]]]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "4"], " ", SuperscriptBox["z", "\[Alpha]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], "2"]]], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["-", "\[Alpha]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], "2"]]], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", "z"]], ")"]], "\[Alpha]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], "2"]]], " ", SuperscriptBox["z", "2"]]], ")"]], "\[Alpha]"], " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]]]], " ", "z"]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Cosh", "[", "b", "]"]], "+", RowBox[List["Sinh", "[", "b", "]"]]]], ")"]]]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]]]], " ", "z"]], ")"]], "\[Alpha]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], "2"]]], " ", SuperscriptBox["z", "2"]]], ")"]], "\[Alpha]"], " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List[RowBox[List["(", RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", "z"]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Cosh", "[", "b", "]"]], "-", RowBox[List["Sinh", "[", "b", "]"]]]], ")"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], "2"]]], " ", SuperscriptBox["z", "2"]]], ")"]], "\[Alpha]"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]]]], " ", "z"]], ")"]], "\[Alpha]"]]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", "z"]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Cosh", "[", "b", "]"]], "-", RowBox[List["Sinh", "[", "b", "]"]]]], ")"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", "z"]], ")"]], "\[Alpha]"], " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]]]], " ", "z"]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Cosh", "[", "b", "]"]], "+", RowBox[List["Sinh", "[", "b", "]"]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]]]]

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

 2002-12-18