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/01.20.21.0689.01

 Input Form

 Integrate[(a + b Sin[d z])^\[Beta] Cosh[c z], z] == (1/(2 (c^2 + d^2 \[Beta]^2))) (((a - ((1/2) I b (-1 + E^(2 I d z)))/E^(I d z))^\[Beta] (E^(2 c z) (c + I d \[Beta]) AppellF1[-((I c)/d) - \[Beta], -\[Beta], -\[Beta], 1 - (I c)/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])] - (c - I d \[Beta]) AppellF1[(I c)/d - \[Beta], -\[Beta], -\[Beta], 1 + (I c)/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])]))/ (E^(c z) (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[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", RowBox[List["2", " ", RowBox[List["(", RowBox[List[SuperscriptBox["c", "2"], "+", RowBox[List[SuperscriptBox["d", "2"], " ", SuperscriptBox["\[Beta]", "2"]]]]], ")"]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "c"]], " ", "z"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", 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"]]]]]]]]], ")"]], 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]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", "b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "d", " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "d", " ", "z"]]]]], ")"]]]]]], ")"]], "\[Beta]"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]], " ", RowBox[List["(", RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "d", " ", "\[Beta]"]]]], ")"]], " ", RowBox[List["AppellF1", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "c"]], "d"]]], "-", "\[Beta]"]], ",", RowBox[List["-", "\[Beta]"]], ",", RowBox[List["-", "\[Beta]"]], ",", RowBox[List["1", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "c"]], "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"]]]]]]], ",", 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[RowBox[List["(", RowBox[List["c", "-", RowBox[List["\[ImaginaryI]", " ", "d", " ", "\[Beta]"]]]], ")"]], " ", RowBox[List["AppellF1", "[", RowBox[List[RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "c"]], "d"], "-", "\[Beta]"]], ",", RowBox[List["-", "\[Beta]"]], ",", RowBox[List["-", "\[Beta]"]], ",", RowBox[List["1", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "c"]], "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"]]]]]]], ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", "b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "d", " ", "z"]]]]], RowBox[List["a", "-", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]]]]], "]"]]]]]], ")"]]]], ")"]]]]]]]]

 MathML Form

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

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[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_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "c"]], " ", "z"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", 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"]]]]]]]]], ")"]], 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]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", "b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "d", " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "d", " ", "z"]]]]], ")"]]]]]], ")"]], "\[Beta]"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]], " ", RowBox[List["(", RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "d", " ", "\[Beta]"]]]], ")"]], " ", RowBox[List["AppellF1", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "c"]], "d"]]], "-", "\[Beta]"]], ",", RowBox[List["-", "\[Beta]"]], ",", RowBox[List["-", "\[Beta]"]], ",", RowBox[List["1", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "c"]], "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"]]]]]]], ",", 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[RowBox[List["(", RowBox[List["c", "-", RowBox[List["\[ImaginaryI]", " ", "d", " ", "\[Beta]"]]]], ")"]], " ", RowBox[List["AppellF1", "[", RowBox[List[RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "c"]], "d"], "-", "\[Beta]"]], ",", RowBox[List["-", "\[Beta]"]], ",", RowBox[List["-", "\[Beta]"]], ",", RowBox[List["1", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "c"]], "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"]]]]]]], ",", 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["2", " ", RowBox[List["(", RowBox[List[SuperscriptBox["c", "2"], "+", RowBox[List[SuperscriptBox["d", "2"], " ", SuperscriptBox["\[Beta]", "2"]]]]], ")"]]]]]]]]]

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

 2002-12-18