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/01.20.21.4726.01

 Input Form

 Integrate[E^(p z) Sinh[d z] (a + b Cosh[c z]^2)^\[Beta], z] == ((a + ((1/4) b (1 + E^(2 c z))^2)/E^(2 c z))^\[Beta] (E^((d + p) z) (d - p + 2 c \[Beta]) AppellF1[(d + p - 2 c \[Beta])/ (2 c), -\[Beta], -\[Beta], 1 + (d + p)/(2 c) - \[Beta], -((b E^(2 c z))/(2 a + b + 2 Sqrt[a (a + b)])), -((b E^(2 c z))/(2 a + b - 2 Sqrt[a (a + b)]))] + E^((-d + p) z) (d + p - 2 c \[Beta]) AppellF1[-((d - p + 2 c \[Beta])/(2 c)), -\[Beta], -\[Beta], 1 + (-d + p)/(2 c) - \[Beta], -((b E^(2 c z))/(2 a + b + 2 Sqrt[a (a + b)])), -((b E^(2 c z))/(2 a + b - 2 Sqrt[a (a + b)]))]))/ ((1 + (b E^(2 c z))/(2 a + b - 2 Sqrt[a (a + b)]))^\[Beta] (1 + (b E^(2 c z))/(2 a + b + 2 Sqrt[a (a + b)]))^\[Beta])/ (2 (d + p - 2 c \[Beta]) (d - p + 2 c \[Beta]))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["p", " ", "z"]]], RowBox[List["Sinh", "[", RowBox[List["d", " ", "z"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]], ")"]], "\[Beta]"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "-", RowBox[List["2", " ", SqrtBox[RowBox[List["a", " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]]]]]]]]]]]], ")"]], RowBox[List["-", "\[Beta]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "+", RowBox[List["2", " ", SqrtBox[RowBox[List["a", " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]]]]]]]]]]]], ")"]], RowBox[List["-", "\[Beta]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List[FractionBox["1", "4"], " ", "b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "c", " ", "z"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], ")"]], "2"]]]]], ")"]], "\[Beta]"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["d", "+", "p"]], ")"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List["d", "-", "p", "+", RowBox[List["2", " ", "c", " ", "\[Beta]"]]]], ")"]], " ", RowBox[List["AppellF1", "[", RowBox[List[FractionBox[RowBox[List["d", "+", "p", "-", RowBox[List["2", " ", "c", " ", "\[Beta]"]]]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["-", "\[Beta]"]], ",", RowBox[List["-", "\[Beta]"]], ",", RowBox[List["1", "+", FractionBox[RowBox[List["d", "+", "p"]], RowBox[List["2", " ", "c"]]], "-", "\[Beta]"]], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "+", RowBox[List["2", " ", SqrtBox[RowBox[List["a", " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]]]]]]]]]]]], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "-", RowBox[List["2", " ", SqrtBox[RowBox[List["a", " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]]]]]]]]]]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "d"]], "+", "p"]], ")"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List["d", "+", "p", "-", RowBox[List["2", " ", "c", " ", "\[Beta]"]]]], ")"]], " ", RowBox[List["AppellF1", "[", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["d", "-", "p", "+", RowBox[List["2", " ", "c", " ", "\[Beta]"]]]], RowBox[List["2", " ", "c"]]]]], ",", RowBox[List["-", "\[Beta]"]], ",", RowBox[List["-", "\[Beta]"]], ",", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["-", "d"]], "+", "p"]], RowBox[List["2", " ", "c"]]], "-", "\[Beta]"]], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "+", RowBox[List["2", " ", SqrtBox[RowBox[List["a", " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]]]]]]]]]]]], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "-", RowBox[List["2", " ", SqrtBox[RowBox[List["a", " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]]]]]]]]]]]]]], "]"]]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List["2", " ", RowBox[List["(", RowBox[List["d", "+", "p", "-", RowBox[List["2", " ", "c", " ", "\[Beta]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["d", "-", "p", "+", RowBox[List["2", " ", "c", " ", "\[Beta]"]]]], ")"]]]], ")"]]]]]]]]

 MathML Form

 p z sinh ( d z ) ( a + b cosh 2 ( c z ) ) β z 1 2 ( d + p - 2 c β ) ( d - p + 2 c β ) ( ( 2 c z b 2 a + b - 2 a ( a + b ) + 1 ) - β ( 2 c z b 2 a + b + 2 a ( a + b ) + 1 ) - β ( 1 4 b - 2 c z ( 1 + 2 c z ) 2 + a ) β ( ( d + p ) z ( d - p + 2 c β ) F 1 AppellF1 ( d + p - 2 c β 2 c ; - β , - β ; d + p 2 c - β + 1 ; - b 2 c z 2 a + b + 2 a ( a + b ) , - b 2 c z 2 a + b - 2 a ( a + b ) ) + ( p - d ) z ( d + p - 2 c β ) F 1 AppellF1 ( - d - p + 2 c β 2 c ; - β , - β ; p - d 2 c - β + 1 ; - b 2 c z 2 a + b + 2 a ( a + b ) , - b 2 c z 2 a + b - 2 a ( a + b ) ) ) ) z p z d z a b c z 2 β 1 2 d p -1 2 c β d -1 p 2 c β -1 2 c z b 2 a b -1 2 a a b 1 2 -1 1 -1 β 2 c z b 2 a b 2 a a b 1 2 -1 1 -1 β 1 4 b -2 c z 1 2 c z 2 a β d p z d -1 p 2 c β AppellF1 d p -1 2 c β 2 c -1 -1 β -1 β d p 2 c -1 -1 β 1 -1 b 2 c z 2 a b 2 a a b 1 2 -1 -1 b 2 c z 2 a b -1 2 a a b 1 2 -1 p -1 d z d p -1 2 c β AppellF1 -1 d -1 p 2 c β 2 c -1 -1 β -1 β p -1 d 2 c -1 -1 β 1 -1 b 2 c z 2 a b 2 a a b 1 2 -1 -1 b 2 c z 2 a b -1 2 a a b 1 2 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["p_", " ", "z_"]]], " ", RowBox[List["Sinh", "[", RowBox[List["d_", " ", "z_"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a_", "+", RowBox[List["b_", " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["c_", " ", "z_"]], "]"]], "2"]]]]], ")"]], "\[Beta]_"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "-", RowBox[List["2", " ", SqrtBox[RowBox[List["a", " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]]]]]]]]]]]], ")"]], RowBox[List["-", "\[Beta]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "+", RowBox[List["2", " ", SqrtBox[RowBox[List["a", " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]]]]]]]]]]]], ")"]], RowBox[List["-", "\[Beta]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List[FractionBox["1", "4"], " ", "b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "c", " ", "z"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], ")"]], "2"]]]]], ")"]], "\[Beta]"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["d", "+", "p"]], ")"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List["d", "-", "p", "+", RowBox[List["2", " ", "c", " ", "\[Beta]"]]]], ")"]], " ", RowBox[List["AppellF1", "[", RowBox[List[FractionBox[RowBox[List["d", "+", "p", "-", RowBox[List["2", " ", "c", " ", "\[Beta]"]]]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["-", "\[Beta]"]], ",", RowBox[List["-", "\[Beta]"]], ",", RowBox[List["1", "+", FractionBox[RowBox[List["d", "+", "p"]], RowBox[List["2", " ", "c"]]], "-", "\[Beta]"]], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "+", RowBox[List["2", " ", SqrtBox[RowBox[List["a", " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]]]]]]]]]]]], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "-", RowBox[List["2", " ", SqrtBox[RowBox[List["a", " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]]]]]]]]]]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "d"]], "+", "p"]], ")"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List["d", "+", "p", "-", RowBox[List["2", " ", "c", " ", "\[Beta]"]]]], ")"]], " ", RowBox[List["AppellF1", "[", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["d", "-", "p", "+", RowBox[List["2", " ", "c", " ", "\[Beta]"]]]], RowBox[List["2", " ", "c"]]]]], ",", RowBox[List["-", "\[Beta]"]], ",", RowBox[List["-", "\[Beta]"]], ",", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["-", "d"]], "+", "p"]], RowBox[List["2", " ", "c"]]], "-", "\[Beta]"]], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "+", RowBox[List["2", " ", SqrtBox[RowBox[List["a", " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]]]]]]]]]]]], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "-", RowBox[List["2", " ", SqrtBox[RowBox[List["a", " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]]]]]]]]]]]]]], "]"]]]]]], ")"]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["d", "+", "p", "-", RowBox[List["2", " ", "c", " ", "\[Beta]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["d", "-", "p", "+", RowBox[List["2", " ", "c", " ", "\[Beta]"]]]], ")"]]]]]]]]]

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

 2002-12-18