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 Cosh

 http://functions.wolfram.com/01.20.21.4311.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

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

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

 2002-12-18