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 Cosh

 http://functions.wolfram.com/01.20.21.1249.01

 Input Form

 Integrate[E^(p z) (a + b Sin[d z]^2)^\[Beta] Cosh[c z], z] == (1/2) ((1/(-c + p - 2 I d \[Beta])) ((E^((-c + p) z) (a - ((1/4) b (-1 + E^(2 I d z))^2)/E^(2 I d z))^\[Beta] AppellF1[-((I (-c + p))/(2 d)) - \[Beta], -\[Beta], -\[Beta], 1 - (I (-c + p))/(2 d) - \[Beta], -((b E^(2 I d z))/(-2 a - b - 2 Sqrt[a (a + b)])), -((b E^(2 I d z))/(-2 a - b + 2 Sqrt[a (a + b)]))])/ ((1 + (b E^(2 I d z))/(-2 a - b + 2 Sqrt[a (a + b)]))^\[Beta] (1 - (b E^(2 I d z))/(2 a + b + 2 Sqrt[a (a + b)]))^\[Beta])) + (1/(c + p - 2 I d \[Beta])) ((E^((c + p) z) (a - ((1/4) b (-1 + E^(2 I d z))^2)/E^(2 I d z))^\[Beta] AppellF1[-((I (c + p))/(2 d)) - \[Beta], -\[Beta], -\[Beta], 1 - (I (c + p))/(2 d) - \[Beta], -((b E^(2 I d z))/(-2 a - b - 2 Sqrt[a (a + b)])), -((b E^(2 I d z))/(-2 a - b + 2 Sqrt[a (a + b)]))])/ ((1 + (b E^(2 I d z))/(-2 a - b + 2 Sqrt[a (a + b)]))^\[Beta] (1 - (b E^(2 I d z))/(2 a + b + 2 Sqrt[a (a + b)]))^\[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", " ", SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["d", " ", "z"]], "]"]], "2"]]]]], ")"]], "\[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["2", " ", "\[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["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "d", " ", "z"]]]]], 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"b"]], ")"]]]]]]]]]]]]]], "]"]]]], ")"]]]]]], ")"]]]]]]]]

 MathML Form

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

 Rule Form

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

 2002-12-18