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 Cosh

 http://functions.wolfram.com/01.20.21.3899.01

 Input Form

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

 Standard Form

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")"]]]]]]]]]]]]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "d"]], "+", "e", "+", "p", "-", RowBox[List["2", " ", "c", " ", "\[Beta]"]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "d"]], "-", "e", "+", "p"]], ")"]], " ", "z"]]], " ", RowBox[List["AppellF1", "[", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "d"]], "+", "e", "-", "p", "+", RowBox[List["2", " ", "c", " ", "\[Beta]"]]]], RowBox[List["2", " ", "c"]]]]], ",", RowBox[List["-", "\[Beta]"]], ",", RowBox[List["-", "\[Beta]"]], ",", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "d"]], "+", "e", "-", "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[RowBox[List["\[ImaginaryI]", " ", "d"]], "+", "e", "-", "p", "+", RowBox[List["2", " ", "c", " ", "\[Beta]"]]]], ")"]]]]]], ")"]]]]]]]]

 MathML Form

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

 Rule Form

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

 2002-12-18