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 Cosh

 http://functions.wolfram.com/01.20.21.0687.01

 Input Form

 Integrate[(Sin[e z] Cosh[c z])/(a + b Sin[d z]^2), z] == (-(1/4)) I ((E^((-c - 2 I d - I e) z) ((-2 a - b + 2 Sqrt[a] Sqrt[a + b]) Hypergeometric2F1[1 + (I (-c - I e))/(2 d), 1, 2 + (I (-c - I e))/(2 d), -(b/(E^(2 I d z) (-2 a - b - 2 Sqrt[a] Sqrt[a + b])))] + (2 a + b + 2 Sqrt[a] Sqrt[a + b]) Hypergeometric2F1[ 1 + (I (-c - I e))/(2 d), 1, 2 + (I (-c - I e))/(2 d), -(b/(E^(2 I d z) (-2 a - b + 2 Sqrt[a] Sqrt[a + b])))]))/ (Sqrt[a] b Sqrt[a + b] (-c - 2 I d - I e)) + (E^((c - 2 I d - I e) z) ((-2 a - b + 2 Sqrt[a] Sqrt[a + b]) Hypergeometric2F1[1 + (I (c - I e))/(2 d), 1, 2 + (I (c - I e))/(2 d), -(b/(E^(2 I d z) (-2 a - b - 2 Sqrt[a] Sqrt[a + b])))] + (2 a + b + 2 Sqrt[a] Sqrt[a + b]) Hypergeometric2F1[ 1 + (I (c - I e))/(2 d), 1, 2 + (I (c - I e))/(2 d), -(b/(E^(2 I d z) (-2 a - b + 2 Sqrt[a] Sqrt[a + b])))]))/ (Sqrt[a] b Sqrt[a + b] (c - 2 I d - I e)) - (E^((-c - 2 I d + I e) z) ((-2 a - b + 2 Sqrt[a] Sqrt[a + b]) Hypergeometric2F1[1 + (I (-c + I e))/(2 d), 1, 2 + (I (-c + I e))/(2 d), -(b/(E^(2 I d z) (-2 a - b - 2 Sqrt[a] Sqrt[a + b])))] + (2 a + b + 2 Sqrt[a] Sqrt[a + b]) Hypergeometric2F1[ 1 + (I (-c + I e))/(2 d), 1, 2 + (I (-c + I e))/(2 d), -(b/(E^(2 I d z) (-2 a - b + 2 Sqrt[a] Sqrt[a + b])))]))/ (Sqrt[a] b Sqrt[a + b] (-c - 2 I d + I e)) - (E^((c - 2 I d + I e) z) ((-2 a - b + 2 Sqrt[a] Sqrt[a + b]) Hypergeometric2F1[1 + (I (c + I e))/(2 d), 1, 2 + (I (c + I e))/(2 d), -(b/(E^(2 I d z) (-2 a - b - 2 Sqrt[a] Sqrt[a + b])))] + (2 a + b + 2 Sqrt[a] Sqrt[a + b]) Hypergeometric2F1[ 1 + (I (c + I e))/(2 d), 1, 2 + (I (c + I e))/(2 d), -(b/(E^(2 I d z) (-2 a - b + 2 Sqrt[a] Sqrt[a + b])))]))/ (Sqrt[a] b Sqrt[a + b] (c - 2 I d + I e)))

 Standard Form

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 MathML Form

 sin ( e z ) cosh ( c z ) a + b sin 2 ( d z ) z - 1 4 ( ( ( - c - 2 d - e ) z ( ( 2 a + 2 a + b a + b ) 2 F 1 ( 1 + ( - c - e ) 2 d , 1 ; 2 + ( - c - e ) 2 d ; - b - 2 d z - 2 a + 2 a + b a - b ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "-", RowBox[List["\[ImaginaryI]", " ", "e"]]]], ")"]]]], RowBox[List["2", " ", "d"]]]]], Hypergeometric2F1], ",", TagBox["1", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List["2", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "-", RowBox[List["\[ImaginaryI]", " ", "e"]]]], ")"]]]], RowBox[List["2", " ", "d"]]]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "d", " ", "z"]]]]], RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "a"]], "+", RowBox[List["2", " ", SqrtBox[RowBox[List["a", "+", "b"]]], " ", SqrtBox["a"]]], "-", "b"]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] + ( - 2 a + 2 a + b a - b ) 2 F 1 ( 1 + ( - c - e ) 2 d , 1 ; 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 Date Added to functions.wolfram.com (modification date)

 2002-12-18