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 Cosh

 http://functions.wolfram.com/01.20.21.1316.01

 Input Form

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

 Standard Form

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 MathML Form

 p z cosh ( c z ) a + b cos 2 ( d z ) z 1 2 a b a + b ( - 1 c + 2 d + p ( ( c + 2 d + p ) z ( ( 2 a - 2 a + b a + b ) 2 F 1 ( 1 - ( c + p ) 2 d , 1 ; 2 - ( c + p ) 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["c", "+", "p"]], ")"]]]], 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["c", "+", "p"]], ")"]]]], RowBox[List["2", " ", "d"]]]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "d", " ", "z"]]]]], 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 + p ) 2 d , 1 ; 2 - ( c + p ) 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["c", "+", "p"]], ")"]]]], 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["c", "+", "p"]], ")"]]]], RowBox[List["2", " ", "d"]]]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "d", " ", "z"]]]]], 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] ) ) - 1 - c + 2 d + p ( ( - c + 2 d + p ) z ( ( 2 a - 2 a + b a + b ) 2 F 1 ( 1 - ( p - c ) 2 d , 1 ; 2 - ( p - c ) 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["p", "-", "c"]], ")"]]]], 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["p", "-", "c"]], ")"]]]], RowBox[List["2", " ", "d"]]]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "d", " ", "z"]]]]], 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 - ( p - c ) 2 d , 1 ; 2 - ( p - c ) 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["p", "-", "c"]], ")"]]]], 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["p", "-", "c"]], ")"]]]], RowBox[List["2", " ", "d"]]]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "d", " ", "z"]]]]], 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] ) ) ) z p z c z a b d z 2 -1 1 2 a 1 2 b a b 1 2 -1 -1 1 c 2 d p -1 c 2 d p z 2 a -1 2 a b 1 2 a 1 2 b Hypergeometric2F1 1 -1 c p 2 d -1 1 2 -1 c p 2 d -1 -1 b 2 d z 2 a 2 a b 1 2 a 1 2 b -1 -2 a -1 2 a b 1 2 a 1 2 -1 b Hypergeometric2F1 1 -1 c p 2 d -1 1 2 -1 c p 2 d -1 -1 b 2 d z 2 a -1 2 a b 1 2 a 1 2 b -1 -1 1 -1 c 2 d p -1 -1 c 2 d p z 2 a -1 2 a b 1 2 a 1 2 b Hypergeometric2F1 1 -1 p -1 c 2 d -1 1 2 -1 p -1 c 2 d -1 -1 b 2 d z 2 a 2 a b 1 2 a 1 2 b -1 -2 a -1 2 a b 1 2 a 1 2 -1 b Hypergeometric2F1 1 -1 p -1 c 2 d -1 1 2 -1 p -1 c 2 d -1 -1 b 2 d z 2 a -1 2 a b 1 2 a 1 2 b -1 [/itex]

 Rule Form

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

 2002-12-18