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 Cosh

 http://functions.wolfram.com/01.20.21.0850.01

 Input Form

 Integrate[Cosh[c z]/(a + b Cos[d z])^2, z] == (1/2) ((E^((-c + I d) z) ((-a^2 + b^2 - a Sqrt[a^2 - b^2]) Hypergeometric2F1[1 + (I c)/d, 2, 2 + (I c)/d, (b E^(I d z))/(-a + Sqrt[a^2 - b^2])] - (-a^2 + b^2 + a Sqrt[a^2 - b^2]) Hypergeometric2F1[1 + (I c)/d, 2, 2 + (I c)/d, -((b E^(I d z))/(a + Sqrt[a^2 - b^2]))] + a (a + Sqrt[a^2 - b^2]) Hypergeometric2F1[-((I (-c + I d))/d), 1, 2 + (I c)/d, (b E^(I d z))/(-a + Sqrt[a^2 - b^2])] + a (-a + Sqrt[a^2 - b^2]) Hypergeometric2F1[-((I (-c + I d))/d), 1, 2 + (I c)/d, -((b E^(I d z))/(a + Sqrt[a^2 - b^2]))]))/ (b (a^2 - b^2)^(3/2) (-c + I d)) + (E^((c + I d) z) (a (a + Sqrt[a^2 - b^2]) Hypergeometric2F1[ -((I (c + I d))/d), 1, 2 - (I c)/d, (b E^(I d z))/ (-a + Sqrt[a^2 - b^2])] + a (-a + Sqrt[a^2 - b^2]) Hypergeometric2F1[-((I (c + I d))/d), 1, 2 - (I c)/d, -((b E^(I d z))/(a + Sqrt[a^2 - b^2]))] + (-a^2 + b^2 - a Sqrt[a^2 - b^2]) Hypergeometric2F1[-((I (c + I d))/d), 2, 2 - (I c)/d, (b E^(I d z))/(-a + Sqrt[a^2 - b^2])] - (-a^2 + b^2 + a Sqrt[a^2 - b^2]) Hypergeometric2F1[-((I (c + I d))/d), 2, 2 - (I c)/d, -((b E^(I d z))/(a + Sqrt[a^2 - b^2]))]))/ (b (a^2 - b^2)^(3/2) (c + I d)))

 Standard Form

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 MathML Form

 cosh ( c z ) ( a + b cos ( d z ) ) 2 z 1 2 ( 1 b ( a 2 - b 2 ) 3 / 2 ( d - c ) ( ( d - c ) z ( ( - a 2 - a 2 - b 2 a + b 2 ) 2 F 1 ( 1 + c d , 2 ; 2 + c d ; b d z a 2 - b 2 - a ) 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]", " ", "c"]], "d"]]], Hypergeometric2F1], ",", TagBox["2", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List["2", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "c"]], "d"]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "d", " ", "z"]]]]], RowBox[List[SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]], "-", "a"]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] - ( - a 2 + a 2 - b 2 a + b 2 ) 2 F 1 ( 1 + c d , 2 ; 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2 - c d ; - b d z a + a 2 - b 2 ) 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["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "d"]]]], " ", ")"]]]], "d"]]], Hypergeometric2F1], ",", TagBox["2", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List["2", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "c"]], "d"]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "d", " ", "z"]]]]], RowBox[List["a", "+", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] ) ) ) z c z a b d z 2 -1 1 2 1 b a 2 -1 b 2 3 2 d -1 c -1 d -1 c z -1 a 2 -1 a 2 -1 b 2 1 2 a b 2 Hypergeometric2F1 1 c d -1 2 2 c d -1 b d z a 2 -1 b 2 1 2 -1 a -1 -1 -1 a 2 a 2 -1 b 2 1 2 a b 2 Hypergeometric2F1 1 c d -1 2 2 c d -1 -1 b d z a a 2 -1 b 2 1 2 -1 a a a 2 -1 b 2 1 2 Hypergeometric2F1 -1 d -1 c d -1 1 2 c d -1 b d z a 2 -1 b 2 1 2 -1 a -1 a a 2 -1 b 2 1 2 -1 a Hypergeometric2F1 -1 d -1 c d -1 1 2 c d -1 -1 b d z a a 2 -1 b 2 1 2 -1 1 b a 2 -1 b 2 3 2 c d -1 c d z a a a 2 -1 b 2 1 2 Hypergeometric2F1 -1 c d d -1 1 2 -1 c d -1 b d z a 2 -1 b 2 1 2 -1 a -1 a a 2 -1 b 2 1 2 -1 a Hypergeometric2F1 -1 c d d -1 1 2 -1 c d -1 -1 b d z a a 2 -1 b 2 1 2 -1 -1 a 2 -1 a 2 -1 b 2 1 2 a b 2 Hypergeometric2F1 -1 c d d -1 2 2 -1 c d -1 b d z a 2 -1 b 2 1 2 -1 a -1 -1 -1 a 2 a 2 -1 b 2 1 2 a b 2 Hypergeometric2F1 -1 c d d -1 2 2 -1 c d -1 -1 b d z a a 2 -1 b 2 1 2 -1 [/itex]

 Rule Form

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

 2002-12-18