html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 Cosh

 http://functions.wolfram.com/01.20.21.0851.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]], RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["d", " ", "z"]], "]"]], "2"]]]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["2", " ", SqrtBox["a"], " ", "b", " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]], RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List[FractionBox["1", RowBox[List["c", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "d"]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "d"]]]], ")"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "a"]], "-", "b", "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "c"]], RowBox[List["2", " ", "d"]]]]], ",", "1", ",", RowBox[List["2", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "c"]], RowBox[List["2", " ", "d"]]]]], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "d", " ", "z"]]]]], RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "c"]], RowBox[List["2", " ", "d"]]]]], ",", "1", ",", RowBox[List["2", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "c"]], RowBox[List["2", " ", "d"]]]]], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "d", " ", "z"]]]]], RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]]]]]]], "]"]]]]]], ")"]]]], ")"]]]]]], "-", RowBox[List[FractionBox["1", RowBox[List[RowBox[List["-", "c"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "d"]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "d"]]]], ")"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "a"]], "-", "b", "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "c"]], RowBox[List["2", " ", "d"]]]]], ",", "1", ",", RowBox[List["2", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "c"]], RowBox[List["2", " ", "d"]]]]], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "d", " ", "z"]]]]], RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "c"]], RowBox[List["2", " ", "d"]]]]], ",", "1", ",", RowBox[List["2", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "c"]], RowBox[List["2", " ", "d"]]]]], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "d", " ", "z"]]]]], RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]]]]]]], "]"]]]]]], ")"]]]], ")"]]]]]], ")"]]]]]]]]

 MathML Form

 cosh ( c z ) a + b cos 2 ( d z ) z 1 2 a b a + b ( - 1 2 d - c ( ( 2 d - c ) z ( ( 2 a - 2 a + b a + b ) 2 F 1 ( 1 + c 2 d , 1 ; 2 + 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]", " ", "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]", " ", "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 + c 2 d , 1 ; 2 + 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]", " ", "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]", " ", "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] ) ) - 1 c + 2 d ( ( c + 2 d ) z ( ( 2 a - 2 a + b a + b ) 2 F 1 ( 1 - c 2 d , 1 ; 2 - 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]", " ", "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]", " ", "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 - c 2 d , 1 ; 2 - 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]", " ", "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]", " ", "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 c z a b d z 2 -1 1 2 a 1 2 b a b 1 2 -1 -1 1 2 d -1 c -1 2 d -1 c z 2 a -1 2 a b 1 2 a 1 2 b Hypergeometric2F1 1 c 2 d -1 1 2 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 c 2 d -1 1 2 c 2 d -1 -1 b 2 d z 2 a -1 2 a b 1 2 a 1 2 b -1 -1 1 c 2 d -1 c 2 d z 2 a -1 2 a b 1 2 a 1 2 b Hypergeometric2F1 1 -1 c 2 d -1 1 2 -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 c 2 d -1 1 2 -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

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["Cosh", "[", RowBox[List["c_", " ", "z_"]], "]"]], RowBox[List["a_", "+", RowBox[List["b_", " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["d_", " ", "z_"]], "]"]], "2"]]]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "d"]]]], ")"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "a"]], "-", "b", "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "c"]], RowBox[List["2", " ", "d"]]]]], ",", "1", ",", RowBox[List["2", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "c"]], RowBox[List["2", " ", "d"]]]]], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "d", " ", "z"]]]]], RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "c"]], RowBox[List["2", " ", "d"]]]]], ",", "1", ",", RowBox[List["2", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "c"]], RowBox[List["2", " ", "d"]]]]], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "d", " ", "z"]]]]], RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]]]]]]], "]"]]]]]], ")"]]]], RowBox[List["c", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "d"]]]]]]], "-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "d"]]]], ")"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "a"]], "-", "b", "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "c"]], RowBox[List["2", " ", "d"]]]]], ",", "1", ",", RowBox[List["2", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "c"]], RowBox[List["2", " ", "d"]]]]], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "d", " ", "z"]]]]], RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "-", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "c"]], RowBox[List["2", " ", "d"]]]]], ",", "1", ",", RowBox[List["2", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "c"]], RowBox[List["2", " ", "d"]]]]], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "d", " ", "z"]]]]], RowBox[List[RowBox[List["2", " ", "a"]], "+", "b", "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]]]]]]], "]"]]]]]], ")"]]]], RowBox[List[RowBox[List["-", "c"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "d"]]]]]]], RowBox[List["2", " ", SqrtBox["a"], " ", "b", " ", SqrtBox[RowBox[List["a", "+", "b"]]]]]]]]]]

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

 2002-12-18