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/01.20.21.0865.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

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

 2002-12-18