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; }

 Coth

 http://functions.wolfram.com/01.22.21.0341.01

 Input Form

 Integrate[Sqrt[a + b Sinh[c z]] Coth[c z]^2, z] == (1/(4 c)) ((1/(a Sqrt[1/(-a + I b)] b)) (6 (I a (a + I b) EllipticE[I ArcSinh[Sqrt[1/(-a + I b)] Sqrt[a + b Sinh[c z]]], (a - I b)/(a + I b)] + b (a EllipticF[I ArcSinh[Sqrt[1/(-a + I b)] Sqrt[a + b Sinh[c z]]], (a - I b)/(a + I b)] + I b EllipticPi[1 - (I b)/a, I ArcSinh[Sqrt[1/(-a + I b)] Sqrt[a + b Sinh[c z]]], (a - I b)/(a + I b)])) Sech[c z] Sqrt[(b - I b Sinh[c z])/(I a + b)] Sqrt[(b + I b Sinh[c z])/((-I) a + b)]) - 4 Coth[c z] Sqrt[a + b Sinh[c z]] + (8 I a EllipticF[(1/4) (Pi - 2 I c z), -((2 I b)/(a - I b))] Sqrt[(a + b Sinh[c z])/(a - I b)])/Sqrt[a + b Sinh[c z]] + (1/Sqrt[a + b Sinh[c z]]) (6 ((I a + b) EllipticE[(1/4) (Pi - 2 I c z), -((2 I b)/(a - I b))] - I a EllipticF[(1/4) (Pi - 2 I c z), -((2 I b)/(a - I b))]) Sqrt[(a + b Sinh[c z])/(a - I b)]) + (2 b EllipticPi[2, (1/4) (Pi - 2 I c z), -((2 I b)/(a - I b))] Sqrt[(a + b Sinh[c z])/(a - I b)])/Sqrt[a + b Sinh[c z]])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]], SuperscriptBox[RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]], "2"], " ", RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["4", " ", "c"]]], RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", RowBox[List["a", " ", SqrtBox[FractionBox["1", RowBox[List[RowBox[List["-", "a"]], "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]], " ", "b"]]], RowBox[List["(", RowBox[List["6", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSinh", "[", RowBox[List[SqrtBox[FractionBox["1", RowBox[List[RowBox[List["-", "a"]], "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]], " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]]], "]"]]]], ",", FractionBox[RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]], RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["a", " ", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSinh", "[", RowBox[List[SqrtBox[FractionBox["1", RowBox[List[RowBox[List["-", "a"]], "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]], " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]]], "]"]]]], ",", FractionBox[RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]], RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]], "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["EllipticPi", "[", RowBox[List[RowBox[List["1", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "b"]], "a"]]], ",", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSinh", "[", RowBox[List[SqrtBox[FractionBox["1", RowBox[List[RowBox[List["-", "a"]], "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]], " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]]], "]"]]]], ",", FractionBox[RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]], RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]], "]"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["Sech", "[", RowBox[List["c", " ", "z"]], "]"]], " ", SqrtBox[FractionBox[RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "+", "b"]]]], " ", SqrtBox[FractionBox[RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "+", "b"]]]]]], ")"]]]], "-", RowBox[List["4", " ", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]], " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]]], "+", FractionBox[RowBox[List["8", " ", "\[ImaginaryI]", " ", "a", " ", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]], ")"]]]], ",", RowBox[List["-", FractionBox[RowBox[List["2", " ", "\[ImaginaryI]", " ", "b"]], RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]]]], "]"]], " ", SqrtBox[FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]]], SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]], "+", RowBox[List[FractionBox["1", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]], RowBox[List["(", RowBox[List["6", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "+", "b"]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]], ")"]]]], ",", RowBox[List["-", FractionBox[RowBox[List["2", " ", "\[ImaginaryI]", " ", "b"]], RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]]]], "]"]]]], "-", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]], ")"]]]], ",", RowBox[List["-", FractionBox[RowBox[List["2", " ", "\[ImaginaryI]", " ", "b"]], RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]]]], "]"]]]]]], ")"]], " ", SqrtBox[FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]]], ")"]]]], "+", FractionBox[RowBox[List["2", " ", "b", " ", RowBox[List["EllipticPi", "[", RowBox[List["2", ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]], ")"]]]], ",", RowBox[List["-", FractionBox[RowBox[List["2", " ", "\[ImaginaryI]", " ", "b"]], RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]]]], "]"]], " ", SqrtBox[FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]]], SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]]]], ")"]]]]]]]]

 MathML Form

 a + b sinh ( c z ) coth 2 ( c z ) z 1 4 c ( - 4 a + b sinh ( c z ) coth ( c z ) + 1 a 1 b - a b ( 6 ( a ( a + b ) E ( sinh - 1 ( 1 b - a a + b sinh ( c z ) ) a - b a + b ) + b ( a F ( sinh - 1 ( 1 b - a a + b sinh ( c z ) ) a - b a + b ) + b Π ( 1 - b a ; sinh - 1 ( 1 b - a a + b sinh ( c z ) ) a - b a + b ) ) ) sech ( c z ) b - b sinh ( c z ) b + a sinh ( c z ) b + b b - a ) + 6 ( ( b + a ) E ( 1 4 ( π - 2 c z ) - 2 b a - b ) - a F ( 1 4 ( π - 2 c z ) - 2 b a - b ) ) a + b sinh ( c z ) a - b a + b sinh ( c z ) + 8 a a + b sinh ( c z ) a - b F ( 1 4 ( π - 2 c z ) - 2 b a - b ) a + b sinh ( c z ) + 2 b a + b sinh ( c z ) a - b Π ( 2 ; 1 4 ( π - 2 c z ) - 2 b a - b ) a + b sinh ( c z ) ) z a b c z 1 2 c z 2 1 4 c -1 -4 a b c z 1 2 c z 1 a 1 b -1 a -1 1 2 b -1 6 a a b EllipticE 1 b -1 a -1 1 2 a b c z 1 2 a -1 b a b -1 b a EllipticF 1 b -1 a -1 1 2 a b c z 1 2 a -1 b a b -1 b EllipticPi 1 -1 b a -1 1 b -1 a -1 1 2 a b c z 1 2 a -1 b a b -1 c z b -1 b c z b a -1 1 2 c z b b b -1 a -1 1 2 6 b a EllipticE 1 4 -1 2 c z -1 2 b a -1 b -1 -1 a EllipticF 1 4 -1 2 c z -1 2 b a -1 b -1 a b c z a -1 b -1 1 2 a b c z 1 2 -1 8 a a b c z a -1 b -1 1 2 EllipticF 1 4 -1 2 c z -1 2 b a -1 b -1 a b c z 1 2 -1 2 b a b c z a -1 b -1 1 2 EllipticPi 2 1 4 -1 2 c z -1 2 b a -1 b -1 a b c z 1 2 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SqrtBox[RowBox[List["a_", "+", RowBox[List["b_", " ", RowBox[List["Sinh", "[", RowBox[List["c_", " ", "z_"]], "]"]]]]]]], " ", SuperscriptBox[RowBox[List["Coth", "[", RowBox[List["c_", " ", "z_"]], "]"]], "2"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[FractionBox[RowBox[List["6", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSinh", "[", RowBox[List[SqrtBox[FractionBox["1", RowBox[List[RowBox[List["-", "a"]], "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]], " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]]], "]"]]]], ",", FractionBox[RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]], RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["a", " ", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSinh", "[", RowBox[List[SqrtBox[FractionBox["1", RowBox[List[RowBox[List["-", "a"]], "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]], " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]]], "]"]]]], ",", FractionBox[RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]], RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]], "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["EllipticPi", "[", RowBox[List[RowBox[List["1", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "b"]], "a"]]], ",", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSinh", "[", RowBox[List[SqrtBox[FractionBox["1", RowBox[List[RowBox[List["-", "a"]], "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]], " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]]], "]"]]]], ",", FractionBox[RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]], RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]], "]"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["Sech", "[", RowBox[List["c", " ", "z"]], "]"]], " ", SqrtBox[FractionBox[RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "+", "b"]]]], " ", SqrtBox[FractionBox[RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]], "+", "b"]]]]]], RowBox[List["a", " ", SqrtBox[FractionBox["1", RowBox[List[RowBox[List["-", "a"]], "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]], " ", "b"]]], "-", RowBox[List["4", " ", RowBox[List["Coth", "[", RowBox[List["c", " ", "z"]], "]"]], " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]]], "+", FractionBox[RowBox[List["8", " ", "\[ImaginaryI]", " ", "a", " ", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]], ")"]]]], ",", RowBox[List["-", FractionBox[RowBox[List["2", " ", "\[ImaginaryI]", " ", "b"]], RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]]]], "]"]], " ", SqrtBox[FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]]], SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]], "+", FractionBox[RowBox[List["6", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a"]], "+", "b"]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]], ")"]]]], ",", RowBox[List["-", FractionBox[RowBox[List["2", " ", "\[ImaginaryI]", " ", "b"]], RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]]]], "]"]]]], "-", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]], ")"]]]], ",", RowBox[List["-", FractionBox[RowBox[List["2", " ", "\[ImaginaryI]", " ", "b"]], RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]]]], "]"]]]]]], ")"]], " ", SqrtBox[FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]]], SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]], "+", FractionBox[RowBox[List["2", " ", "b", " ", RowBox[List["EllipticPi", "[", RowBox[List["2", ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]], ")"]]]], ",", RowBox[List["-", FractionBox[RowBox[List["2", " ", "\[ImaginaryI]", " ", "b"]], RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]]]], "]"]], " ", SqrtBox[FractionBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]]], SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]]]]]], RowBox[List["4", " ", "c"]]]]]]]

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

 2002-12-18