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/01.20.21.4256.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["Sinh", "[", RowBox[List["d", " ", "z"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]], "2"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["c", "-", "d"]], ")"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["a", " ", RowBox[List["(", RowBox[List["a", "+", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["c", "-", "d"]], "c"], ",", "1", ",", RowBox[List["2", "-", FractionBox["d", "c"]]], ",", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]]]], RowBox[List[RowBox[List["-", "a"]], "+", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]]]]], "]"]]]], "+", RowBox[List["a", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["c", "-", "d"]], "c"], ",", "1", ",", RowBox[List["2", "-", FractionBox["d", "c"]]], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]]]], RowBox[List["a", "+", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["c", "-", "d"]], "c"], ",", "2", ",", RowBox[List["2", "-", FractionBox["d", "c"]]], ",", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]]]], RowBox[List[RowBox[List["-", "a"]], "+", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]]]]], "]"]], "-", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["c", "-", "d"]], "c"], ",", "2", ",", RowBox[List["2", "-", FractionBox["d", "c"]]], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]]]], RowBox[List["a", "+", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]]]]]]], "]"]]]], ")"]]]], "-", RowBox[List["a", " ", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["c", "-", "d"]], "c"], ",", "2", ",", RowBox[List["2", "-", FractionBox["d", "c"]]], ",", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]]]], RowBox[List[RowBox[List["-", "a"]], "+", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]]]]], "]"]], "+", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["c", "-", "d"]], "c"], ",", "2", ",", RowBox[List["2", "-", FractionBox["d", "c"]]], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]]]], RowBox[List["a", "+", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]]]]]]], "]"]]]], ")"]]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List["b", " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]], ")"]], RowBox[List["3", "/", "2"]]], " ", RowBox[List["(", RowBox[List["c", "-", "d"]], ")"]]]], ")"]]]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["c", "+", "d"]], ")"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["a", " ", RowBox[List["(", RowBox[List["a", "+", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["c", "+", "d"]], "c"], ",", "1", ",", RowBox[List["2", "+", FractionBox["d", "c"]]], ",", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]]]], RowBox[List[RowBox[List["-", "a"]], "+", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]]]]], "]"]]]], "+", RowBox[List["a", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["c", "+", "d"]], "c"], ",", "1", ",", RowBox[List["2", "+", FractionBox["d", "c"]]], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]]]], RowBox[List["a", "+", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["c", "+", "d"]], "c"], ",", "2", ",", RowBox[List["2", "+", FractionBox["d", "c"]]], ",", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]]]], RowBox[List[RowBox[List["-", "a"]], "+", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]]]]], "]"]], "-", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["c", "+", "d"]], "c"], ",", "2", ",", RowBox[List["2", "+", FractionBox["d", "c"]]], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]]]], RowBox[List["a", "+", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]]]]]]], "]"]]]], ")"]]]], "-", RowBox[List["a", " ", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["c", "+", "d"]], "c"], ",", "2", ",", RowBox[List["2", "+", FractionBox["d", "c"]]], ",", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]]]], RowBox[List[RowBox[List["-", "a"]], "+", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]]]]], "]"]], "+", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["c", "+", "d"]], "c"], ",", "2", ",", RowBox[List["2", "+", FractionBox["d", "c"]]], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]]]], RowBox[List["a", "+", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]]]]]]], "]"]]]], ")"]]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List["b", " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]], ")"]], RowBox[List["3", "/", "2"]]], " ", RowBox[List["(", RowBox[List["c", "+", "d"]], ")"]]]], ")"]]]]]], ")"]]]]]]]]

 MathML Form

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["Sinh", "[", RowBox[List["d_", " ", "z_"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["a_", "+", RowBox[List["b_", " ", RowBox[List["Cosh", "[", RowBox[List["c_", " ", "z_"]], "]"]]]]]], ")"]], "2"]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["c", "-", "d"]], ")"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["a", " ", RowBox[List["(", RowBox[List["a", "+", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["c", "-", "d"]], "c"], ",", "1", ",", RowBox[List["2", "-", FractionBox["d", "c"]]], ",", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]]]], RowBox[List[RowBox[List["-", "a"]], "+", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]]]]], "]"]]]], "+", RowBox[List["a", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["c", "-", "d"]], "c"], ",", "1", ",", RowBox[List["2", "-", FractionBox["d", "c"]]], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]]]], RowBox[List["a", "+", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["c", "-", "d"]], "c"], ",", "2", ",", RowBox[List["2", "-", FractionBox["d", "c"]]], ",", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]]]], RowBox[List[RowBox[List["-", "a"]], "+", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]]]]], "]"]], "-", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["c", "-", "d"]], "c"], ",", "2", ",", RowBox[List["2", "-", FractionBox["d", "c"]]], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]]]], RowBox[List["a", "+", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]]]]]]], "]"]]]], ")"]]]], "-", RowBox[List["a", " ", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["c", "-", "d"]], "c"], ",", "2", ",", RowBox[List["2", "-", FractionBox["d", "c"]]], ",", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]]]], RowBox[List[RowBox[List["-", "a"]], "+", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]]]]], "]"]], "+", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["c", "-", "d"]], "c"], ",", "2", ",", RowBox[List["2", "-", FractionBox["d", "c"]]], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]]]], RowBox[List["a", "+", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]]]]]]], "]"]]]], ")"]]]]]], ")"]]]], RowBox[List["b", " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]], ")"]], RowBox[List["3", "/", "2"]]], " ", RowBox[List["(", RowBox[List["c", "-", "d"]], ")"]]]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["c", "+", "d"]], ")"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["a", " ", RowBox[List["(", RowBox[List["a", "+", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["c", "+", "d"]], "c"], ",", "1", ",", RowBox[List["2", "+", FractionBox["d", "c"]]], ",", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]]]], RowBox[List[RowBox[List["-", "a"]], "+", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]]]]], "]"]]]], "+", RowBox[List["a", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["c", "+", "d"]], "c"], ",", "1", ",", RowBox[List["2", "+", FractionBox["d", "c"]]], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]]]], RowBox[List["a", "+", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["c", "+", "d"]], "c"], ",", "2", ",", RowBox[List["2", "+", FractionBox["d", "c"]]], ",", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]]]], RowBox[List[RowBox[List["-", "a"]], "+", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]]]]], "]"]], "-", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["c", "+", "d"]], "c"], ",", "2", ",", RowBox[List["2", "+", FractionBox["d", "c"]]], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]]]], RowBox[List["a", "+", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]]]]]]], "]"]]]], ")"]]]], "-", RowBox[List["a", " ", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["c", "+", "d"]], "c"], ",", "2", ",", RowBox[List["2", "+", FractionBox["d", "c"]]], ",", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]]]], RowBox[List[RowBox[List["-", "a"]], "+", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]]]]], "]"]], "+", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["c", "+", "d"]], "c"], ",", "2", ",", RowBox[List["2", "+", FractionBox["d", "c"]]], ",", RowBox[List["-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]]]], RowBox[List["a", "+", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]]]]]]], "]"]]]], ")"]]]]]], ")"]]]], RowBox[List["b", " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]], ")"]], RowBox[List["3", "/", "2"]]], " ", RowBox[List["(", RowBox[List["c", "+", "d"]], ")"]]]]]]], ")"]]]]]]]]

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

 2002-12-18