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 Cosh

 http://functions.wolfram.com/01.20.21.1490.01

 Input Form

 Integrate[Sinh[b Sqrt[z] + d z + e] Cosh[c Sqrt[z]], z] == (1/8) (E^e ((4 Cosh[e + (b - c) Sqrt[z] + d z])/(E^e d) + ((-b + c) E^((-b + c)^2/(4 d) - 2 e) Sqrt[Pi] Erfi[(-b + c - 2 d Sqrt[z])/(2 Sqrt[-d])])/(-d)^(3/2) - ((-b + c) Sqrt[Pi] Erfi[(-b + c - 2 d Sqrt[z])/(2 Sqrt[d])])/ (E^((b - c)^2/(4 d)) d^(3/2))) - (-((4 E^e Cosh[e + (b + c) Sqrt[z] + d z])/d) - ((b + c) E^((-b - c)^2/(4 d)) Sqrt[Pi] Erfi[(b + c + 2 d Sqrt[z])/ (2 Sqrt[-d])])/(-d)^(3/2) + ((b + c) E^(-((b + c)^2/(4 d)) + 2 e) Sqrt[Pi] Erfi[(b + c + 2 d Sqrt[z])/(2 Sqrt[d])])/d^(3/2))/E^e)

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["Sinh", "[", RowBox[List[RowBox[List["b", " ", SqrtBox["z"]]], "+", RowBox[List["d", " ", "z"]], "+", "e"]], "]"]], RowBox[List["Cosh", "[", RowBox[List["c", " ", SqrtBox["z"]]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", "e"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["4", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", "e"]]], " ", RowBox[List["Cosh", "[", RowBox[List["e", "+", RowBox[List[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], " ", SqrtBox["z"]]], "+", RowBox[List["d", " ", "z"]]]], "]"]]]], "d"], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", "c"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", "c"]], ")"]], "2"], RowBox[List["4", " ", "d"]]], "-", RowBox[List["2", " ", "e"]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", "c", "-", RowBox[List["2", " ", "d", " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", "d"]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "d"]], ")"]], RowBox[List["3", "/", "2"]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", "c"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "2"], RowBox[List["4", " ", "d"]]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", "c", "-", RowBox[List["2", " ", "d", " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox["d"]]]], "]"]]]], SuperscriptBox["d", RowBox[List["3", "/", "2"]]]]]], ")"]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", "e"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["4", " ", SuperscriptBox["\[ExponentialE]", "e"], " ", RowBox[List["Cosh", "[", RowBox[List["e", "+", RowBox[List[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], " ", SqrtBox["z"]]], "+", RowBox[List["d", " ", "z"]]]], "]"]]]], "d"]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "-", "c"]], ")"]], "2"], RowBox[List["4", " ", "d"]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["b", "+", "c", "+", RowBox[List["2", " ", "d", " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", "d"]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "d"]], ")"]], RowBox[List["3", "/", "2"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], "2"], RowBox[List["4", " ", "d"]]]]], "+", RowBox[List["2", " ", "e"]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["b", "+", "c", "+", RowBox[List["2", " ", "d", " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox["d"]]]], "]"]]]], SuperscriptBox["d", RowBox[List["3", "/", "2"]]]]]], ")"]]]]]], ")"]]]]]]]]

 MathML Form

 sinh ( z b + e + d z ) cosh ( c z ) z 1 8 ( e ( 4 - e cosh ( z ( b - c ) + e + d z ) d + ( c - b ) ( c - b ) 2 4 d - 2 e π erfi ( - b + c - 2 d z 2 - d ) ( - d ) 3 / 2 - ( c - b ) - ( b - c ) 2 4 d π erfi ( - b + c - 2 d z 2 d ) d 3 / 2 ) - - e ( - 4 e cosh ( z ( b + c ) + e + d z ) d + ( b + c ) 2 e - ( b + c ) 2 4 d π erfi ( b + c + 2 d z 2 d ) d 3 / 2 - ( b + c ) ( - b - c ) 2 4 d π erfi ( b + c + 2 d z 2 - d ) ( - d ) 3 / 2 ) ) z z 1 2 b e d z c z 1 2 1 8 e 4 -1 e z 1 2 b -1 c e d z d -1 c -1 b c -1 b 2 4 d -1 -1 2 e 1 2 Erfi -1 b c -1 2 d z 1 2 2 -1 d 1 2 -1 -1 d 3 2 -1 -1 c -1 b -1 b -1 c 2 4 d -1 1 2 Erfi -1 b c -1 2 d z 1 2 2 d 1 2 -1 d 3 2 -1 -1 -1 e -1 4 e z 1 2 b c e d z d -1 b c 2 e -1 b c 2 4 d -1 1 2 Erfi b c 2 d z 1 2 2 d 1 2 -1 d 3 2 -1 -1 b c -1 b -1 c 2 4 d -1 1 2 Erfi b c 2 d z 1 2 2 -1 d 1 2 -1 -1 d 3 2 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[RowBox[List["Sinh", "[", RowBox[List[RowBox[List["b_", " ", SqrtBox["z_"]]], "+", RowBox[List["d_", " ", "z_"]], "+", "e_"]], "]"]], " ", RowBox[List["Cosh", "[", RowBox[List["c_", " ", SqrtBox["z_"]]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", "e"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["4", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", "e"]]], " ", RowBox[List["Cosh", "[", RowBox[List["e", "+", RowBox[List[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], " ", SqrtBox["z"]]], "+", RowBox[List["d", " ", "z"]]]], "]"]]]], "d"], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", "c"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", "c"]], ")"]], "2"], RowBox[List["4", " ", "d"]]], "-", RowBox[List["2", " ", "e"]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", "c", "-", RowBox[List["2", " ", "d", " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", "d"]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "d"]], ")"]], RowBox[List["3", "/", "2"]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", "c"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "2"], RowBox[List["4", " ", "d"]]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", "c", "-", RowBox[List["2", " ", "d", " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox["d"]]]], "]"]]]], SuperscriptBox["d", RowBox[List["3", "/", "2"]]]]]], ")"]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", "e"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["4", " ", SuperscriptBox["\[ExponentialE]", "e"], " ", RowBox[List["Cosh", "[", RowBox[List["e", "+", RowBox[List[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], " ", SqrtBox["z"]]], "+", RowBox[List["d", " ", "z"]]]], "]"]]]], "d"]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "-", "c"]], ")"]], "2"], RowBox[List["4", " ", "d"]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["b", "+", "c", "+", RowBox[List["2", " ", "d", " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", "d"]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "d"]], ")"]], RowBox[List["3", "/", "2"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], "2"], RowBox[List["4", " ", "d"]]]]], "+", RowBox[List["2", " ", "e"]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["b", "+", "c", "+", RowBox[List["2", " ", "d", " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox["d"]]]], "]"]]]], SuperscriptBox["d", RowBox[List["3", "/", "2"]]]]]], ")"]]]]]], ")"]]]]]]]]

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

 2002-12-18