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 Cosh

 http://functions.wolfram.com/01.20.21.1524.01

 Input Form

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

 Standard Form

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

 MathML Form

 sinh ( z b + e ) cosh ( z c + g + f z ) z 1 8 ( e - g ( - ( c - b ) ( b - c ) 2 4 f π erfi ( - b + c + 2 f z 2 - f ) ( - f ) 3 / 2 + ( c - b ) 2 ( g - e ) - ( c - b ) 2 4 f π erfi ( - b + c + 2 f z 2 f ) f 3 / 2 - 2 z ( c - b ) + 2 ( g - e ) + f z f - 2 ( b - c ) z - f z f ) - - e - g ( - ( b + c ) ( - b - c ) 2 4 f π erfi ( b + c + 2 f z 2 - f ) ( - f ) 3 / 2 + ( b + c ) 2 ( e + g ) - ( b + c ) 2 4 f π erfi ( b + c + 2 f z 2 f ) f 3 / 2 - 2 z ( b + c ) + 2 ( e + g ) + f z f - 2 ( - b - c ) z - f z f ) ) z z 1 2 b e z 1 2 c g f z 1 8 e -1 g -1 c -1 b b -1 c 2 4 f -1 1 2 Erfi -1 b c 2 f z 1 2 2 -1 f 1 2 -1 -1 f 3 2 -1 c -1 b 2 g -1 e -1 c -1 b 2 4 f -1 1 2 Erfi -1 b c 2 f z 1 2 2 f 1 2 -1 f 3 2 -1 -1 2 z 1 2 c -1 b 2 g -1 e f z f -1 -1 2 b -1 c z 1 2 -1 f z f -1 -1 -1 e -1 g -1 b c -1 b -1 c 2 4 f -1 1 2 Erfi b c 2 f z 1 2 2 -1 f 1 2 -1 -1 f 3 2 -1 b c 2 e g -1 b c 2 4 f -1 1 2 Erfi b c 2 f z 1 2 2 f 1 2 -1 f 3 2 -1 -1 2 z 1 2 b c 2 e g f z f -1 -1 2 -1 b -1 c z 1 2 -1 f z f -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_"]]], "+", "e_"]], "]"]], " ", RowBox[List["Cosh", "[", RowBox[List[RowBox[List["c_", " ", SqrtBox["z_"]]], "+", RowBox[List["f_", " ", "z_"]], "+", "g_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["e", "-", "g"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], " ", SqrtBox["z"]]], "-", RowBox[List["f", " ", "z"]]]]]]], "f"]]], "-", FractionBox[RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "e"]], "+", "g"]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", "c"]], ")"]], " ", SqrtBox["z"]]], "+", RowBox[List["f", " ", "z"]]]]]]], "f"], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", "c"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "2"], RowBox[List["4", " ", "f"]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", "c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", "f"]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "f"]], ")"]], RowBox[List["3", "/", "2"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", "c"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", "c"]], ")"]], "2"], RowBox[List["4", " ", "f"]]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "e"]], "+", "g"]], ")"]]]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", "c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox["f"]]]], "]"]]]], SuperscriptBox["f", RowBox[List["3", "/", "2"]]]]]], ")"]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "e"]], "-", "g"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "-", "c"]], ")"]], " ", SqrtBox["z"]]], "-", RowBox[List["f", " ", "z"]]]]]]], "f"]]], "-", FractionBox[RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["e", "+", "g"]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], " ", SqrtBox["z"]]], "+", RowBox[List["f", " ", "z"]]]]]]], "f"], "-", 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", " ", "f"]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["b", "+", "c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", "f"]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "f"]], ")"]], 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", " ", "f"]]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["e", "+", "g"]], ")"]]]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["b", "+", "c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox["f"]]]], "]"]]]], SuperscriptBox["f", RowBox[List["3", "/", "2"]]]]]], ")"]]]]]], ")"]]]]]]]]

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

 2002-12-18