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 Cosh

 http://functions.wolfram.com/01.20.21.1510.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[RowBox[List["Sinh", "[", RowBox[List["b_", " ", SqrtBox["z_"]]], "]"]], " ", RowBox[List["Cosh", "[", RowBox[List[RowBox[List["c_", " ", SqrtBox["z_"]]], "+", RowBox[List["f_", " ", "z_"]]]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "8"], " ", 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[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], " ", SqrtBox["z"]]], "-", RowBox[List["f", " ", "z"]]]]]]], "f"], "-", 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[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["-", FractionBox[SuperscriptBox[RowBox[List["(", 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["f"]]]], "]"]]]], SuperscriptBox["f", RowBox[List["3", "/", "2"]]]], "+", 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["-", FractionBox[SuperscriptBox[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["f"]]]], "]"]]]], SuperscriptBox["f", RowBox[List["3", "/", "2"]]]]]], ")"]]]]]]]]

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

 2002-12-18