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 SinhIntegral

 http://functions.wolfram.com/06.39.21.0019.01

 Input Form

 Integrate[z^3 E^(b z) SinhIntegral[a z], z] == (1/b^4) (3 (-ExpIntegralEi[(-a + b) z] + ExpIntegralEi[(a + b) z]) - (1/((-a + b)^3 (a + b)^3)) (b E^(b z) ((-a) (b^4 (18 - 7 b z + b^2 z^2) - 2 a^2 b^2 (8 - 5 b z + b^2 z^2) + a^4 (6 - 3 b z + b^2 z^2)) Cosh[a z] + b (b^4 (11 - 5 b z + b^2 z^2) - 2 a^2 b^2 (3 - 3 b z + b^2 z^2) + a^4 (3 - b z + b^2 z^2)) Sinh[a z])) + E^(b z) (-6 + 6 b z - 3 b^2 z^2 + b^3 z^3) SinhIntegral[a z])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "3"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["b", " ", "z"]]], " ", RowBox[List["SinhIntegral", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", SuperscriptBox["b", "4"]], RowBox[List["(", RowBox[List[RowBox[List["3", " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "b"]], ")"]], " ", "z"]], "]"]]]], "+", RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", "z"]], "]"]]]], ")"]]]], "-", RowBox[List[FractionBox["1", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "b"]], ")"]], "3"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], "3"]]]], RowBox[List["(", RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["b", " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "a"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["b", "4"], " ", RowBox[List["(", RowBox[List["18", "-", RowBox[List["7", " ", "b", " ", "z"]], "+", RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox["z", "2"]]]]], ")"]]]], "-", RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["b", "2"], " ", RowBox[List["(", RowBox[List["8", "-", RowBox[List["5", " ", "b", " ", "z"]], "+", RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox["z", "2"]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["a", "4"], " ", RowBox[List["(", RowBox[List["6", "-", RowBox[List["3", " ", "b", " ", "z"]], "+", RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox["z", "2"]]]]], ")"]]]]]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["a", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["b", "4"], " ", RowBox[List["(", RowBox[List["11", "-", RowBox[List["5", " ", "b", " ", "z"]], "+", RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox["z", "2"]]]]], ")"]]]], "-", RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["b", "2"], " ", RowBox[List["(", RowBox[List["3", "-", RowBox[List["3", " ", "b", " ", "z"]], "+", RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox["z", "2"]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["a", "4"], " ", RowBox[List["(", RowBox[List["3", "-", RowBox[List["b", " ", "z"]], "+", RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox["z", "2"]]]]], ")"]]]]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List["a", " ", "z"]], "]"]]]]]], ")"]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["b", " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "6"]], "+", RowBox[List["6", " ", "b", " ", "z"]], "-", RowBox[List["3", " ", SuperscriptBox["b", "2"], " ", SuperscriptBox["z", "2"]]], "+", RowBox[List[SuperscriptBox["b", "3"], " ", SuperscriptBox["z", "3"]]]]], ")"]], " ", RowBox[List["SinhIntegral", "[", RowBox[List["a", " ", "z"]], "]"]]]]]], ")"]]]]]]]]

 MathML Form

 z 3 b z Shi ( a z ) z 1 b 4 ( 3 ( Ei ( ( a + b ) z ) - Ei ( ( b - a ) z ) ) - 1 ( b - a ) 3 ( a + b ) 3 ( b b z ( b ( ( b 2 z 2 - b z + 3 ) a 4 - 2 b 2 ( b 2 z 2 - 3 b z + 3 ) a 2 + b 4 ( b 2 z 2 - 5 b z + 11 ) ) sinh ( a z ) - a ( ( b 2 z 2 - 3 b z + 6 ) a 4 - 2 b 2 ( b 2 z 2 - 5 b z + 8 ) a 2 + b 4 ( b 2 z 2 - 7 b z + 18 ) ) cosh ( a z ) ) ) + b z ( b 3 z 3 - 3 b 2 z 2 + 6 b z - 6 ) Shi ( a z ) ) z z 3 b z SinhIntegral a z 1 b 4 -1 3 ExpIntegralEi a b z -1 ExpIntegralEi b -1 a z -1 1 b -1 a 3 a b 3 -1 b b z b b 2 z 2 -1 b z 3 a 4 -1 2 b 2 b 2 z 2 -1 3 b z 3 a 2 b 4 b 2 z 2 -1 5 b z 11 a z -1 a b 2 z 2 -1 3 b z 6 a 4 -1 2 b 2 b 2 z 2 -1 5 b z 8 a 2 b 4 b 2 z 2 -1 7 b z 18 a z b z b 3 z 3 -1 3 b 2 z 2 6 b z -6 SinhIntegral a z [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", "3"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["b_", " ", "z_"]]], " ", RowBox[List["SinhIntegral", "[", RowBox[List["a_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["3", " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "b"]], ")"]], " ", "z"]], "]"]]]], "+", RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", "z"]], "]"]]]], ")"]]]], "-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["b", " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "a"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["b", "4"], " ", RowBox[List["(", RowBox[List["18", "-", RowBox[List["7", " ", "b", " ", "z"]], "+", RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox["z", "2"]]]]], ")"]]]], "-", RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["b", "2"], " ", RowBox[List["(", RowBox[List["8", "-", RowBox[List["5", " ", "b", " ", "z"]], "+", RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox["z", "2"]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["a", "4"], " ", RowBox[List["(", RowBox[List["6", "-", RowBox[List["3", " ", "b", " ", "z"]], "+", RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox["z", "2"]]]]], ")"]]]]]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["a", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["b", "4"], " ", RowBox[List["(", RowBox[List["11", "-", RowBox[List["5", " ", "b", " ", "z"]], "+", RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox["z", "2"]]]]], ")"]]]], "-", RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["b", "2"], " ", RowBox[List["(", RowBox[List["3", "-", RowBox[List["3", " ", "b", " ", "z"]], "+", RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox["z", "2"]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["a", "4"], " ", RowBox[List["(", RowBox[List["3", "-", RowBox[List["b", " ", "z"]], "+", RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox["z", "2"]]]]], ")"]]]]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List["a", " ", "z"]], "]"]]]]]], ")"]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "b"]], ")"]], "3"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], "3"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["b", " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "6"]], "+", RowBox[List["6", " ", "b", " ", "z"]], "-", RowBox[List["3", " ", SuperscriptBox["b", "2"], " ", SuperscriptBox["z", "2"]]], "+", RowBox[List[SuperscriptBox["b", "3"], " ", SuperscriptBox["z", "3"]]]]], ")"]], " ", RowBox[List["SinhIntegral", "[", RowBox[List["a", " ", "z"]], "]"]]]]]], SuperscriptBox["b", "4"]]]]]]

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

 2001-10-29