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 SinhIntegral

 http://functions.wolfram.com/06.39.21.0042.01

 Input Form

 Integrate[z^3 Cosh[b z] SinhIntegral[a z], z] == (1/(4 b^4)) ((1/((a - b)^3 (a + b)^3)) (6 (a^2 - b^2)^3 (ExpIntegralEi[(a - b) z] - ExpIntegralEi[(-a + b) z] - ExpIntegralEi[(-(a + b)) z] + ExpIntegralEi[(a + b) z]) + 4 b^2 Cosh[b z] (a (3 a^4 - 10 a^2 b^2 + 7 b^4) z Cosh[a z] + (3 a^4 - 6 a^2 b^2 + 11 b^4 + b^2 (a^2 - b^2)^2 z^2) Sinh[a z]) + 4 b ((-a) (2 (3 a^4 - 8 a^2 b^2 + 9 b^4) + b^2 (a^2 - b^2)^2 z^2) Cosh[a z] - b^2 (a^4 - 6 a^2 b^2 + 5 b^4) z Sinh[a z]) Sinh[b z]) + 4 (-3 (2 + b^2 z^2) Cosh[b z] + b z (6 + b^2 z^2) Sinh[b z]) SinhIntegral[a z])

 Standard Form

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 MathML Form

 z 3 cosh ( b z ) Shi ( a z ) z 1 4 b 4 ( 1 ( a - b ) 3 ( a + b ) 3 ( 6 ( Ei ( ( a - b ) z ) - Ei ( ( b - a ) z ) - Ei ( - ( a + b ) z ) + Ei ( ( a + b ) z ) ) ( a 2 - b 2 ) 3 + 4 b 2 cosh ( b z ) ( a ( 3 a 4 - 10 b 2 a 2 + 7 b 4 ) z cosh ( a z ) + ( 3 a 4 - 6 b 2 a 2 + 11 b 4 + b 2 ( a 2 - b 2 ) 2 z 2 ) sinh ( a z ) ) + 4 b ( - ( a 4 - 6 b 2 a 2 + 5 b 4 ) z sinh ( a z ) b 2 - a ( b 2 ( a 2 - b 2 ) 2 z 2 + 2 ( 3 a 4 - 8 b 2 a 2 + 9 b 4 ) ) cosh ( a z ) ) sinh ( b z ) ) + 4 ( b z ( b 2 z 2 + 6 ) sinh ( b z ) - 3 ( b 2 z 2 + 2 ) cosh ( b z ) ) Shi ( a z ) ) z z 3 b z SinhIntegral a z 1 4 b 4 -1 1 a -1 b 3 a b 3 -1 6 ExpIntegralEi a -1 b z -1 ExpIntegralEi b -1 a z -1 ExpIntegralEi -1 a b z ExpIntegralEi a b z a 2 -1 b 2 3 4 b 2 b z a 3 a 4 -1 10 b 2 a 2 7 b 4 z a z 3 a 4 -1 6 b 2 a 2 11 b 4 b 2 a 2 -1 b 2 2 z 2 a z 4 b -1 a 4 -1 6 b 2 a 2 5 b 4 z a z b 2 -1 a b 2 a 2 -1 b 2 2 z 2 2 3 a 4 -1 8 b 2 a 2 9 b 4 a z b z 4 b z b 2 z 2 6 b z -1 3 b 2 z 2 2 b z SinhIntegral a z [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", "3"], " ", RowBox[List["Cosh", "[", RowBox[List["b_", " ", "z_"]], "]"]], " ", RowBox[List["SinhIntegral", "[", RowBox[List["a_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[FractionBox[RowBox[List[RowBox[List["6", " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]], ")"]], "3"], " ", RowBox[List["(", RowBox[List[RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", "z"]], "]"]], "-", RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "b"]], ")"]], " ", "z"]], "]"]], "-", 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["4", " ", SuperscriptBox["b", "2"], " ", RowBox[List["Cosh", "[", RowBox[List["b", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["a", " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", SuperscriptBox["a", "4"]]], "-", RowBox[List["10", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["b", "2"]]], "+", RowBox[List["7", " ", SuperscriptBox["b", "4"]]]]], ")"]], " ", "z", " ", RowBox[List["Cosh", "[", RowBox[List["a", " ", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["3", " ", SuperscriptBox["a", "4"]]], "-", RowBox[List["6", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["b", "2"]]], "+", RowBox[List["11", " ", SuperscriptBox["b", "4"]]], "+", RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]], ")"]], "2"], " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List["a", " ", "z"]], "]"]]]]]], ")"]]]], "+", RowBox[List["4", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "a"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", SuperscriptBox["a", "4"]]], "-", RowBox[List["8", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["b", "2"]]], "+", RowBox[List["9", " ", SuperscriptBox["b", "4"]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]], ")"]], "2"], " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["a", " ", "z"]], "]"]]]], "-", RowBox[List[SuperscriptBox["b", "2"], " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "4"], "-", RowBox[List["6", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["b", "2"]]], "+", RowBox[List["5", " ", SuperscriptBox["b", "4"]]]]], ")"]], " ", "z", " ", RowBox[List["Sinh", "[", RowBox[List["a", " ", "z"]], "]"]]]]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List["b", " ", "z"]], "]"]]]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], "3"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], "3"]]]], "+", RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "3"]], " ", RowBox[List["(", RowBox[List["2", "+", RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["b", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", "z", " ", RowBox[List["(", RowBox[List["6", "+", RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List["b", " ", "z"]], "]"]]]]]], ")"]], " ", RowBox[List["SinhIntegral", "[", RowBox[List["a", " ", "z"]], "]"]]]]]], RowBox[List["4", " ", SuperscriptBox["b", "4"]]]]]]]]

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

 2001-10-29