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 SinhIntegral

 http://functions.wolfram.com/06.39.21.0041.01

 Input Form

 Integrate[z^2 Cosh[b z] SinhIntegral[a z], z] == (1/(2 b^3)) (ExpIntegralEi[(a - b) z] + ExpIntegralEi[(-a + b) z] + (1/((a - b)^2 (a + b)^2)) ((-(a^2 - b^2)^2) (ExpIntegralEi[(-(a + b)) z] + ExpIntegralEi[(a + b) z]) + 2 b Cosh[b z] (2 a (a^2 - 2 b^2) Cosh[a z] + (a - b) b^2 (a + b) z Sinh[a z]) + 2 b^2 (a (-a^2 + b^2) z Cosh[a z] - (a^2 - 3 b^2) Sinh[a z]) Sinh[b z]) + 2 (-2 b z Cosh[b z] + (2 + b^2 z^2) Sinh[b z]) SinhIntegral[a z])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "2"], " ", RowBox[List["Cosh", "[", RowBox[List["b", " ", "z"]], "]"]], RowBox[List["SinhIntegral", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["2", " ", SuperscriptBox["b", "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[FractionBox["1", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], "2"]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]], ")"]], "2"]]], " ", 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["2", " ", "b", " ", RowBox[List["Cosh", "[", RowBox[List["b", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a", " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", RowBox[List["2", " ", SuperscriptBox["b", "2"]]]]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["a", " ", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", SuperscriptBox["b", "2"], " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", "z", " ", RowBox[List["Sinh", "[", RowBox[List["a", " ", "z"]], "]"]]]]]], ")"]]]], "+", RowBox[List["2", " ", SuperscriptBox["b", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["a", " ", RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]], ")"]], " ", "z", " ", RowBox[List["Cosh", "[", RowBox[List["a", " ", "z"]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", RowBox[List["3", " ", SuperscriptBox["b", "2"]]]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List["a", " ", "z"]], "]"]]]]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List["b", " ", "z"]], "]"]]]]]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "b", " ", "z", " ", RowBox[List["Cosh", "[", RowBox[List["b", " ", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["2", "+", RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List["b", " ", "z"]], "]"]]]]]], ")"]], " ", RowBox[List["SinhIntegral", "[", RowBox[List["a", " ", "z"]], "]"]]]]]], ")"]]]]]]]]

 MathML Form

 z 2 cosh ( b z ) Shi ( a z ) z 1 2 b 3 ( Ei ( ( a - b ) z ) + Ei ( ( b - a ) z ) + 1 ( a - b ) 2 ( a + b ) 2 ( 2 ( a ( b 2 - a 2 ) z cosh ( a z ) - ( a 2 - 3 b 2 ) sinh ( a z ) ) sinh ( b z ) b 2 + 2 cosh ( b z ) ( ( a - b ) ( a + b ) z sinh ( a z ) b 2 + 2 a ( a 2 - 2 b 2 ) cosh ( a z ) ) b - ( a 2 - b 2 ) 2 ( Ei ( - ( a + b ) z ) + Ei ( ( a + b ) z ) ) ) + 2 ( ( b 2 z 2 + 2 ) sinh ( b z ) - 2 b z cosh ( b z ) ) Shi ( a z ) ) z z 2 b z SinhIntegral a z 1 2 b 3 -1 ExpIntegralEi a -1 b z ExpIntegralEi b -1 a z 1 a -1 b 2 a b 2 -1 2 a b 2 -1 a 2 z a z -1 a 2 -1 3 b 2 a z b z b 2 2 b z a -1 b a b z a z b 2 2 a a 2 -1 2 b 2 a z b -1 a 2 -1 b 2 2 ExpIntegralEi -1 a b z ExpIntegralEi a b z 2 b 2 z 2 2 b z -1 2 b z b z SinhIntegral a z [/itex]

 Rule Form

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

 2001-10-29