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 SinhIntegral

 http://functions.wolfram.com/06.39.21.0066.01

 Input Form

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

 Standard Form

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

 MathML Form

 z 2 Ei ( b z ) Shi ( a z ) z 1 3 ( 1 2 b 2 ( b Γ ( 2 , ( a - b ) z ) ( a - b ) 2 + 2 ( b - a ) z b - a - 2 ( a + b ) z a + b - b Γ ( 2 , - ( a + b ) z ) ( a + b ) 2 ) + Ei ( ( a + b ) z ) - Ei ( ( b - a ) z ) b 3 - 1 2 a 3 ( Γ ( 2 , ( a - b ) z ) a 2 ( a - b ) 2 + Γ ( 2 , - ( a + b ) z ) a 2 ( a + b ) 2 + 2 ( b - a ) z a a - b + 2 ( a + b ) z a a + b - 2 Ei ( ( b - a ) z ) - 2 Ei ( ( a + b ) z ) + Ei ( b z ) Γ ( 3 , - a z ) + Ei ( b z ) Γ ( 3 , a z ) ) + ( z 3 Ei ( b z ) - Γ ( 3 , - b z ) b 3 ) Shi ( a z ) ) z z 2 ExpIntegralEi b z SinhIntegral a z 1 3 1 2 b 2 -1 b Gamma 2 a -1 b z a -1 b 2 -1 2 b -1 a z b -1 a -1 -1 2 a b z a b -1 -1 b Gamma 2 -1 a b z a b 2 -1 ExpIntegralEi a b z -1 ExpIntegralEi b -1 a z b 3 -1 -1 1 2 a 3 -1 Gamma 2 a -1 b z a 2 a -1 b 2 -1 Gamma 2 -1 a b z a 2 a b 2 -1 2 b -1 a z a a -1 b -1 2 a b z a a b -1 -1 2 ExpIntegralEi b -1 a z -1 2 ExpIntegralEi a b z ExpIntegralEi b z Gamma 3 -1 a z ExpIntegralEi b z Gamma 3 a z z 3 ExpIntegralEi b z -1 Gamma 3 -1 b z b 3 -1 SinhIntegral a z [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", "2"], " ", RowBox[List["ExpIntegralEi", "[", RowBox[List["b_", " ", "z_"]], "]"]], " ", RowBox[List["SinhIntegral", "[", RowBox[List["a_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "3"], " ", RowBox[List["(", RowBox[List[FractionBox[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"]], "]"]]]], SuperscriptBox["b", "3"]], "+", FractionBox[RowBox[List[FractionBox[RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "b"]], ")"]], " ", "z"]]]]], RowBox[List[RowBox[List["-", "a"]], "+", "b"]]], "-", FractionBox[RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", "z"]]]]], RowBox[List["a", "+", "b"]]], "+", FractionBox[RowBox[List["b", " ", RowBox[List["Gamma", "[", RowBox[List["2", ",", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", "z"]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], "2"]], "-", FractionBox[RowBox[List["b", " ", RowBox[List["Gamma", "[", RowBox[List["2", ",", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]]]], " ", "z"]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], "2"]]]], RowBox[List["2", " ", SuperscriptBox["b", "2"]]]], "-", FractionBox[RowBox[List[FractionBox[RowBox[List["2", " ", "a", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "b"]], ")"]], " ", "z"]]]]], RowBox[List["a", "-", "b"]]], "+", FractionBox[RowBox[List["2", " ", "a", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", "z"]]]]], RowBox[List["a", "+", "b"]]], "-", RowBox[List["2", " ", RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "b"]], ")"]], " ", "z"]], "]"]]]], "-", RowBox[List["2", " ", RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", "z"]], "]"]]]], "+", FractionBox[RowBox[List[SuperscriptBox["a", "2"], " ", RowBox[List["Gamma", "[", RowBox[List["2", ",", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", "z"]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], "2"]], "+", FractionBox[RowBox[List[SuperscriptBox["a", "2"], " ", RowBox[List["Gamma", "[", RowBox[List["2", ",", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]]]], " ", "z"]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], "2"]], "+", RowBox[List[RowBox[List["ExpIntegralEi", "[", RowBox[List["b", " ", "z"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["3", ",", RowBox[List[RowBox[List["-", "a"]], " ", "z"]]]], "]"]]]], "+", RowBox[List[RowBox[List["ExpIntegralEi", "[", RowBox[List["b", " ", "z"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["3", ",", RowBox[List["a", " ", "z"]]]], "]"]]]]]], RowBox[List["2", " ", SuperscriptBox["a", "3"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["z", "3"], " ", RowBox[List["ExpIntegralEi", "[", RowBox[List["b", " ", "z"]], "]"]]]], "-", FractionBox[RowBox[List["Gamma", "[", RowBox[List["3", ",", RowBox[List[RowBox[List["-", "b"]], " ", "z"]]]], "]"]], SuperscriptBox["b", "3"]]]], ")"]], " ", RowBox[List["SinhIntegral", "[", RowBox[List["a", " ", "z"]], "]"]]]]]], ")"]]]]]]]]

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

 2001-10-29