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 CoshIntegral

 http://functions.wolfram.com/06.40.21.0064.01

 Input Form

 Integrate[z^n ExpIntegralEi[b z] CoshIntegral[a z], z] == (1/(1 + n)) (CoshIntegral[a z] (z^(1 + n) ExpIntegralEi[b z] + (-b)^(-1 - n) Gamma[1 + n, (-b) z])) + (1/(2 a (1 + n))) (((-(-1)^n) ExpIntegralEi[(-a + b) z] n! + ExpIntegralEi[(a + b) z] n! - ExpIntegralEi[b z] Gamma[1 + n, (-a) z] + (-1)^n ExpIntegralEi[b z] Gamma[1 + n, a z] + (-1)^n n! Sum[(a^k Gamma[k, (a - b) z])/ ((a - b)^k k!), {k, 1, n}] - n! Sum[(a^k Gamma[k, (-(a + b)) z])/((a + b)^k k!), {k, 1, n}])/ (-a)^n) + (1/(2 b (n + 1))) ((n! Sum[(1/k!) (b^k ((-(b - a)^(-k)) Gamma[k, (a - b) z] - Gamma[k, (-(a + b)) z]/(a + b)^k)), {k, 0, n}])/(-b)^n) /; Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "n"], " ", RowBox[List["ExpIntegralEi", "[", RowBox[List["b", " ", "z"]], "]"]], " ", RowBox[List["CoshIntegral", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["1", RowBox[List["1", "+", "n"]]], RowBox[List["(", RowBox[List[RowBox[List["CoshIntegral", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["z", RowBox[List["1", "+", "n"]]], " ", RowBox[List["ExpIntegralEi", "[", RowBox[List["b", " ", "z"]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "b"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List[RowBox[List["-", "b"]], " ", "z"]]]], "]"]]]]]], ")"]]]], ")"]]]], "+", RowBox[List[FractionBox["1", RowBox[List["2", " ", "a", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "a"]], ")"]], RowBox[List["-", "n"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"]]], " ", RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "b"]], ")"]], " ", "z"]], "]"]], " ", RowBox[List["n", "!"]]]], "+", RowBox[List[RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", "z"]], "]"]], " ", RowBox[List["n", "!"]]]], "-", RowBox[List[RowBox[List["ExpIntegralEi", "[", RowBox[List["b", " ", "z"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List[RowBox[List["-", "a"]], " ", "z"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["ExpIntegralEi", "[", RowBox[List["b", " ", "z"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List["a", " ", "z"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["n", "!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], FractionBox[RowBox[List[SuperscriptBox["a", "k"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], RowBox[List["-", "k"]]], " ", RowBox[List["Gamma", "[", RowBox[List["k", ",", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", "z"]]]], "]"]]]], RowBox[List["k", "!"]]]]]]], "-", RowBox[List[RowBox[List["n", "!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], FractionBox[RowBox[List[SuperscriptBox["a", "k"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], RowBox[List["-", "k"]]], " ", RowBox[List["Gamma", "[", RowBox[List["k", ",", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]]]], " ", "z"]]]], "]"]]]], RowBox[List["k", "!"]]]]]]]]], ")"]]]], ")"]]]], "+", RowBox[List[FractionBox["1", RowBox[List["2", " ", "b", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]]]]], RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "b"]], ")"]], RowBox[List["-", "n"]]], " ", RowBox[List["n", "!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[FractionBox["1", RowBox[List["k", "!"]]], RowBox[List["(", RowBox[List[SuperscriptBox["b", "k"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "a"]], ")"]], RowBox[List["-", "k"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List["k", ",", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", "z"]]]], "]"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], RowBox[List["-", "k"]]], " ", RowBox[List["Gamma", "[", RowBox[List["k", ",", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]]]], " ", "z"]]]], "]"]]]]]], ")"]]]], ")"]]]]]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 z n Ei ( b z ) Chi ( a z ) z Chi ( a z ) n + 1 ( Γ ( n + 1 , - b z ) ( - b ) - n - 1 + z n + 1 Ei ( b z ) ) + 1 2 a ( n + 1 ) ( ( - a ) - n ( - ( - 1 ) n Ei ( ( b - a ) z ) n ! + Ei ( ( a + b ) z ) n ! + ( - 1 ) n ( k = 1 n a k ( a - b ) - k Γ ( k , ( a - b ) z ) k ! ) n ! - ( k = 1 n a k ( a + b ) - k Γ ( k , - ( a + b ) z ) k ! ) n ! - Ei ( b z ) Γ ( n + 1 , - a z ) + ( - 1 ) n Ei ( b z ) Γ ( n + 1 , a z ) ) ) + ( - b ) - n n ! 2 b ( n + 1 ) k = 0 n 1 k ! ( b k ( - ( b - a ) - k Γ ( k , ( a - b ) z ) - ( a + b ) - k Γ ( k , - ( a + b ) z ) ) ) /; n Condition z z n ExpIntegralEi b z CoshIntegral a z CoshIntegral a z n 1 -1 Gamma n 1 -1 b z -1 b -1 n -1 z n 1 ExpIntegralEi b z 1 2 a n 1 -1 -1 a -1 n -1 -1 n ExpIntegralEi b -1 a z n ExpIntegralEi a b z n -1 n k 1 n a k a -1 b -1 k Gamma k a -1 b z k -1 n -1 k 1 n a k a b -1 k Gamma k -1 a b z k -1 n -1 ExpIntegralEi b z Gamma n 1 -1 a z -1 n ExpIntegralEi b z Gamma n 1 a z -1 b -1 n n 2 b n 1 -1 k 0 n 1 k -1 b k -1 b -1 a -1 k Gamma k a -1 b z -1 a b -1 k Gamma k -1 a b z n [/itex]

 Rule Form

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

 2001-10-29