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 CoshIntegral

 http://functions.wolfram.com/06.40.21.0050.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

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

 2001-10-29