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 Cosh

 http://functions.wolfram.com/01.20.21.0099.01

 Input Form

 Integrate[z^n Cosh[a Sqrt[z]], z] == (ExpIntegralEi[(-a) Sqrt[z]]/(-2 - 2 n)! + ExpIntegralEi[a Sqrt[z]]/ (-2 - 2 n)! - E^(a Sqrt[z]) Sum[((-a) Sqrt[z])^k/ Pochhammer[2 + 2 n, -1 + k - 2 n], {k, 0, 1 + 2 n}] + E^(a Sqrt[z]) Sum[((-a) Sqrt[z])^k/Pochhammer[2 + 2 n, -1 + k - 2 n], {k, 2 + 2 n, -1}] - Sum[(a Sqrt[z])^k/Pochhammer[2 + 2 n, -1 + k - 2 n], {k, 0, 1 + 2 n}]/ E^(a Sqrt[z]) + Sum[(a Sqrt[z])^k/Pochhammer[2 + 2 n, -1 + k - 2 n], {k, 2 + 2 n, -1}]/E^(a Sqrt[z]))/a^(2 (1 + n)) /; Element[n, Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "n"], RowBox[List["Cosh", "[", RowBox[List["a", " ", SqrtBox["z"]]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[SuperscriptBox["a", RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List["-", "a"]], " ", SqrtBox["z"]]], "]"]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "-", RowBox[List["2", " ", "n"]]]], ")"]], "!"]]], "+", FractionBox[RowBox[List["ExpIntegralEi", "[", RowBox[List["a", " ", SqrtBox["z"]]], "]"]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "-", RowBox[List["2", " ", "n"]]]], ")"]], "!"]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["a", " ", SqrtBox["z"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["1", "+", RowBox[List["2", "n"]]]]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], " ", SqrtBox["z"]]], ")"]], "k"], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["2", "+", RowBox[List["2", " ", "n"]]]], ",", RowBox[List[RowBox[List["-", "1"]], "+", "k", "-", RowBox[List["2", " ", "n"]]]]]], "]"]]]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["a", " ", SqrtBox["z"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List["2", "+", RowBox[List["2", "n"]]]]]], RowBox[List["-", "1"]]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], " ", SqrtBox["z"]]], ")"]], "k"], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["2", "+", RowBox[List["2", " ", "n"]]]], ",", RowBox[List[RowBox[List["-", "1"]], "+", "k", "-", RowBox[List["2", " ", "n"]]]]]], "]"]]]]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "a"]], " ", SqrtBox["z"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["1", "+", RowBox[List["2", "n"]]]]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SqrtBox["z"]]], ")"]], "k"], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["2", "+", RowBox[List["2", " ", "n"]]]], ",", RowBox[List[RowBox[List["-", "1"]], "+", "k", "-", RowBox[List["2", " ", "n"]]]]]], "]"]]]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "a"]], " ", SqrtBox["z"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List["2", "+", RowBox[List["2", "n"]]]]]], RowBox[List["-", "1"]]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SqrtBox["z"]]], ")"]], "k"], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["2", "+", RowBox[List["2", " ", "n"]]]], ",", RowBox[List[RowBox[List["-", "1"]], "+", "k", "-", RowBox[List["2", " ", "n"]]]]]], "]"]]]]]]]]], ")"]]]]]], "/;", RowBox[List["n", "\[Element]", "Integers"]]]]]]

 MathML Form

 z n cosh ( a z ) z a - 2 ( n + 1 ) ( Ei ( - a z ) ( - 2 n - 2 ) ! - a z k = 0 2 n + 1 ( - a z ) k ( 2 n + 2 ) k - 2 n - 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "+", "2"]], ")"]], RowBox[List["k", "-", RowBox[List["2", " ", "n"]], "-", "1"]]], Pochhammer] + a z k = 2 n + 2 - 1 ( - a z ) k ( 2 n + 2 ) k - 2 n - 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "+", "2"]], ")"]], RowBox[List["k", "-", RowBox[List["2", " ", "n"]], "-", "1"]]], Pochhammer] - - a z k = 0 2 n + 1 ( a z ) k ( 2 n + 2 ) k - 2 n - 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "+", "2"]], ")"]], RowBox[List["k", "-", RowBox[List["2", " ", "n"]], "-", "1"]]], Pochhammer] + - a z k = 2 n + 2 - 1 ( a z ) k ( 2 n + 2 ) k - 2 n - 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "+", "2"]], ")"]], RowBox[List["k", "-", RowBox[List["2", " ", "n"]], "-", "1"]]], Pochhammer] + Ei ( a z ) ( - 2 n - 2 ) ! ) /; n TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] Condition z z n a z 1 2 a -2 n 1 ExpIntegralEi -1 a z 1 2 -2 n -2 -1 -1 a z 1 2 k 0 2 n 1 -1 a z 1 2 k Pochhammer 2 n 2 k -1 2 n -1 -1 a z 1 2 k 2 n 2 -1 -1 a z 1 2 k Pochhammer 2 n 2 k -1 2 n -1 -1 -1 -1 a z 1 2 k 0 2 n 1 a z 1 2 k Pochhammer 2 n 2 k -1 2 n -1 -1 -1 a z 1 2 k 2 n 2 -1 a z 1 2 k Pochhammer 2 n 2 k -1 2 n -1 -1 ExpIntegralEi a z 1 2 -2 n -2 -1 n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", "n_"], " ", RowBox[List["Cosh", "[", RowBox[List["a_", " ", SqrtBox["z_"]]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["a", RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List["-", "a"]], " ", SqrtBox["z"]]], "]"]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "-", RowBox[List["2", " ", "n"]]]], ")"]], "!"]]], "+", FractionBox[RowBox[List["ExpIntegralEi", "[", RowBox[List["a", " ", SqrtBox["z"]]], "]"]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "-", RowBox[List["2", " ", "n"]]]], ")"]], "!"]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["a", " ", SqrtBox["z"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["1", "+", RowBox[List["2", " ", "n"]]]]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], " ", SqrtBox["z"]]], ")"]], "k"], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["2", "+", RowBox[List["2", " ", "n"]]]], ",", RowBox[List[RowBox[List["-", "1"]], "+", "k", "-", RowBox[List["2", " ", "n"]]]]]], "]"]]]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["a", " ", SqrtBox["z"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List["2", "+", RowBox[List["2", " ", "n"]]]]]], RowBox[List["-", "1"]]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], " ", SqrtBox["z"]]], ")"]], "k"], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["2", "+", RowBox[List["2", " ", "n"]]]], ",", RowBox[List[RowBox[List["-", "1"]], "+", "k", "-", RowBox[List["2", " ", "n"]]]]]], "]"]]]]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "a"]], " ", SqrtBox["z"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["1", "+", RowBox[List["2", " ", "n"]]]]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SqrtBox["z"]]], ")"]], "k"], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["2", "+", RowBox[List["2", " ", "n"]]]], ",", RowBox[List[RowBox[List["-", "1"]], "+", "k", "-", RowBox[List["2", " ", "n"]]]]]], "]"]]]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "a"]], " ", SqrtBox["z"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List["2", "+", RowBox[List["2", " ", "n"]]]]]], RowBox[List["-", "1"]]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SqrtBox["z"]]], ")"]], "k"], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["2", "+", RowBox[List["2", " ", "n"]]]], ",", RowBox[List[RowBox[List["-", "1"]], "+", "k", "-", RowBox[List["2", " ", "n"]]]]]], "]"]]]]]]]]], ")"]]]], "/;", RowBox[List["n", "\[Element]", "Integers"]]]]]]]]

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

 2002-12-18