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 Cosh

 http://functions.wolfram.com/01.20.21.0113.01

 Input Form

 Integrate[z^(2 n) Cosh[a z^2 + b], z] == (-(1/4)) z ((1/Sqrt[(-a) z^2]) ((E^b (Erfc[Sqrt[(-a) z^2]] Gamma[1/2 + n] + E^(a z^2) Sum[((-a) z^2)^(1/2 + k)/Pochhammer[1/2 + n, 1 + k - n], {k, 0, -1 + n}] - E^(a z^2) Sum[((-a) z^2)^(1/2 + k)/ Pochhammer[1/2 + n, 1 + k - n], {k, n, -1}]))/(-a)^n) + (1/Sqrt[a z^2]) ((Erfc[Sqrt[a z^2]] Gamma[1/2 + n] + Sum[(a z^2)^(1/2 + k)/Pochhammer[1/2 + n, 1 + k - n], {k, 0, -1 + n}]/ E^(a z^2) - Sum[(a z^2)^(1/2 + k)/Pochhammer[1/2 + n, 1 + k - n], {k, n, -1}]/E^(a z^2))/(a^n E^b))) /; Element[n, Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", RowBox[List["2", "n"]]], " ", RowBox[List["Cosh", "[", RowBox[List[RowBox[List["a", " ", SuperscriptBox["z", "2"]]], "+", "b"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], "z", " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", SqrtBox[RowBox[List[RowBox[List["-", "a"]], " ", SuperscriptBox["z", "2"]]]]], RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "a"]], ")"]], RowBox[List["-", "n"]]], " ", SuperscriptBox["\[ExponentialE]", "b"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Erfc", "[", SqrtBox[RowBox[List[RowBox[List["-", "a"]], " ", SuperscriptBox["z", "2"]]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "n"]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["a", " ", SuperscriptBox["z", "2"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List[FractionBox["1", "2"], "+", "k"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "n"]], ",", RowBox[List["1", "+", "k", "-", "n"]]]], "]"]]]]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["a", " ", SuperscriptBox["z", "2"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "n"]], RowBox[List["-", "1"]]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List[FractionBox["1", "2"], "+", "k"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "n"]], ",", RowBox[List["1", "+", "k", "-", "n"]]]], "]"]]]]]]]]], ")"]]]], ")"]]]], "+", RowBox[List[FractionBox["1", SqrtBox[RowBox[List["a", " ", SuperscriptBox["z", "2"]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["a", RowBox[List["-", "n"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", "b"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Erfc", "[", SqrtBox[RowBox[List["a", " ", SuperscriptBox["z", "2"]]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "n"]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "a"]], " ", SuperscriptBox["z", "2"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List[FractionBox["1", "2"], "+", "k"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "n"]], ",", RowBox[List["1", "+", "k", "-", "n"]]]], "]"]]]]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "a"]], " ", SuperscriptBox["z", "2"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "n"]], RowBox[List["-", "1"]]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List[FractionBox["1", "2"], "+", "k"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "n"]], ",", RowBox[List["1", "+", "k", "-", "n"]]]], "]"]]]]]]]]], ")"]]]], ")"]]]]]], ")"]]]]]], "/;", RowBox[List["n", "\[Element]", "Integers"]]]]]]

 MathML Form

 z 2 n cosh ( a z 2 + b ) z - 1 4 z ( ( - a ) - n b - a z 2 ( erfc ( - a z 2 ) Γ ( n + 1 2 ) + a z 2 k = 0 n - 1 ( - a z 2 ) k + 1 2 ( n + 1 2 ) k - n + 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["n", "+", FractionBox["1", "2"]]], ")"]], RowBox[List["k", "-", "n", "+", "1"]]], Pochhammer] - a z 2 k = n - 1 ( - a z 2 ) k + 1 2 ( n + 1 2 ) k - n + 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["n", "+", FractionBox["1", "2"]]], ")"]], RowBox[List["k", "-", "n", "+", "1"]]], Pochhammer] ) + a - n - b a z 2 ( erfc ( a z 2 ) Γ ( n + 1 2 ) + - a z 2 k = 0 n - 1 ( a z 2 ) k + 1 2 ( n + 1 2 ) k - n + 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["n", "+", FractionBox["1", "2"]]], ")"]], RowBox[List["k", "-", "n", "+", "1"]]], Pochhammer] - - a z 2 k = n - 1 ( a z 2 ) k + 1 2 ( n + 1 2 ) k - n + 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["n", "+", FractionBox["1", "2"]]], ")"]], RowBox[List["k", "-", "n", "+", "1"]]], Pochhammer] ) ) /; n TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] Condition z z 2 n a z 2 b -1 1 4 z -1 a -1 n b -1 a z 2 1 2 -1 Erfc -1 a z 2 1 2 Gamma n 1 2 a z 2 k 0 n -1 -1 a z 2 k 1 2 Pochhammer n 1 2 k -1 n 1 -1 -1 a z 2 k n -1 -1 a z 2 k 1 2 Pochhammer n 1 2 k -1 n 1 -1 a -1 n -1 b a z 2 1 2 -1 Erfc a z 2 1 2 Gamma n 1 2 -1 a z 2 k 0 n -1 a z 2 k 1 2 Pochhammer n 1 2 k -1 n 1 -1 -1 -1 a z 2 k n -1 a z 2 k 1 2 Pochhammer n 1 2 k -1 n 1 -1 n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", RowBox[List["2", " ", "n_"]]], " ", RowBox[List["Cosh", "[", RowBox[List[RowBox[List["a_", " ", SuperscriptBox["z_", "2"]]], "+", "b_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", "z", " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "a"]], ")"]], RowBox[List["-", "n"]]], " ", SuperscriptBox["\[ExponentialE]", "b"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Erfc", "[", SqrtBox[RowBox[List[RowBox[List["-", "a"]], " ", SuperscriptBox["z", "2"]]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "n"]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["a", " ", SuperscriptBox["z", "2"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List[FractionBox["1", "2"], "+", "k"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "n"]], ",", RowBox[List["1", "+", "k", "-", "n"]]]], "]"]]]]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["a", " ", SuperscriptBox["z", "2"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "n"]], RowBox[List["-", "1"]]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List[FractionBox["1", "2"], "+", "k"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "n"]], ",", RowBox[List["1", "+", "k", "-", "n"]]]], "]"]]]]]]]]], ")"]]]], SqrtBox[RowBox[List[RowBox[List["-", "a"]], " ", SuperscriptBox["z", "2"]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["a", RowBox[List["-", "n"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", "b"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Erfc", "[", SqrtBox[RowBox[List["a", " ", SuperscriptBox["z", "2"]]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "n"]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "a"]], " ", SuperscriptBox["z", "2"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List[FractionBox["1", "2"], "+", "k"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "n"]], ",", RowBox[List["1", "+", "k", "-", "n"]]]], "]"]]]]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "a"]], " ", SuperscriptBox["z", "2"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "n"]], RowBox[List["-", "1"]]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List[FractionBox["1", "2"], "+", "k"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "n"]], ",", RowBox[List["1", "+", "k", "-", "n"]]]], "]"]]]]]]]]], ")"]]]], SqrtBox[RowBox[List["a", " ", SuperscriptBox["z", "2"]]]]]]], ")"]]]], "/;", RowBox[List["n", "\[Element]", "Integers"]]]]]]]]

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

 2002-12-18