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 Cosh

 http://functions.wolfram.com/01.20.21.0062.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

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

 2002-12-18