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 Cosh

 http://functions.wolfram.com/01.20.21.0350.01

 Input Form

 Integrate[z^(n + 1/2) E^(b z) Cosh[c z + d], z] == (-(1/2)) E^d z^(3/2 + n) (((-c - b) z)^(-(3/2) - n) (Erfc[Sqrt[(-c - b) z]] Gamma[3/2 + n] + E^((c + b) z) Sum[((-c - b) z)^(1/2 + k)/Pochhammer[3/2 + n, k - n], {k, 0, n}] - E^((c + b) z) Sum[((-c - b) z)^(1/2 + k)/Pochhammer[3/2 + n, k - n], {k, 1 + n, -1}]) + (((c - b) z)^(-(3/2) - n) (Erfc[Sqrt[(c - b) z]] Gamma[3/2 + n] + E^((-c + b) z) Sum[((c - b) z)^(1/2 + k)/Pochhammer[3/2 + n, k - n], {k, 0, n}] - E^((-c + b) z) Sum[((c - b) z)^(1/2 + k)/Pochhammer[3/2 + n, k - n], {k, 1 + n, -1}]))/E^(2 d)) /; Element[n, Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", RowBox[List["n", "+", FractionBox["1", "2"]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["b", " ", "z"]]], " ", RowBox[List["Cosh", "[", RowBox[List[RowBox[List["c", " ", "z"]], "+", "d"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", SuperscriptBox["\[ExponentialE]", "d"], " ", SuperscriptBox["z", RowBox[List[FractionBox["3", "2"], "+", "n"]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "-", "b"]], ")"]], " ", "z"]], ")"]], RowBox[List[RowBox[List["-", FractionBox["3", "2"]]], "-", "n"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Erfc", "[", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "-", "b"]], ")"]], " ", "z"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "+", "n"]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["c", "+", "b"]], ")"]], " ", "z"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "-", "b"]], ")"]], " ", "z"]], ")"]], RowBox[List[FractionBox["1", "2"], "+", "k"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "n"]], ",", RowBox[List["k", "-", "n"]]]], "]"]]]]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["c", "+", "b"]], ")"]], " ", "z"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List["1", "+", "n"]]]], RowBox[List["-", "1"]]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "-", "b"]], ")"]], " ", "z"]], ")"]], RowBox[List[FractionBox["1", "2"], "+", "k"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "n"]], ",", RowBox[List["k", "-", "n"]]]], "]"]]]]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "d"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["c", "-", "b"]], ")"]], " ", "z"]], ")"]], RowBox[List[RowBox[List["-", FractionBox["3", "2"]]], "-", "n"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Erfc", "[", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["c", "-", "b"]], ")"]], " ", "z"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "+", "n"]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", "b"]], ")"]], " ", "z"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["c", "-", "b"]], ")"]], " ", "z"]], ")"]], RowBox[List[FractionBox["1", "2"], "+", "k"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "n"]], ",", RowBox[List["k", "-", "n"]]]], "]"]]]]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", "b"]], ")"]], " ", "z"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List["1", "+", "n"]]]], RowBox[List["-", "1"]]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["c", "-", "b"]], ")"]], " ", "z"]], ")"]], RowBox[List[FractionBox["1", "2"], "+", "k"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "n"]], ",", RowBox[List["k", "-", "n"]]]], "]"]]]]]]]]], ")"]]]]]], ")"]]]]]], "/;", RowBox[List["n", "\[Element]", "Integers"]]]]]]

 MathML Form

 z n + 1 2 b z cosh ( d + c z ) z - 1 2 d z n + 3 2 ( ( erfc ( ( - b - c ) z ) Γ ( n + 3 2 ) + ( b + c ) z k = 0 n ( ( - b - c ) z ) k + 1 2 ( n + 3 2 ) k - n TagBox[SubscriptBox[RowBox[List["(", RowBox[List["n", "+", FractionBox["3", "2"]]], ")"]], RowBox[List["k", "-", "n"]]], Pochhammer] - ( b + c ) z k = n + 1 - 1 ( ( - b - c ) z ) k + 1 2 ( n + 3 2 ) k - n TagBox[SubscriptBox[RowBox[List["(", RowBox[List["n", "+", FractionBox["3", "2"]]], ")"]], RowBox[List["k", "-", "n"]]], Pochhammer] ) ( ( - b - c ) z ) - n - 3 2 + - 2 d ( ( c - b ) z ) - n - 3 2 ( erfc ( ( c - b ) z ) Γ ( n + 3 2 ) + ( b - c ) z k = 0 n ( ( c - b ) z ) k + 1 2 ( n + 3 2 ) k - n TagBox[SubscriptBox[RowBox[List["(", RowBox[List["n", "+", FractionBox["3", "2"]]], ")"]], RowBox[List["k", "-", "n"]]], Pochhammer] - ( b - c ) z k = n + 1 - 1 ( ( c - b ) z ) k + 1 2 ( n + 3 2 ) k - n TagBox[SubscriptBox[RowBox[List["(", RowBox[List["n", "+", FractionBox["3", "2"]]], ")"]], RowBox[List["k", "-", "n"]]], Pochhammer] ) ) /; n TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] Condition z z n 1 2 b z d c z -1 1 2 d z n 3 2 Erfc -1 b -1 c z 1 2 Gamma n 3 2 b c z k 0 n -1 b -1 c z k 1 2 Pochhammer n 3 2 k -1 n -1 -1 b c z k n 1 -1 -1 b -1 c z k 1 2 Pochhammer n 3 2 k -1 n -1 -1 b -1 c z -1 n -1 3 2 -2 d c -1 b z -1 n -1 3 2 Erfc c -1 b z 1 2 Gamma n 3 2 b -1 c z k 0 n c -1 b z k 1 2 Pochhammer n 3 2 k -1 n -1 -1 b -1 c z k n 1 -1 c -1 b z k 1 2 Pochhammer n 3 2 k -1 n -1 n [/itex]

 Rule Form

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

 2002-12-18