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 Cosh

 http://functions.wolfram.com/01.20.21.4640.01

 Input Form

 Integrate[E^(p z) Sinh[d + c z]^\[Mu] Cosh[a z]^v, z] == (Sinh[d + c z]^\[Mu] (-((1/(p - c \[Mu])) (E^(p z) Binomial[v, v/2] Hypergeometric2F1[(p - c \[Mu])/(2 c), -\[Mu], (1/2) (2 + p/c - \[Mu]), E^(2 (d + c z))] (-1 + Mod[v, 2]))) + Sum[Binomial[v, s] ((E^((p + 2 a s - a v) z) Hypergeometric2F1[ (p + 2 a s - a v - c \[Mu])/(2 c), -\[Mu], (2 c + p + 2 a s - a v - c \[Mu])/(2 c), E^(2 (d + c z))])/ (p + 2 a s - a v - c \[Mu]) + (E^((p + a (-2 s + v)) z) Hypergeometric2F1[(p - 2 a s + a v - c \[Mu])/(2 c), -\[Mu], (1/2) (2 + (p - 2 a s + a v)/c - \[Mu]), E^(2 (d + c z))])/ (p - 2 a s + a v - c \[Mu])), {s, 0, Floor[(1/2) (-1 + v)]}]))/ (2^v (1 - E^(2 (d + c z)))^\[Mu]) /; Element[v, Integers] && v > 0

 Standard Form

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 MathML Form

 p z sinh μ ( d + c z ) cosh v ( a z ) z 2 - v ( 1 - 2 ( d + c z ) ) - μ sinh μ ( d + c z ) ( s = 0 v - 1 2 ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( p + a ( v - 2 s ) ) z p - 2 a s + a v - c μ 2 F 1 ( p - 2 a s + a v - c μ 2 c , - μ ; 1 2 ( p - 2 a s + a v c - μ + 2 ) ; 2 ( d + c z ) ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["p", "-", RowBox[List["2", " ", "a", " ", "s"]], "+", RowBox[List["a", " ", "v"]], "-", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], ",", TagBox[RowBox[List["-", "\[Mu]"]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["p", "-", RowBox[List["2", " ", "a", " ", "s"]], "+", RowBox[List["a", " ", "v"]]]], "c"], "-", "\[Mu]", "+", "2"]], ")"]]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", RowBox[List["(", RowBox[List["d", "+", RowBox[List["c", " ", "z"]]]], ")"]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] + ( p + 2 a s - a v ) z p + 2 a s - a v - c μ 2 F 1 ( p + 2 a s - a v - c μ 2 c , - μ ; - μ c + 2 c + p + 2 a s - a v 2 c ; 2 ( d + c z ) ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["p", "+", RowBox[List["2", " ", "a", " ", "s"]], "-", RowBox[List["a", " ", "v"]], "-", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], ",", TagBox[RowBox[List["-", "\[Mu]"]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[Mu]"]], " ", "c"]], "+", RowBox[List["2", " ", "c"]], "+", "p", "+", RowBox[List["2", " ", "a", " ", "s"]], "-", RowBox[List["a", " ", "v"]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", RowBox[List["(", RowBox[List["d", "+", RowBox[List["c", " ", "z"]]]], ")"]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] ) - p z ( v mod 2 \$CellContext`v 2 - 1 ) p - c μ ( v v 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox[FractionBox["v", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] 2 F 1 ( p - c μ 2 c , - μ ; 1 2 ( p c - μ + 2 ) ; 2 ( d + c z ) ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["p", "-", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], ",", TagBox[RowBox[List["-", "\[Mu]"]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[FractionBox["p", "c"], "-", "\[Mu]", "+", "2"]], ")"]]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", RowBox[List["(", RowBox[List["d", "+", RowBox[List["c", " ", "z"]]]], ")"]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] ) /; v + Condition z p z d c z μ a z v 2 -1 v 1 -1 2 d c z -1 μ d c z μ s 0 v -1 2 -1 Binomial v s p a v -1 2 s z p -1 2 a s a v -1 c μ -1 Hypergeometric2F1 p -1 2 a s a v -1 c μ 2 c -1 -1 μ 1 2 p -1 2 a s a v c -1 -1 μ 2 2 d c z p 2 a s -1 a v z p 2 a s -1 a v -1 c μ -1 Hypergeometric2F1 p 2 a s -1 a v -1 c μ 2 c -1 -1 μ -1 μ c 2 c p 2 a s -1 a v 2 c -1 2 d c z -1 p z \$CellContext`v 2 -1 p -1 c μ -1 Binomial v v 2 -1 Hypergeometric2F1 p -1 c μ 2 c -1 -1 μ 1 2 p c -1 -1 μ 2 2 d c z v SuperPlus [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["p_", " ", "z_"]]], " ", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["d_", "+", RowBox[List["c_", " ", "z_"]]]], "]"]], "\[Mu]_"], " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["a_", " ", "z_"]], "]"]], "v_"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List["-", "v"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", RowBox[List["(", RowBox[List["d", "+", RowBox[List["c", " ", "z"]]]], ")"]]]]]]], ")"]], RowBox[List["-", "\[Mu]"]]], " ", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["d", "+", RowBox[List["c", " ", "z"]]]], "]"]], "\[Mu]"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["p", " ", "z"]]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", FractionBox["v", "2"]]], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["p", "-", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["-", "\[Mu]"]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", FractionBox["p", "c"], "-", "\[Mu]"]], ")"]]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", RowBox[List["(", RowBox[List["d", "+", RowBox[List["c", " ", "z"]]]], ")"]]]]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Mod", "[", RowBox[List["v", ",", "2"]], "]"]]]], ")"]]]], RowBox[List["p", "-", RowBox[List["c", " ", "\[Mu]"]]]]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "v"]], ")"]]]], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["v", ",", "s"]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["p", "+", RowBox[List["2", " ", "a", " ", "s"]], "-", RowBox[List["a", " ", "v"]]]], ")"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["p", "+", RowBox[List["2", " ", "a", " ", "s"]], "-", RowBox[List["a", " ", "v"]], "-", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["-", "\[Mu]"]], ",", FractionBox[RowBox[List[RowBox[List["2", " ", "c"]], "+", "p", "+", RowBox[List["2", " ", "a", " ", "s"]], "-", RowBox[List["a", " ", "v"]], "-", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", RowBox[List["(", RowBox[List["d", "+", RowBox[List["c", " ", "z"]]]], ")"]]]]]]], "]"]]]], RowBox[List["p", "+", RowBox[List["2", " ", "a", " ", "s"]], "-", RowBox[List["a", " ", "v"]], "-", RowBox[List["c", " ", "\[Mu]"]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["p", "+", RowBox[List["a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", "z"]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["p", "-", RowBox[List["2", " ", "a", " ", "s"]], "+", RowBox[List["a", " ", "v"]], "-", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["-", "\[Mu]"]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", FractionBox[RowBox[List["p", "-", RowBox[List["2", " ", "a", " ", "s"]], "+", RowBox[List["a", " ", "v"]]]], "c"], "-", "\[Mu]"]], ")"]]]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", RowBox[List["(", RowBox[List["d", "+", RowBox[List["c", " ", "z"]]]], ")"]]]]]]], "]"]]]], RowBox[List["p", "-", RowBox[List["2", " ", "a", " ", "s"]], "+", RowBox[List["a", " ", "v"]], "-", RowBox[List["c", " ", "\[Mu]"]]]]]]], ")"]]]]]]]], ")"]]]], "/;", RowBox[List[RowBox[List["v", "\[Element]", "Integers"]], "&&", RowBox[List["v", ">", "0"]]]]]]]]]]

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

 2002-12-18