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 Cosh

 http://functions.wolfram.com/01.20.21.3765.01

 Input Form

 Integrate[E^(p z) Sin[c z]^m Cosh[b + a z]^\[Nu], z] == (1/(p - a \[Nu])) ((E^(p z) Binomial[m, m/2] Hypergeometric2F1[ (p - a \[Nu])/(2 a), -\[Nu], (1/2) (2 + p/a - \[Nu]), -E^(2 (b + a z))] (1 - Mod[m, 2]) Cosh[b + a z]^\[Nu])/ (2^m (1 + E^(2 (b + a z)))^\[Nu])) + (Cosh[b + a z]^\[Nu] Sum[(-1)^k Binomial[m, k] ((E^(I m Pi + ((-I) c (-2 k + m) + p) z) Hypergeometric2F1[ ((-I) c (-2 k + m) + p - a \[Nu])/(2 a), -\[Nu], (1/2) (2 + ((-I) c (-2 k + m) + p)/a - \[Nu]), -E^(2 (b + a z))])/ ((-I) c (-2 k + m) + p - a \[Nu]) + (E^((I c (-2 k + m) + p) z) Hypergeometric2F1[(I c (-2 k + m) + p - a \[Nu])/(2 a), -\[Nu], (1/2) (2 + (I c (-2 k + m) + p)/a - \[Nu]), -E^(2 (b + a z))])/ (I c (-2 k + m) + p - a \[Nu])), {k, 0, Floor[(1/2) (-1 + m)]}])/ (2^m I^m (1 + E^(2 (b + a z)))^\[Nu]) /; Element[m, Integers] && m > 0

 Standard Form

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 MathML Form

 p z sin m ( c z ) cosh ν ( b + a z ) z 2 - m p z ( 1 + 2 ( b + a z ) ) - ν ( 1 - m mod 2 \$CellContext`m 2 ) cosh ν ( b + a z ) p - a ν ( m m 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox[FractionBox["m", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] 2 F 1 ( p - a ν 2 a , - ν ; 1 2 ( p a - ν + 2 ) ; - 2 ( b + a 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["a", " ", "\[Nu]"]]]], RowBox[List["2", " ", "a"]]], Hypergeometric2F1], ",", TagBox[RowBox[List["-", "\[Nu]"]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[FractionBox["p", "a"], "-", "\[Nu]", "+", "2"]], ")"]]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], ")"]]]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] + 2 - m - m ( 1 + 2 ( b + a z ) ) - ν cosh ν ( b + a z ) k = 0 m - 1 2 ( - 1 ) k ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( π m + ( p - c ( m - 2 k ) ) z - c ( m - 2 k ) + p - a ν 2 F 1 ( - c ( m - 2 k ) + p - a ν 2 a , - ν ; 1 2 ( p - c ( m - 2 k ) a - ν + 2 ) ; - 2 ( b + a 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[RowBox[List["-", "\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", "p", "-", RowBox[List["a", " ", "\[Nu]"]]]], RowBox[List["2", " ", "a"]]], Hypergeometric2F1], ",", TagBox[RowBox[List["-", "\[Nu]"]], 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["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]]]], "a"], "-", "\[Nu]", "+", "2"]], ")"]]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], ")"]]]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] + ( c ( m - 2 k ) + p ) z c ( m - 2 k ) + p - a ν 2 F 1 ( c ( m - 2 k ) + p - a ν 2 a , - ν ; 1 2 ( c ( m - 2 k ) + p a - ν + 2 ) ; - 2 ( b + a 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[RowBox[List["c", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", "p", "-", RowBox[List["a", " ", "\[Nu]"]]]], RowBox[List["2", " ", "a"]]], Hypergeometric2F1], ",", TagBox[RowBox[List["-", "\[Nu]"]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["c", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", "p"]], "a"], "-", "\[Nu]", "+", "2"]], ")"]]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], ")"]]]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] ) /; m + Condition z p z c z m b a z ν 2 -1 m p z 1 2 b a z -1 ν 1 -1 \$CellContext`m 2 b a z ν p -1 a ν -1 Binomial m m 2 -1 Hypergeometric2F1 p -1 a ν 2 a -1 -1 ν 1 2 p a -1 -1 ν 2 -1 2 b a z 2 -1 m -1 m 1 2 b a z -1 ν b a z ν k 0 m -1 2 -1 -1 k Binomial m k m p -1 c m -1 2 k z -1 c m -1 2 k p -1 a ν -1 Hypergeometric2F1 -1 c m -1 2 k p -1 a ν 2 a -1 -1 ν 1 2 p -1 c m -1 2 k a -1 -1 ν 2 -1 2 b a z c m -1 2 k p z c m -1 2 k p -1 a ν -1 Hypergeometric2F1 c m -1 2 k p -1 a ν 2 a -1 -1 ν 1 2 c m -1 2 k p a -1 -1 ν 2 -1 2 b a z m 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["Sin", "[", RowBox[List["c_", " ", "z_"]], "]"]], "m_"], " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["b_", "+", RowBox[List["a_", " ", "z_"]]]], "]"]], "\[Nu]_"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["-", "m"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["p", " ", "z"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], ")"]]]]]]], ")"]], RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", 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 Date Added to functions.wolfram.com (modification date)

 2002-12-18