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 Csch

 http://functions.wolfram.com/01.23.21.0211.01

 Input Form

 Integrate[Cos[a z]^m Cosh[b z]^u Csch[c z], z] == (-((2^(1 - m - u) E^(c z) Binomial[m, m/2] Binomial[u, u/2])/c)) HypergeometricPFQ[{1/2, 1}, {3/2}, E^(2 c z)] (1 - Mod[m, 2]) (1 - Mod[u, 2]) - 2^(1 - m - u) E^(c z) Binomial[u, u/2] (1 - Mod[u, 2]) Sum[Binomial[m, s] ((1/(I a (m - 2 s) + c)) (E^(I a (m - 2 s) z) HypergeometricPFQ[{(I a m)/(2 c) - (I a s)/c + 1/2, 1}, {(I a m)/(2 c) - (I a s)/c + 3/2}, E^(2 c z)]) + HypergeometricPFQ[{-((I a m)/(2 c)) + (I a s)/c + 1/2, 1}, {-((I a m)/(2 c)) + (I a s)/c + 3/2}, E^(2 c z)]/ E^(I a (m - 2 s) z)/((-I) a (m - 2 s) + c)), {s, 0, Floor[(1/2) (-1 + m)]}] - 2^(1 - m - u) E^(c z) Binomial[m, m/2] (1 - Mod[m, 2]) Sum[Binomial[u, k] (HypergeometricPFQ[{(b k)/c - (b u)/(2 c) + 1/2, 1}, {(b k)/c - (b u)/(2 c) + 3/2}, E^(2 c z)]/E^(b (-2 k + u) z)/ ((-b) (-2 k + u) + c) + (1/(b (-2 k + u) + c)) (E^(b (-2 k + u) z) HypergeometricPFQ[{-((b k)/c) + (b u)/(2 c) + 1/2, 1}, {-((b k)/c) + (b u)/(2 c) + 3/2}, E^(2 c z)])), {k, 0, Floor[(1/2) (-1 + u)]}] - 2^(1 - m - u) E^(c z) Sum[Binomial[m, s] Binomial[u, k] ((E^((I a (m - 2 s) - b (-2 k + u)) z) HypergeometricPFQ[ {(b k)/c + (I a m)/(2 c) - (I a s)/c - (b u)/(2 c) + 1/2, 1}, {(b k)/c + (I a m)/(2 c) - (I a s)/c - (b u)/(2 c) + 3/2}, E^(2 c z)])/(I a (m - 2 s) - b (-2 k + u) + c) + (E^(((-I) a (m - 2 s) - b (-2 k + u)) z) HypergeometricPFQ[ {(b k)/c - (I a m)/(2 c) + (I a s)/c - (b u)/(2 c) + 1/2, 1}, {(b k)/c - (I a m)/(2 c) + (I a s)/c - (b u)/(2 c) + 3/2}, E^(2 c z)])/((-I) a (m - 2 s) - b (-2 k + u) + c) + (E^((I a (m - 2 s) + b (-2 k + u)) z) HypergeometricPFQ[ {-((b k)/c) + (I a m)/(2 c) - (I a s)/c + (b u)/(2 c) + 1/2, 1}, {-((b k)/c) + (I a m)/(2 c) - (I a s)/c + (b u)/(2 c) + 3/2}, E^(2 c z)])/(I a (m - 2 s) + b (-2 k + u) + c) + (E^(((-I) a (m - 2 s) + b (-2 k + u)) z) HypergeometricPFQ[ {-((b k)/c) - (I a m)/(2 c) + (I a s)/c + (b u)/(2 c) + 1/2, 1}, {-((b k)/c) - (I a m)/(2 c) + (I a s)/c + (b u)/(2 c) + 3/2}, E^(2 c z)])/((-I) a (m - 2 s) + b (-2 k + u) + c)), {k, 0, Floor[(1/2) (-1 + u)]}, {s, 0, Floor[(1/2) (-1 + m)]}] /; Element[m, Integers] && m > 0 && Element[u, Integers] && u > 0

 Standard Form

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 MathML Form

 cos m ( a z ) cosh u ( b z ) csch ( c z ) z - 2 - m - u + 1 c z ( 1 - m mod 2 \$CellContext`m 2 ) ( 1 - u mod 2 \$CellContext`u 2 ) c ( 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]]]]] ( u u 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["u", Identity]], List[TagBox[FractionBox["u", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] 2 F 1 ( 1 2 , 1 ; 3 2 ; 2 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["1", "2"], HypergeometricPFQ], ",", TagBox["1", HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[TagBox[FractionBox["3", "2"], HypergeometricPFQ], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] - 2 - m - u + 1 c z ( u u 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["u", Identity]], List[TagBox[FractionBox["u", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 1 - u mod 2 \$CellContext`u 2 ) s = 0 m - 1 2 ( m s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( - a ( m - 2 s ) z 2 F 1 ( - a m 2 c + a s c + 1 2 , 1 ; - a m 2 c + a s c + 3 2 ; 2 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[RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"], "+", FractionBox["1", "2"]]], HypergeometricPFQ], ",", TagBox["1", HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[TagBox[RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"], "+", FractionBox["3", "2"]]], HypergeometricPFQ], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] c - a ( m - 2 s ) + a ( m - 2 s ) z 2 F 1 ( a m 2 c + 1 2 - a s c , 1 ; a m 2 c + 3 2 - a s c ; 2 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[RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "+", FractionBox["1", "2"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"]]], HypergeometricPFQ], ",", TagBox["1", HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "+", FractionBox["3", "2"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"]]], HypergeometricPFQ], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] c + a ( m - 2 s ) ) - 2 - m - u + 1 c z ( 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]]]]] ( 1 - m mod 2 \$CellContext`m 2 ) k = 0 u - 1 2 ( u k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["u", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( b ( u - 2 k ) z 2 F 1 ( - b k c + b u 2 c + 1 2 , 1 ; - b k c + b u 2 c + 3 2 ; 2 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[RowBox[List[RowBox[List["-", FractionBox[RowBox[List["b", " ", "k"]], "c"]]], "+", FractionBox[RowBox[List["b", " ", "u"]], RowBox[List["2", " ", "c"]]], "+", FractionBox["1", "2"]]], HypergeometricPFQ], ",", TagBox["1", HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[TagBox[RowBox[List[RowBox[List["-", FractionBox[RowBox[List["b", " ", "k"]], "c"]]], "+", FractionBox[RowBox[List["b", " ", "u"]], RowBox[List["2", " ", "c"]]], "+", FractionBox["3", "2"]]], HypergeometricPFQ], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] c + b ( u - 2 k ) + - b ( u - 2 k ) z 2 F 1 ( b k c + 1 2 - b u 2 c , 1 ; 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b k c + a s c + 3 2 - a m 2 c - b u 2 c ; 2 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[RowBox[List[FractionBox[RowBox[List["b", " ", "k"]], "c"], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"], "+", FractionBox["1", "2"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "-", FractionBox[RowBox[List["b", " ", "u"]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ], ",", TagBox["1", HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox[RowBox[List["b", " ", "k"]], "c"], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "s"]], "c"], "+", FractionBox["3", "2"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], RowBox[List["2", " ", "c"]]], "-", FractionBox[RowBox[List["b", " ", "u"]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] ) / ( c - a ( m - 2 s ) - b ( u - 2 k ) ) + ( ( a ( m - 2 s ) - b ( u - 2 k ) ) z 2 F 1 ( b k c + a m 2 c + 1 2 - a s c - b u 2 c , 1 ; 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m + u + Condition z a z m b z u c z -1 2 -1 m -1 u 1 c z 1 -1 \$CellContext`m 2 1 -1 \$CellContext`u 2 c -1 Binomial m m 2 -1 Binomial u u 2 -1 HypergeometricPFQ 1 2 1 3 2 2 c z -1 2 -1 m -1 u 1 c z Binomial u u 2 -1 1 -1 \$CellContext`u 2 s 0 m -1 2 -1 Binomial m s -1 a m -1 2 s z HypergeometricPFQ -1 a m 2 c -1 a s c -1 1 2 1 -1 a m 2 c -1 a s c -1 3 2 2 c z c -1 a m -1 2 s -1 a m -1 2 s z HypergeometricPFQ a m 2 c -1 1 2 -1 a s c -1 1 a m 2 c -1 3 2 -1 a s c -1 2 c z c a m -1 2 s -1 -1 2 -1 m -1 u 1 c z Binomial m m 2 -1 1 -1 \$CellContext`m 2 k 0 u -1 2 -1 Binomial u k b u -1 2 k z HypergeometricPFQ -1 b k c -1 b u 2 c -1 1 2 1 -1 b k c -1 b u 2 c -1 3 2 2 c z c b u -1 2 k -1 -1 b u -1 2 k z HypergeometricPFQ b k c -1 1 2 -1 b u 2 c -1 1 b k c -1 3 2 -1 b u 2 c -1 2 c z c -1 b u -1 2 k -1 -1 2 -1 m -1 u 1 c z s 0 m -1 2 -1 k 0 u -1 2 -1 Binomial m s Binomial u k b u -1 2 k -1 a m -1 2 s z HypergeometricPFQ -1 b k c -1 a s c -1 b u 2 c -1 1 2 -1 a m 2 c -1 1 -1 b k c -1 a s c -1 b u 2 c -1 3 2 -1 a m 2 c -1 2 c z c -1 a m -1 2 s b u -1 2 k -1 a m -1 2 s b u -1 2 k z HypergeometricPFQ -1 b k c -1 a m 2 c -1 b u 2 c -1 1 2 -1 a s c -1 1 -1 b k c -1 a m 2 c -1 b u 2 c -1 3 2 -1 a s c -1 2 c z c a m -1 2 s b u -1 2 k -1 -1 a m -1 2 s -1 b u -1 2 k z HypergeometricPFQ b k c -1 a s c -1 1 2 -1 a m 2 c -1 -1 b u 2 c -1 1 b k c -1 a s c -1 3 2 -1 a m 2 c -1 -1 b u 2 c -1 2 c z c -1 a m -1 2 s -1 b u -1 2 k -1 a m -1 2 s -1 b u -1 2 k z HypergeometricPFQ b k c -1 a m 2 c -1 1 2 -1 a s c -1 -1 b u 2 c -1 1 b k c -1 a m 2 c -1 3 2 -1 a s c -1 -1 b u 2 c -1 2 c z c a m -1 2 s -1 b u -1 2 k -1 m SuperPlus u SuperPlus [/itex]

 Rule Form

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

 2002-12-18