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 Tanh

 http://functions.wolfram.com/01.21.21.0170.01

 Input Form

 Integrate[Cos[a z]^m Cosh[c z]^\[Mu] Tanh[c z], z] == (Binomial[m, m/2] Cosh[c z]^\[Mu] ((E^(2 c z) Hypergeometric2F1[(2 - \[Mu])/2, 1 - \[Mu], (4 - \[Mu])/2, -E^(2 c z)])/(c (2 - \[Mu])) + Hypergeometric2F1[-(\[Mu]/2), 1 - \[Mu], (2 - \[Mu])/2, -E^(2 c z)]/(c \[Mu])) (1 - Mod[m, 2]))/ (2^m (1 + E^(2 c z))^\[Mu]) + (Cosh[c z]^\[Mu] Sum[Binomial[m, k] ((E^((2 c - I a (-2 k + m)) z) Hypergeometric2F1[ ((-I) a (-2 k + m) + c (2 - \[Mu]))/(2 c), 1 - \[Mu], (1/2) (4 - (I a (-2 k + m))/c - \[Mu]), -E^(2 c z)])/ ((-I) a (-2 k + m) + c (2 - \[Mu])) + (E^((2 c + I a (-2 k + m)) z) Hypergeometric2F1[ (I a (-2 k + m) + c (2 - \[Mu]))/(2 c), 1 - \[Mu], (1/2) (4 + (I a (-2 k + m))/c - \[Mu]), -E^(2 c z)])/ (I a (-2 k + m) + c (2 - \[Mu])) - Hypergeometric2F1[((-I) a (-2 k + m) - c \[Mu])/(2 c), 1 - \[Mu], (1/2) (2 - (I a (-2 k + m))/c - \[Mu]), -E^(2 c z)]/ E^(I a (-2 k + m) z)/((-I) a (-2 k + m) - c \[Mu]) - (E^(I a (-2 k + m) z) Hypergeometric2F1[(I a (-2 k + m) - c \[Mu])/ (2 c), 1 - \[Mu], (1/2) (2 + (I a (-2 k + m))/c - \[Mu]), -E^(2 c z)])/(I a (-2 k + m) - c \[Mu])), {k, 0, Floor[(1/2) (-1 + m)]}])/(2^m (1 + E^(2 c z))^\[Mu]) /; Element[m, Integers] && m > 0

 Standard Form

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RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "c"], "-", "\[Mu]"]], ")"]]]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "-", RowBox[List["c", " ", "\[Mu]"]]]], ")"]]]]]], ")"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]]]]]]]]

 MathML Form

 cos m ( a z ) cosh μ ( c z ) tanh ( c z ) z 2 - m cosh μ ( c z ) ( 1 - m mod 2 \$CellContext`m 2 ) ( 1 + 2 c z ) - μ ( 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 c z 2 F 1 ( 2 - μ 2 , 1 - μ ; 4 - μ 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[RowBox[List["2", "-", "\[Mu]"]], "2"], Hypergeometric2F1], ",", TagBox[RowBox[List["1", "-", "\[Mu]"]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List["4", "-", "\[Mu]"]], "2"], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] c ( 2 - μ ) + 2 F 1 ( - μ 2 , 1 - μ ; 2 - μ 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["-", FractionBox["\[Mu]", "2"]]], Hypergeometric2F1], ",", TagBox[RowBox[List["1", "-", "\[Mu]"]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List["2", "-", "\[Mu]"]], "2"], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] c μ ) + 2 - m cosh μ ( c z ) ( 1 + 2 c z ) - μ k = 0 m - 1 2 ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( 2 c - a ( m - 2 k ) ) z 2 F 1 ( c ( 2 - μ ) - a ( m - 2 k ) 2 c , 1 - μ ; 1 2 ( - a ( m - 2 k ) c - μ + 4 ) ; - 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[RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List["2", "-", "\[Mu]"]], ")"]]]], "-", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], ",", TagBox[RowBox[List["1", "-", "\[Mu]"]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "c"]]], "-", "\[Mu]", "+", "4"]], ")"]]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] c ( 2 - μ ) - a ( m - 2 k ) + ( 2 c + a ( m - 2 k ) ) z 2 F 1 ( a ( m - 2 k ) + c ( 2 - μ ) 2 c , 1 - μ ; 1 2 ( a ( m - 2 k ) c - μ + 4 ) ; - 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[RowBox[List[RowBox[List["a", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List["2", "-", "\[Mu]"]], ")"]]]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], ",", TagBox[RowBox[List["1", "-", "\[Mu]"]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["a", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "c"], "-", "\[Mu]", "+", "4"]], ")"]]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] a ( m - 2 k ) + c ( 2 - μ ) - - a ( m - 2 k ) z 2 F 1 ( - a ( m - 2 k ) - c μ 2 c , 1 - μ ; 1 2 ( - a ( m - 2 k ) c - μ + 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[RowBox[List[RowBox[List["-", "\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "-", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], ",", TagBox[RowBox[List["1", "-", "\[Mu]"]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "c"]]], "-", "\[Mu]", "+", "2"]], ")"]]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] - a ( m - 2 k ) - c μ - a ( m - 2 k ) z 2 F 1 ( a ( m - 2 k ) - c μ 2 c , 1 - μ ; 1 2 ( a ( m - 2 k ) c - μ + 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[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "-", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]], Hypergeometric2F1], ",", TagBox[RowBox[List["1", "-", "\[Mu]"]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["a", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "c"], "-", "\[Mu]", "+", "2"]], ")"]]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], Hypergeometric2F1]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] a ( m - 2 k ) - c μ ) /; m + Condition z a z m c z μ c z 2 -1 m c z μ 1 -1 \$CellContext`m 2 1 2 c z -1 μ Binomial m m 2 -1 2 c z Hypergeometric2F1 2 -1 μ 2 -1 1 -1 μ 4 -1 μ 2 -1 -1 2 c z c 2 -1 μ -1 Hypergeometric2F1 -1 μ 2 -1 1 -1 μ 2 -1 μ 2 -1 -1 2 c z c μ -1 2 -1 m c z μ 1 2 c z -1 μ k 0 m -1 2 -1 Binomial m k 2 c -1 a m -1 2 k z Hypergeometric2F1 c 2 -1 μ -1 a m -1 2 k 2 c -1 1 -1 μ 1 2 -1 a m -1 2 k c -1 -1 μ 4 -1 2 c z c 2 -1 μ -1 a m -1 2 k -1 2 c a m -1 2 k z Hypergeometric2F1 a m -1 2 k c 2 -1 μ 2 c -1 1 -1 μ 1 2 a m -1 2 k c -1 -1 μ 4 -1 2 c z a m -1 2 k c 2 -1 μ -1 -1 -1 a m -1 2 k z Hypergeometric2F1 -1 a m -1 2 k -1 c μ 2 c -1 1 -1 μ 1 2 -1 a m -1 2 k c -1 -1 μ 2 -1 2 c z -1 a m -1 2 k -1 c μ -1 -1 a m -1 2 k z Hypergeometric2F1 a m -1 2 k -1 c μ 2 c -1 1 -1 μ 1 2 a m -1 2 k c -1 -1 μ 2 -1 2 c z a m -1 2 k -1 c μ -1 m SuperPlus [/itex]

 Rule Form

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

 2002-12-18