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 Tanh

 http://functions.wolfram.com/01.21.21.0191.01

 Input Form

 Integrate[E^(p z) Cos[a z]^m Cosh[c z]^\[Mu] Tanh[c z], z] == (Binomial[m, m/2] Cosh[c z]^\[Mu] (1 - Mod[m, 2]) (-(E^(p z) Hypergeometric2F1[(p - c \[Mu])/(2 c), 1 - \[Mu], (1/2) (2 + p/c - \[Mu]), -E^(2 c z)])/(p - c \[Mu]) + (E^((2 c + p) z) Hypergeometric2F1[(p + c (2 - \[Mu]))/(2 c), 1 - \[Mu], (1/2) (4 + p/c - \[Mu]), -E^(2 c z)])/ (p + c (2 - \[Mu]))))/(2^m (1 + E^(2 c z))^\[Mu]) + (Cosh[c z]^\[Mu] Sum[Binomial[m, k] (-(E^(((-I) a (-2 k + m) + p) z) Hypergeometric2F1[ ((-I) a (-2 k + m) + p - c \[Mu])/(2 c), 1 - \[Mu], (1/2) (2 + ((-I) a (-2 k + m) + p)/c - \[Mu]), -E^(2 c z)])/ ((-I) a (-2 k + m) + p - c \[Mu]) + (E^((2 c - I a (-2 k + m) + p) z) Hypergeometric2F1[ ((-I) a (-2 k + m) + p + c (2 - \[Mu]))/(2 c), 1 - \[Mu], (1/2) (4 + ((-I) a (-2 k + m) + p)/c - \[Mu]), -E^(2 c z)])/ ((-I) a (-2 k + m) + p + c (2 - \[Mu])) - (E^((I a (-2 k + m) + p) z) Hypergeometric2F1[ (I a (-2 k + m) + p - c \[Mu])/(2 c), 1 - \[Mu], (1/2) (2 + (I a (-2 k + m) + p)/c - \[Mu]), -E^(2 c z)])/ (I a (-2 k + m) + p - c \[Mu]) + (E^((2 c + I a (-2 k + m) + p) z) Hypergeometric2F1[(I a (-2 k + m) + p + c (2 - \[Mu]))/(2 c), 1 - \[Mu], (1/2) (4 + (I a (-2 k + m) + p)/c - \[Mu]), -E^(2 c z)])/(I a (-2 k + m) + p + c (2 - \[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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 MathML Form

 p z cos m ( a z ) cosh μ ( c z ) tanh ( c z ) z 2 - m ( 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]]]]] cosh μ ( c z ) ( 1 - m mod 2 \$CellContext`m 2 ) ( 1 + 2 c z ) - μ ( ( 2 c + p ) z 2 F 1 ( p + c ( 2 - μ ) 2 c , 1 - μ ; 1 2 ( p 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["p", "+", 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["p", "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] p + c ( 2 - μ ) - p z 2 F 1 ( p - c μ 2 c , 1 - μ ; 1 2 ( p 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["p", "-", 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["p", "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] p - 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 ) + p ) z 2 F 1 ( - a ( m - 2 k ) + p + c ( 2 - μ ) 2 c , 1 - μ ; 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m + Condition z p z a z m c z μ c z 2 -1 m Binomial m m 2 -1 c z μ 1 -1 \$CellContext`m 2 1 2 c z -1 μ 2 c p z Hypergeometric2F1 p c 2 -1 μ 2 c -1 1 -1 μ 1 2 p c -1 -1 μ 4 -1 2 c z p c 2 -1 μ -1 -1 p z Hypergeometric2F1 p -1 c μ 2 c -1 1 -1 μ 1 2 p c -1 -1 μ 2 -1 2 c z p -1 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 p z Hypergeometric2F1 -1 a m -1 2 k p c 2 -1 μ 2 c -1 1 -1 μ 1 2 p -1 a m -1 2 k c -1 -1 μ 4 -1 2 c z -1 a m -1 2 k p c 2 -1 μ -1 2 c a m -1 2 k p z Hypergeometric2F1 a m -1 2 k p c 2 -1 μ 2 c -1 1 -1 μ 1 2 a m -1 2 k p c -1 -1 μ 4 -1 2 c z a m -1 2 k p c 2 -1 μ -1 -1 p -1 a m -1 2 k z Hypergeometric2F1 -1 a m -1 2 k p -1 c μ 2 c -1 1 -1 μ 1 2 p -1 a m -1 2 k c -1 -1 μ 2 -1 2 c z -1 a m -1 2 k p -1 c μ -1 -1 a m -1 2 k p z Hypergeometric2F1 a m -1 2 k p -1 c μ 2 c -1 1 -1 μ 1 2 a m -1 2 k p c -1 -1 μ 2 -1 2 c z a m -1 2 k p -1 c μ -1 m SuperPlus [/itex]

 Rule Form

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

 2002-12-18