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 Cosh

 http://functions.wolfram.com/01.20.21.4148.01

 Input Form

 Integrate[Sinh[d + c z]^m Cosh[b + a z]^v, z] == I^m 2^(-m - v) Binomial[m, m/2] Binomial[v, v/2] (1 - Mod[m, 2]) (1 - Mod[v, 2]) z + ((I^m 2^(1 - m - v))/a) Binomial[m, m/2] (1 - Mod[m, 2]) Sum[Binomial[v, s] (Sinh[(2 s - v) (b + a z)]/(2 s - v)), {s, 0, Floor[(1/2) (-1 + v)]}] + ((I^(m - 1) 2^(1 - m - v))/c) Binomial[v, v/2] (1 - Mod[v, 2]) Sum[(-1)^k Binomial[m, k] (Sin[(m Pi)/2 + I (-2 k + m) (d + c z)]/(-2 k + m)), {k, 0, Floor[(1/2) (-1 + m)]}] + I^m 2^(1 - m - v) Sum[(-1)^k Binomial[m, k] Sum[Binomial[v, s] (((c (-2 k + m) + a (-2 s + v)) Sinh[d (-2 k + m) - (I m Pi)/2 - b (-2 s + v) + c (-2 k + m) z - a (-2 s + v) z] + (c (-2 k + m) - a (-2 s + v)) Sinh[d (-2 k + m) - (I m Pi)/2 + b (-2 s + v) + (c (-2 k + m) + a (-2 s + v)) z])/ ((c (-2 k + m) - a (-2 s + v)) (c (-2 k + m) + a (-2 s + v)))), {s, 0, Floor[(1/2) (-1 + v)]}], {k, 0, Floor[(1/2) (-1 + m)]}] /; Element[m, Integers] && m > 0 && Element[v, Integers] && v > 0

 Standard Form

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 MathML Form

 sinh m ( d + c z ) cosh v ( b + a z ) z m 2 - m - v ( 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]]]]] ( 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]]]]] ( 1 - m mod 2 \$CellContext`m 2 ) ( 1 - v mod 2 \$CellContext`v 2 ) z + m 2 - m - v + 1 ( 1 - m mod 2 \$CellContext`m 2 ) a ( 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]]]]] 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]]]]] sinh ( ( 2 s - v ) ( b + a z ) ) 2 s - v + m - 1 2 - m - v + 1 ( 1 - v mod 2 \$CellContext`v 2 ) c ( 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]]]]] 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]]]]] sin ( π m 2 + ( m - 2 k ) ( d + c z ) ) m - 2 k + m 2 - m - v + 1 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]]]]] 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]]]]] ( ( c ( m - 2 k ) - a ( v - 2 s ) ) sinh ( - 1 2 π m + d ( m - 2 k ) + b ( v - 2 s ) + ( c ( m - 2 k ) + a ( v - 2 s ) ) z ) + ( c ( m - 2 k ) + a ( v - 2 s ) ) sinh ( - 1 2 π m + d ( m - 2 k ) - b ( v - 2 s ) + c ( m - 2 k ) z - a ( v - 2 s ) z ) ) / ( ( c ( m - 2 k ) - a ( v - 2 s ) ) ( c ( m - 2 k ) + a ( v - 2 s ) ) ) /; m + v + Condition z d c z m b a z v m 2 -1 m -1 v Binomial m m 2 -1 Binomial v v 2 -1 1 -1 \$CellContext`m 2 1 -1 \$CellContext`v 2 z m 2 -1 m -1 v 1 1 -1 \$CellContext`m 2 a -1 Binomial m m 2 -1 s 0 v -1 2 -1 Binomial v s 2 s -1 v b a z 2 s -1 v -1 m -1 2 -1 m -1 v 1 1 -1 \$CellContext`v 2 c -1 Binomial v v 2 -1 k 0 m -1 2 -1 -1 k Binomial m k m 2 -1 m -1 2 k d c z m -1 2 k -1 m 2 -1 m -1 v 1 k 0 m -1 2 -1 -1 k Binomial m k s 0 v -1 2 -1 Binomial v s c m -1 2 k -1 a v -1 2 s -1 1 2 m d m -1 2 k b v -1 2 s c m -1 2 k a v -1 2 s z c m -1 2 k a v -1 2 s -1 1 2 m d m -1 2 k -1 b v -1 2 s c m -1 2 k z -1 a v -1 2 s z c m -1 2 k -1 a v -1 2 s c m -1 2 k a v -1 2 s -1 m SuperPlus v SuperPlus [/itex]

 Rule Form

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

 2002-12-18