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 Cosh

 http://functions.wolfram.com/01.20.21.3935.01

 Input Form

 Integrate[z^(\[Alpha] - 1) E^(p z) Sin[d + c z]^m Cosh[a z]^v, z] == ((-2^(-m - v)) z^\[Alpha] Binomial[m, m/2] Binomial[v, v/2] Gamma[\[Alpha], (-p) z] (1 - Mod[m, 2]) (1 - Mod[v, 2]))/ ((-p) z)^\[Alpha] - (2^(-m - v) z^\[Alpha] Binomial[v, v/2] (1 - Mod[v, 2]) Sum[((-1)^k Binomial[m, k] (E^(2 I d m) ExpIntegralE[1 - \[Alpha], (2 I c k - I c m - p) z] + E^(4 I d k + I m Pi) ExpIntegralE[1 - \[Alpha], (-2 I c k + I c m - p) z]))/E^(I d (2 k + m)), {k, 0, Floor[(1/2) (-1 + m)]}])/I^m - 2^(-m - v) z^\[Alpha] Binomial[m, m/2] (1 - Mod[m, 2]) Sum[Binomial[v, s] (ExpIntegralE[1 - \[Alpha], (-p - 2 a s + a v) z] + ExpIntegralE[1 - \[Alpha], (-p - a (-2 s + v)) z]), {s, 0, Floor[(1/2) (-1 + v)]}] - (2^(-m - v) z^\[Alpha] Sum[Binomial[v, s] Sum[(-1)^k E^(-2 I d k - I d m) Binomial[m, k] (E^(2 I d m) (ExpIntegralE[1 - \[Alpha], (2 I c k - I c m - p + 2 a s - a v) z] + ExpIntegralE[ 1 - \[Alpha], (2 I c k - I c m - p - 2 a s + a v) z]) + E^(4 I d k + I m Pi) (ExpIntegralE[1 - \[Alpha], (-2 I c k + I c m - p + 2 a s - a v) z] + ExpIntegralE[ 1 - \[Alpha], (I c (-2 k + m) - p - 2 a s + a v) z])), {k, 0, Floor[(1/2) (-1 + m)]}], {s, 0, Floor[(1/2) (-1 + v)]}])/ I^m /; Element[m, Integers] && m > 0 && Element[v, Integers] && v > 0

 Standard Form

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 MathML Form

 z α - 1 p z sin m ( d + c z ) cosh v ( a z ) z - 2 - m - v z α ( - p z ) - α ( 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]]]]] Γ ( α , - p z ) ( 1 - m mod 2 \$CellContext`m 2 ) ( 1 - v mod 2 \$CellContext`v 2 ) - - m 2 - m - v z α ( 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 - v mod 2 \$CellContext`v 2 ) k = 0 m - 1 2 ( - 1 ) k - d ( 2 k + m ) ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 4 d k + m π E TagBox["E", ExpIntegralE] 1 - α ( ( - 2 c k + c m - p ) z ) + 2 d m E TagBox["E", ExpIntegralE] 1 - α ( ( 2 c k - c m - p ) z ) ) - 2 - m - v z α ( 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 ) 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]]]]] ( E TagBox["E", ExpIntegralE] 1 - α ( ( - p - 2 a s + a v ) z ) + E TagBox["E", ExpIntegralE] 1 - α ( ( - p - a ( v - 2 s ) ) z ) ) - - m 2 - m - v z α 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]]]]] k = 0 m - 1 2 ( - 1 ) k - 2 d k - d m ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 2 d m ( E TagBox["E", ExpIntegralE] 1 - α ( ( 2 c k - c m - p - 2 a s + a v ) z ) + E TagBox["E", ExpIntegralE] 1 - α ( ( 2 c k - c m - p + 2 a s - a v ) z ) ) + 4 d k + m π ( E TagBox["E", ExpIntegralE] 1 - α ( ( c ( m - 2 k ) - p - 2 a s + a v ) z ) + E TagBox["E", ExpIntegralE] 1 - α ( ( - 2 c k + c m - p + 2 a s - a v ) z ) ) ) /; m + v + Condition z z α -1 p z d c z m a z v -1 2 -1 m -1 v z α -1 p z -1 α Binomial m m 2 -1 Binomial v v 2 -1 Gamma α -1 p z 1 -1 \$CellContext`m 2 1 -1 \$CellContext`v 2 -1 -1 m 2 -1 m -1 v z α Binomial v v 2 -1 1 -1 \$CellContext`v 2 k 0 m -1 2 -1 -1 k -1 d 2 k m Binomial m k 4 d k m ExpIntegralE 1 -1 α -2 c k c m -1 p z 2 d m ExpIntegralE 1 -1 α 2 c k -1 c m -1 p z -1 2 -1 m -1 v z α Binomial m m 2 -1 1 -1 \$CellContext`m 2 s 0 v -1 2 -1 Binomial v s ExpIntegralE 1 -1 α -1 p -1 2 a s a v z ExpIntegralE 1 -1 α -1 p -1 a v -1 2 s z -1 -1 m 2 -1 m -1 v z α s 0 v -1 2 -1 Binomial v s k 0 m -1 2 -1 -1 k -2 d k -1 d m Binomial m k 2 d m ExpIntegralE 1 -1 α 2 c k -1 c m -1 p -1 2 a s a v z ExpIntegralE 1 -1 α 2 c k -1 c m -1 p 2 a s -1 a v z 4 d k m ExpIntegralE 1 -1 α c m -1 2 k -1 p -1 2 a s a v z ExpIntegralE 1 -1 α -2 c k c m -1 p 2 a s -1 a v z m SuperPlus v SuperPlus [/itex]

 Rule Form

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

 2002-12-18