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 Cosh

 http://functions.wolfram.com/01.20.21.4176.01

 Input Form

 Integrate[Sinh[b Sqrt[z] + d z]^m Cosh[c Sqrt[z]]^v, z] == I^m 2^(-m - v) z Binomial[m, m/2] Binomial[v, v/2] (1 - Mod[m, 2]) (1 - Mod[v, 2]) + (1/c^2) (I^m 2^(1 - m - v) Binomial[m, m/2] (1 - Mod[m, 2]) Sum[(1/(-2 s + v)^2) (E^((-2 s - v) c Sqrt[z]) (E^(4 s c Sqrt[z]) (-1 + c (2 s - v) Sqrt[z]) + E^(2 v c Sqrt[z]) (-1 + c (-2 s + v) Sqrt[z])) Binomial[v, s]), {s, 0, Floor[(1/2) (-1 + v)]}]) + 2^(-1 - m - v) Binomial[v, v/2] (1 - Mod[v, 2]) Sum[(-1)^k Binomial[m, k] ((2 E^(b (2 k - m) Sqrt[z] + d (2 k - m) z) (-(-1)^m + E^(2 (-2 k + m) (b Sqrt[z] + d z))))/(d (-2 k + m)) - ((-1)^m b E^((b^2 (-2 k + m))/(4 d)) (-2 k + m) Sqrt[Pi] Erfi[(b (-2 k + m) + 2 d (-2 k + m) Sqrt[z])/(2 Sqrt[d (2 k - m)])])/ (d (2 k - m))^(3/2) - (b (-2 k + m) Sqrt[Pi] Erfi[(b (-2 k + m) + 2 d (-2 k + m) Sqrt[z])/ (2 Sqrt[d (-2 k + m)])])/E^((b^2 (-2 k + m))/(4 d))/ (d (-2 k + m))^(3/2)), {k, 0, Floor[(1/2) (-1 + m)]}] + 2^(-1 - m - v) Sum[(-1)^k Binomial[m, k] Sum[Binomial[v, s] ((1/(d (2 k - m))) (2 (-1)^m E^((b (2 k - m) + c (-2 s + v)) Sqrt[z] + d (2 k - m) z)) + (2 E^(((-b) (2 k - m) - c (-2 s + v)) Sqrt[z] + d (-2 k + m) z))/ (d (-2 k + m)) - ((-1)^m Sqrt[Pi] (b (2 k - m) + c (-2 s + v)) Erfi[(b (2 k - m) + c (-2 s + v) + 2 d (2 k - m) Sqrt[z])/ (2 Sqrt[d (2 k - m)])])/E^((b (2 k - m) + c (-2 s + v))^2/ (4 d (2 k - m)))/(d (2 k - m))^(3/2) - (Sqrt[Pi] (b (2 k - m) + c (-2 s + v)) Erfi[(b (2 k - m) + c (-2 s + v) + 2 d (2 k - m) Sqrt[z])/ (2 Sqrt[d (-2 k + m)])])/E^(((-b) (2 k - m) - c (-2 s + v))^2/ (4 d (-2 k + m)))/(d (-2 k + m))^(3/2) + (-1)^m ((2 E^(((-b) (-2 k + m) - c (-2 s + v)) Sqrt[z] + d (2 k - m) z))/(d (2 k - m)) + (1/(d (-2 k + m))) (2 (-1)^m E^((b (-2 k + m) + c (-2 s + v)) Sqrt[z] + d (-2 k + m) z)) - (Sqrt[Pi] (b (-2 k + m) + c (-2 s + v)) Erfi[(b (-2 k + m) + c (-2 s + v) + 2 d (-2 k + m) Sqrt[z])/ (2 Sqrt[d (2 k - m)])])/E^(((-b) (-2 k + m) - c (-2 s + v))^ 2/(4 d (2 k - m)))/(d (2 k - m))^(3/2) - ((-1)^m Sqrt[Pi] (b (-2 k + m) + c (-2 s + v)) Erfi[(b (-2 k + m) + c (-2 s + v) + 2 d (-2 k + m) Sqrt[z])/ (2 Sqrt[d (-2 k + m)])])/E^((b (-2 k + m) + c (-2 s + v))^2/( 4 d (-2 k + m)))/(d (-2 k + m))^(3/2))), {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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"0"]], "\[And]", RowBox[List["v", "\[Element]", "Integers"]], "\[And]", RowBox[List["v", ">", "0"]]]]]]]]

 MathML Form

 sinh m ( z b + d z ) cosh v ( c z ) z m 2 - m - v 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]]]]] ( 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 ) + m 2 - m - v + 1 ( 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 ) c 2 s = 0 v - 1 2 ( - 2 s - v ) c z ( 4 s c z ( c ( 2 s - v ) z - 1 ) + 2 v c z ( c ( v - 2 s ) z - 1 ) ) ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( v - 2 s ) 2 + 2 - m - v - 1 ( 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 ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 2 d z ( 2 k - m ) + b z ( 2 k - m ) ( - ( - 1 ) m + 2 ( m - 2 k ) ( z b + d z ) ) d ( m - 2 k ) - ( - 1 ) m b b 2 ( m - 2 k ) 4 d ( m - 2 k ) π erfi ( b ( m - 2 k ) + 2 d z ( m - 2 k ) 2 d ( 2 k - m ) ) ( d ( 2 k - m ) ) 3 / 2 - b - b 2 ( m - 2 k ) 4 d ( m - 2 k ) π erfi ( b ( m - 2 k ) + 2 d z ( m - 2 k ) 2 d ( m - 2 k ) ) ( d ( m - 2 k ) ) 3 / 2 ) + 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]]]]] ( ( - 1 ) m ( 2 ( - 1 ) m z ( b ( m - 2 k ) + c ( v - 2 s ) ) + d ( m - 2 k ) z d ( m - 2 k ) + 2 z ( - b ( m - 2 k ) - c ( v - 2 s ) ) + d ( 2 k - m ) z d ( 2 k - m ) - 1 ( d ( 2 k - m ) ) 3 / 2 ( - ( - b ( m - 2 k ) - c ( v - 2 s ) ) 2 4 d ( 2 k - m ) π ( b ( m - 2 k ) + c ( v - 2 s ) ) erfi ( b ( m - 2 k ) + 2 d z ( m - 2 k ) + c ( v - 2 s ) 2 d ( 2 k - m ) ) ) - 1 ( d ( m - 2 k ) ) 3 / 2 ( ( - 1 ) m - ( b ( m - 2 k ) + c ( v - 2 s ) ) 2 4 d ( m - 2 k ) π ( b ( m - 2 k ) + c ( v - 2 s ) ) erfi ( b ( m - 2 k ) + 2 d z ( m - 2 k ) + c ( v - 2 s ) 2 d ( m - 2 k ) ) ) ) + 2 ( - 1 ) m z ( b ( 2 k - m ) + c ( v - 2 s ) ) + d ( 2 k - m ) z d ( 2 k - m ) - 1 ( d ( 2 k - m ) ) 3 / 2 ( ( - 1 ) m - ( b ( 2 k - m ) + c ( v - 2 s ) ) 2 4 d ( 2 k - m ) π ( b ( 2 k - m ) + c ( v - 2 s ) ) erfi ( b ( 2 k - m ) + 2 d z ( 2 k - m ) + c ( v - 2 s ) 2 d ( 2 k - m ) ) ) + 2 z ( - b ( 2 k - m ) - c ( v - 2 s ) ) + d ( m - 2 k ) z d ( m - 2 k ) - 1 ( d ( m - 2 k ) ) 3 / 2 ( - ( - b ( 2 k - m ) - c ( v - 2 s ) ) 2 4 d ( m - 2 k ) π ( b ( 2 k - m ) + c ( v - 2 s ) ) erfi ( b ( 2 k - m ) + 2 d z ( 2 k - m ) + c ( v - 2 s ) 2 d ( m - 2 k ) ) ) ) /; m + v + Condition z z 1 2 b d z m c z 1 2 v m 2 -1 m -1 v z Binomial m m 2 -1 Binomial v v 2 -1 1 -1 \$CellContext`m 2 1 -1 \$CellContext`v 2 m 2 -1 m -1 v 1 Binomial m m 2 -1 1 -1 \$CellContext`m 2 c 2 -1 s 0 v -1 2 -1 -2 s -1 v c z 1 2 4 s c z 1 2 c 2 s -1 v z 1 2 -1 2 v c z 1 2 c v -1 2 s z 1 2 -1 Binomial v s v -1 2 s 2 -1 2 -1 m -1 v -1 Binomial v v 2 -1 1 -1 \$CellContext`v 2 k 0 m -1 2 -1 -1 k Binomial m k 2 d z 2 k -1 m b z 1 2 2 k -1 m -1 -1 m 2 m -1 2 k z 1 2 b d z d m -1 2 k -1 -1 -1 m b b 2 m -1 2 k 4 d -1 m -1 2 k 1 2 Erfi b m -1 2 k 2 d z 1 2 m -1 2 k 2 d 2 k -1 m 1 2 -1 d 2 k -1 m 3 2 -1 -1 b -1 b 2 m -1 2 k 4 d -1 m -1 2 k 1 2 Erfi b m -1 2 k 2 d z 1 2 m -1 2 k 2 d m -1 2 k 1 2 -1 d m -1 2 k 3 2 -1 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 -1 m 2 -1 m z 1 2 b m -1 2 k c v -1 2 s d m -1 2 k z d m -1 2 k -1 2 z 1 2 -1 b m -1 2 k -1 c v -1 2 s d 2 k -1 m z d 2 k -1 m -1 -1 1 d 2 k -1 m 3 2 -1 -1 -1 b m -1 2 k -1 c v -1 2 s 2 4 d 2 k -1 m -1 1 2 b m -1 2 k c v -1 2 s Erfi b m -1 2 k 2 d z 1 2 m -1 2 k c v -1 2 s 2 d 2 k -1 m 1 2 -1 -1 1 d m -1 2 k 3 2 -1 -1 m -1 b m -1 2 k c v -1 2 s 2 4 d m -1 2 k -1 1 2 b m -1 2 k c v -1 2 s Erfi b m -1 2 k 2 d z 1 2 m -1 2 k c v -1 2 s 2 d m -1 2 k 1 2 -1 2 -1 m z 1 2 b 2 k -1 m c v -1 2 s d 2 k -1 m z d 2 k -1 m -1 -1 1 d 2 k -1 m 3 2 -1 -1 m -1 b 2 k -1 m c v -1 2 s 2 4 d 2 k -1 m -1 1 2 b 2 k -1 m c v -1 2 s Erfi b 2 k -1 m 2 d z 1 2 2 k -1 m c v -1 2 s 2 d 2 k -1 m 1 2 -1 2 z 1 2 -1 b 2 k -1 m -1 c v -1 2 s d m -1 2 k z d m -1 2 k -1 -1 1 d m -1 2 k 3 2 -1 -1 -1 b 2 k -1 m -1 c v -1 2 s 2 4 d m -1 2 k -1 1 2 b 2 k -1 m c v -1 2 s Erfi b 2 k -1 m 2 d z 1 2 2 k -1 m c v -1 2 s 2 d m -1 2 k 1 2 -1 m SuperPlus v SuperPlus [/itex]

 Rule Form

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

 2002-12-18