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 Cosh

 http://functions.wolfram.com/01.20.21.4156.01

 Input Form

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

 MathML Form

 sinh m ( z b + e + 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 s = 0 v - 1 2 sinh ( ( 2 s - v ) c z ) ( 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 - 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 - e ( m - 2 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 + e + 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 2 e ( m - 2 k ) - 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]]]]] ( - e ( 2 k - m ) ( 2 ( - 1 ) m 2 e ( 2 k - m ) + b z ( 2 k - m ) + ( d ( 2 k - m ) + c ( v - 2 s ) ) z d ( 2 k - m ) + c ( v - 2 s ) - ( ( - 1 ) m b 2 e ( 2 k - m ) - b 2 ( 2 k - m ) 2 4 ( d ( 2 k - m ) + c ( v - 2 s ) ) ( 2 k - m ) π erfi ( b ( 2 k - m ) + 2 ( d ( 2 k - m ) + c ( v - 2 s ) ) z 2 d ( 2 k - m ) + c ( v - 2 s ) ) ) / ( d ( 2 k - m ) + c ( v - 2 s ) ) 3 / 2 + 2 ( d ( m - 2 k ) - c ( v - 2 s ) ) z - b ( 2 k - m ) z d ( m - 2 k ) - c ( v - 2 s ) - ( b - b 2 ( 2 k - m ) 2 4 ( d ( m - 2 k ) - c ( v - 2 s ) ) ( 2 k - m ) π erfi ( b ( 2 k - m ) + 2 ( d ( 2 k - m ) + c ( v - 2 s ) ) z 2 d ( m - 2 k ) - c ( v - 2 s ) ) ) / ( d ( m - 2 k ) - c ( v - 2 s ) ) 3 / 2 ) + ( - 1 ) m - e ( m - 2 k ) ( 2 ( - 1 ) m 2 e ( m - 2 k ) + b z ( m - 2 k ) + ( d ( m - 2 k ) + c ( v - 2 s ) ) z d ( m - 2 k ) + c ( v - 2 s ) - ( ( - 1 ) m b 2 e ( m - 2 k ) - b 2 ( m - 2 k ) 2 4 ( d ( m - 2 k ) + c ( v - 2 s ) ) ( m - 2 k ) π erfi ( b ( m - 2 k ) + 2 ( d ( m - 2 k ) + c ( v - 2 s ) ) z 2 d ( m - 2 k ) + c ( v - 2 s ) ) ) / ( d ( m - 2 k ) + c ( v - 2 s ) ) 3 / 2 + 2 ( d ( 2 k - m ) - c ( v - 2 s ) ) z - b ( m - 2 k ) z d ( 2 k - m ) - c ( v - 2 s ) - ( b - b 2 ( m - 2 k ) 2 4 ( d ( 2 k - m ) - c ( v - 2 s ) ) ( m - 2 k ) π erfi ( b ( m - 2 k ) + 2 ( d ( m - 2 k ) + c ( v - 2 s ) ) z 2 d ( 2 k - m ) - c ( v - 2 s ) ) ) / ( d ( 2 k - m ) - c ( v - 2 s ) ) 3 / 2 ) ) /; m + v + Condition z z 1 2 b e d z m c z 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 -1 m 2 -1 m -1 v 1 Binomial m m 2 -1 1 -1 \$CellContext`m 2 c -1 s 0 v -1 2 -1 2 s -1 v c z Binomial v s v -1 2 s -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 -1 e m -1 2 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 e 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 2 e m -1 2 k -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 e 2 k -1 m 2 -1 m 2 e 2 k -1 m b z 1 2 2 k -1 m d 2 k -1 m c v -1 2 s z d 2 k -1 m c v -1 2 s -1 -1 -1 m b 2 e 2 k -1 m -1 b 2 2 k -1 m 2 4 d 2 k -1 m c v -1 2 s -1 2 k -1 m 1 2 Erfi b 2 k -1 m 2 d 2 k -1 m c v -1 2 s z 1 2 2 d 2 k -1 m c v -1 2 s 1 2 -1 d 2 k -1 m c v -1 2 s 3 2 -1 2 d m -1 2 k -1 c v -1 2 s z -1 b 2 k -1 m z 1 2 d m -1 2 k -1 c v -1 2 s -1 -1 b -1 b 2 2 k -1 m 2 4 d m -1 2 k -1 c v -1 2 s -1 2 k -1 m 1 2 Erfi b 2 k -1 m 2 d 2 k -1 m c v -1 2 s z 1 2 2 d m -1 2 k -1 c v -1 2 s 1 2 -1 d m -1 2 k -1 c v -1 2 s 3 2 -1 -1 m -1 e m -1 2 k 2 -1 m 2 e m -1 2 k b z 1 2 m -1 2 k d m -1 2 k c v -1 2 s z d m -1 2 k c v -1 2 s -1 -1 -1 m b 2 e m -1 2 k -1 b 2 m -1 2 k 2 4 d m -1 2 k c v -1 2 s -1 m -1 2 k 1 2 Erfi b m -1 2 k 2 d m -1 2 k c v -1 2 s z 1 2 2 d m -1 2 k c v -1 2 s 1 2 -1 d m -1 2 k c v -1 2 s 3 2 -1 2 d 2 k -1 m -1 c v -1 2 s z -1 b m -1 2 k z 1 2 d 2 k -1 m -1 c v -1 2 s -1 -1 b -1 b 2 m -1 2 k 2 4 d 2 k -1 m -1 c v -1 2 s -1 m -1 2 k 1 2 Erfi b m -1 2 k 2 d m -1 2 k c v -1 2 s z 1 2 2 d 2 k -1 m -1 c v -1 2 s 1 2 -1 d 2 k -1 m -1 c v -1 2 s 3 2 -1 m SuperPlus v SuperPlus [/itex]

 Rule Form

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

 2002-12-18