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 Cosh

 http://functions.wolfram.com/01.20.21.3821.01

 Input Form

 Integrate[E^(b Sqrt[z] + d z + e) Cos[a Sqrt[z] + p z + q] Cosh[c Sqrt[z] + f z + g]^v, z] == 2^(-2 - v) E^(e - I q) Binomial[v, v/2] ((2 E^(((-I) a + b) Sqrt[z] + (d - I p) z))/(d - I p) + (2 E^(2 I q + (I a + b) Sqrt[z] + (d + I p) z))/(d + I p) + ((I a - b) Sqrt[Pi] Erfi[((-I) a + b + 2 (d - I p) Sqrt[z])/ (2 Sqrt[d - I p])])/(E^(((-I) a + b)^2/(4 (d - I p))) (d - I p)^(3/2)) - ((I a + b) E^(-((I a + b)^2/(4 (d + I p))) + 2 I q) Sqrt[Pi] Erfi[(I a + b + 2 (d + I p) Sqrt[z])/(2 Sqrt[d + I p])])/ (d + I p)^(3/2)) (1 - Mod[v, 2]) + 2^(-2 - v) Sum[Binomial[v, k] (2 E^(e - 2 g k - I q - g v) (E^(4 g k + ((-I) a + b + 2 c k - c v) Sqrt[z] + (d + 2 f k - I p - f v) z)/(d + 2 f k - I p - f v) + E^(4 g k + 2 I q + (I a + b + 2 c k - c v) Sqrt[z] + (d + 2 f k + I p - f v) z)/(d + 2 f k + I p - f v) + E^(2 g v + ((-I) a + b - 2 c k + c v) Sqrt[z] + (d - 2 f k - I p + f v) z)/(d - 2 f k - I p + f v) + E^(2 I q + 2 g v + (I a + b - 2 c k + c v) Sqrt[z] + (d - 2 f k + I p + f v) z)/(d - 2 f k + I p + f v)) - (E^(e + 2 g k - I q - g v - ((-I) a + b + 2 c k - c v)^2/ (4 (d + 2 f k - I p - f v))) Sqrt[Pi] ((-I) a + b + 2 c k - c v) Erfi[((-I) a + b + 2 c k - c v + 2 (d + 2 f k - I p - f v) Sqrt[z])/ (2 Sqrt[d + 2 f k - I p - f v])])/(d + 2 f k - I p - f v)^(3/2) - (E^(e + 2 g k + I q - g v - (I a + b + 2 c k - c v)^2/ (4 (d + 2 f k + I p - f v))) Sqrt[Pi] (I a + b + 2 c k - c v) Erfi[(I a + b + 2 c k - c v + 2 (d + 2 f k + I p - f v) Sqrt[z])/ (2 Sqrt[d + 2 f k + I p - f v])])/(d + 2 f k + I p - f v)^(3/2) - (E^(e - 2 g k - I q + g v - ((-I) a + b - 2 c k + c v)^2/ (4 (d - 2 f k - I p + f v))) Sqrt[Pi] ((-I) a + b - 2 c k + c v) Erfi[((-I) a + b - 2 c k + c v + 2 (d - 2 f k - I p + f v) Sqrt[z])/ (2 Sqrt[d - 2 f k - I p + f v])])/(d - 2 f k - I p + f v)^(3/2) - (E^(e - 2 g k + I q + g v - (I a + b + c (-2 k + v))^2/ (4 (d + I p + f (-2 k + v)))) Sqrt[Pi] (I a + b + c (-2 k + v)) Erfi[(I a + b - 2 c k + c v + 2 (d + I p + f (-2 k + v)) Sqrt[z])/ (2 Sqrt[d + I p + f (-2 k + v)])])/(d + I p + f (-2 k + v))^ (3/2)), {k, 0, Floor[(1/2) (-1 + v)]}] /; Element[v, Integers] && v > 0

 Standard Form

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 MathML Form

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

 Rule Form

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

 2002-12-18