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 Cosh

 http://functions.wolfram.com/01.20.21.3789.01

 Input Form

 Integrate[E^(b Sqrt[z] + d z + e) Sin[a Sqrt[z] + p z + q]^m Cosh[c Sqrt[z] + f z + g]^v, z] == 2^(-2 - m - v) E^e Binomial[m, m/2] Binomial[v, v/2] ((4 E^(b Sqrt[z] + d z))/d - (2 b Sqrt[Pi] Erfi[(b + 2 d Sqrt[z])/(2 Sqrt[d])])/ (E^(b^2/(4 d)) d^(3/2))) (1 - Mod[m, 2]) (1 - Mod[v, 2]) + 2^(-1 - m - v) Binomial[v, v/2] (1 - Mod[v, 2]) Sum[(-1)^k E^(e + (I m Pi)/2 - I (-2 k + m) q) Binomial[m, k] ((2 E^((b - I a (-2 k + m)) Sqrt[z] + (d - I (-2 k + m) p) z))/ (d - I (-2 k + m) p) + (2 E^(2 ((-(1/2)) I m Pi + I (-2 k + m) q) + (b + I a (-2 k + m)) Sqrt[z] + (d + I (-2 k + m) p) z))/ (d + I (-2 k + m) p) + Sqrt[Pi] (-(((b + 2 I a k - I a m) Erfi[(b + I a (2 k - m) + 2 (d + 2 I k p - I m p) Sqrt[z])/(2 Sqrt[d + 2 I k p - I m p])])/E^((b + 2 I a k - I a m)^2/(4 (d + 2 I k p - I m p)))/(d + 2 I k p - I m p)^(3/2)) - (E^(-((b + I a (-2 k + m))^2/(4 (d + I (-2 k + m) p))) - I m Pi + 2 I (-2 k + m) q) (b + I a (-2 k + m)) Erfi[(b + I a (-2 k + m) + 2 (d - 2 I k p + I m p) Sqrt[z])/ (2 Sqrt[d + I (-2 k + m) p])])/(d + I (-2 k + m) p)^(3/2))), {k, 0, Floor[(1/2) (-1 + m)]}] + 2^(-1 - m - v) Binomial[m, m/2] (1 - Mod[m, 2]) Sum[E^(e - g (-2 s + v)) Binomial[v, s] ((2 E^((b - c (-2 s + v)) Sqrt[z] + (d - f (-2 s + v)) z))/ (d - f (-2 s + v)) + (2 E^(2 g (-2 s + v) + (b + c (-2 s + v)) Sqrt[z] + (d + f (-2 s + v)) z))/(d + f (-2 s + v)) + Sqrt[Pi] (-(((b + 2 c s - c v) Erfi[(b + c (2 s - v) + 2 (d + 2 f s - f v) Sqrt[z])/(2 Sqrt[d + 2 f s - f v])])/ E^((b + 2 c s - c v)^2/(4 (d + 2 f s - f v)))/(d + 2 f s - f v)^ (3/2)) - (E^(2 g (-2 s + v) - (b + c (-2 s + v))^2/ (4 (d + f (-2 s + v)))) (b + c (-2 s + v)) Erfi[(b + c (-2 s + v) + 2 (d - 2 f s + f v) Sqrt[z])/ (2 Sqrt[d + f (-2 s + v)])])/(d + f (-2 s + v))^(3/2))), {s, 0, Floor[(1/2) (-1 + v)]}] + 2^(-1 - m - v) Sum[(-1)^k Binomial[m, k] Sum[Binomial[v, s] (E^(e - (I m Pi)/2 - I (2 k - m) q - g (-2 s + v)) ((2 E^((b - 2 I a k + I a m + 2 c s - c v) Sqrt[z] + (d - 2 I k p + I m p + 2 f s - f v) z))/(d - 2 I k p + I m p + 2 f s - f v) + (2 E^(I m Pi + 4 I k q - 2 I m q + 2 g (-2 s + v) + (b + 2 I a k - I a m - 2 c s + c v) Sqrt[z] + (d + 2 I k p - I m p - 2 f s + f v) z))/(d + 2 I k p - I m p - 2 f s + f v) - (Sqrt[Pi] (b - 2 I a k + I a m + 2 c s - c v) Erfi[(b + I a (-2 k + m) + c (2 s - v) + 2 (d - 2 I k p + I m p + 2 f s - f v) Sqrt[z])/(2 Sqrt[d - 2 I k p + I m p + 2 f s - f v])])/E^((b - 2 I a k + I a m + 2 c s - c v)^2/(4 (d - 2 I k p + I m p + 2 f s - f v)))/ (d - 2 I k p + I m p + 2 f s - f v)^(3/2) - (E^(I m Pi + 4 I k q - 2 I m q + 2 g (-2 s + v) - (b + 2 I a k - I a m - 2 c s + c v)^2/(4 (d + 2 I k p - I m p - 2 f s + f v))) Sqrt[Pi] (b + 2 I a k - I a m - 2 c s + c v) Erfi[(b + I a (2 k - m) + c (-2 s + v) + 2 (d + 2 I k p - I m p - 2 f s + f v) Sqrt[z])/(2 Sqrt[d + 2 I k p - I m p - 2 f s + f v])])/ (d + 2 I k p - I m p - 2 f s + f v)^(3/2)) + E^(e + (I m Pi)/2 - I (-2 k + m) q - g (-2 s + v)) ((2 E^((b + 2 I a k - I a m + 2 c s - c v) Sqrt[z] + (d + 2 I k p - I m p + 2 f s - f v) z))/(d + 2 I k p - I m p + 2 f s - f v) + (2 E^((-I) m Pi + 2 I (-2 k + m) q + 2 g (-2 s + v) + (b - 2 I a k + I a m - 2 c s + c v) Sqrt[z] + (d - 2 I k p + I m p - 2 f s + f v) z))/(d - 2 I k p + I m p - 2 f s + f v) - (Sqrt[Pi] (b + 2 I a k - I a m + 2 c s - c v) Erfi[(b + I a (2 k - m) + c (2 s - v) + 2 (d + 2 I k p - I m p + 2 f s - f v) Sqrt[z])/(2 Sqrt[d + 2 I k p - I m p + 2 f s - f v])])/E^((b + 2 I a k - I a m + 2 c s - c v)^2/(4 (d + 2 I k p - I m p + 2 f s - f v)))/ (d + 2 I k p - I m p + 2 f s - f v)^(3/2) - (E^((-I) m Pi + 2 I (-2 k + m) q + 2 g (-2 s + v) - (b + I a (-2 k + m) + c (-2 s + v))^2/(4 (d - 2 I k p + I m p - 2 f s + f v))) Sqrt[Pi] (b + I a (-2 k + m) + c (-2 s + v)) Erfi[(b + I a (-2 k + m) + c (-2 s + v) + 2 (d - 2 I k p + I m p - 2 f s + f v) Sqrt[z])/(2 Sqrt[d - 2 I k p + I m p - 2 f s + f v])])/ (d - 2 I k p + I m p - 2 f s + f v)^(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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 MathML Form

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

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["b_", " ", SqrtBox["z_"]]], "+", RowBox[List["d_", " ", "z_"]], "+", "e_"]]], " ", SuperscriptBox[RowBox[List["Sin", "[", RowBox[List[RowBox[List["a_", " ", SqrtBox["z_"]]], "+", RowBox[List["p_", " ", "z_"]], "+", "q_"]], "]"]], "m_"], " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List[RowBox[List["c_", " ", SqrtBox["z_"]]], "+", RowBox[List["f_", " ", "z_"]], "+", "g_"]], "]"]], "v_"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", "m", "-", "v"]]], " ", SuperscriptBox["\[ExponentialE]", "e"], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", FractionBox["v", "2"]]], "]"]], " ", RowBox[List["(", 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 Date Added to functions.wolfram.com (modification date)

 2002-12-18