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 Cosh

 http://functions.wolfram.com/01.20.21.4667.01

 Input Form

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

 b z 2 + d z + e sinh m ( a z 2 + p z + q ) cosh v ( c z 2 + f z + g ) z m 2 - m - v - 1 - d 2 - 4 b e 4 b π ( 1 - m mod 2 \$CellContext`m 2 ) ( 1 - v mod 2 \$CellContext`v 2 ) b ( 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]]]]] erfi ( d + 2 b z 2 b ) + m 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 ) π 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]]]]] ( - ( d + ( m - 2 k ) p ) 2 - 4 ( b + a ( m - 2 k ) ) ( e + ( m - 2 k ) q - m π 2 ) 4 ( b + a ( m - 2 k ) ) b + a ( m - 2 k ) ( b - a ( m - 2 k ) ) erfi ( d + ( m - 2 k ) p + 2 ( b + a ( m - 2 k ) ) z 2 b + a ( m - 2 k ) ) + - ( d - ( m - 2 k ) p ) 2 - 4 ( b - a ( m - 2 k ) ) ( e - ( m - 2 k ) q + m π 2 ) 4 ( b - a ( m - 2 k ) ) ( b + a ( m - 2 k ) ) b - a ( m - 2 k ) erfi ( d - ( m - 2 k ) p + 2 b z - 2 a ( m - 2 k ) z 2 b - a ( m - 2 k ) ) ) ) / ( ( b - a ( m - 2 k ) ) ( b + a ( m - 2 k ) ) ) + 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 ) π 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]]]]] ( - ( d + f ( v - 2 s ) ) 2 - 4 ( b + c ( v - 2 s ) ) ( e + g ( v - 2 s ) ) 4 ( b + c ( v - 2 s ) ) b + c ( v - 2 s ) ( b - c ( v - 2 s ) ) erfi ( d + f ( v - 2 s ) + 2 ( b + c ( v - 2 s ) ) z 2 b + c ( v - 2 s ) ) + - ( d - f ( v - 2 s ) ) 2 - 4 ( b - c ( v - 2 s ) ) ( e - g ( v - 2 s ) ) 4 ( b - c ( v - 2 s ) ) ( b + c ( v - 2 s ) ) b - c ( v - 2 s ) erfi ( d - f ( v - 2 s ) + 2 b z - 2 c ( v - 2 s ) z 2 b - c ( v - 2 s ) ) ) ) / ( ( b - c ( v - 2 s ) ) ( b + c ( v - 2 s ) ) ) + m 2 - m - v - 1 π 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]]]]] ( ( - ( d + ( 2 k - m ) p + f ( v - 2 s ) ) 2 - 4 ( b + a ( 2 k - m ) + c ( v - 2 s ) ) ( e + ( 2 k - m ) q + g ( v - 2 s ) + m π 2 ) 4 ( b + a ( 2 k - m ) + c ( v - 2 s ) ) b + a ( 2 k - m ) + c ( v - 2 s ) ( b - a ( 2 k - m ) - c ( v - 2 s ) ) erfi ( d + ( 2 k - m ) p + f ( v - 2 s ) + 2 ( b + a ( 2 k - m ) + c ( v - 2 s ) ) z 2 b + a ( 2 k - m ) + c ( v - 2 s ) ) + - ( d - ( 2 k - m ) p - f ( v - 2 s ) ) 2 - 4 ( b - a ( 2 k - m ) - c ( v - 2 s ) ) ( e - ( 2 k - m ) q - g ( v - 2 s ) - m π 2 ) 4 ( b - a ( 2 k - m ) - c ( v - 2 s ) ) ( b + a ( 2 k - m ) + c ( v - 2 s ) ) b - a ( 2 k - m ) - c ( v - 2 s ) erfi ( ( d - ( 2 k - m ) p - f ( v - 2 s ) + 2 b z - 2 ( a ( 2 k - m ) + c ( v - 2 s ) ) z ) / ( 2 b - a ( 2 k - m ) - c ( v - 2 s ) ) ) ) / ( ( b - a ( 2 k - m ) - c ( v - 2 s ) ) ( b + a ( 2 k - m ) + c ( v - 2 s ) ) ) + ( - ( d + ( m - 2 k ) p + f ( v - 2 s ) ) 2 - 4 ( b + a ( m - 2 k ) + c ( v - 2 s ) ) ( e + ( m - 2 k ) q + g ( v - 2 s ) - m π 2 ) 4 ( b + a ( m - 2 k ) + c ( v - 2 s ) ) b + a ( m - 2 k ) + c ( v - 2 s ) ( b - a ( m - 2 k ) - c ( v - 2 s ) ) erfi ( d + ( m - 2 k ) p + f ( v - 2 s ) + 2 ( b + a ( m - 2 k ) + c ( v - 2 s ) ) z 2 b + a ( m - 2 k ) + c ( v - 2 s ) ) + - ( d - ( m - 2 k ) p - f ( v - 2 s ) ) 2 - 4 ( b - a ( m - 2 k ) - c ( v - 2 s ) ) ( e - ( m - 2 k ) q - g ( v - 2 s ) + m π 2 ) 4 ( b - a ( m - 2 k ) - c ( v - 2 s ) ) ( b + a ( m - 2 k ) + c ( v - 2 s ) ) b - a ( m - 2 k ) - c ( v - 2 s ) erfi ( ( d - ( m - 2 k ) p - f ( v - 2 s ) + 2 b z - 2 ( a ( m - 2 k ) + c ( v - 2 s ) ) z ) / ( 2 b - a ( m - 2 k ) - c ( v - 2 s ) ) ) ) / ( ( b - a ( m - 2 k ) - c ( v - 2 s ) ) ( b + a ( m - 2 k ) + c ( v - 2 s ) ) ) ) /; m + v + Condition z b z 2 d z e a z 2 p z q m c z 2 f z g v m 2 -1 m -1 v -1 -1 d 2 -1 4 b e 4 b -1 1 2 1 -1 \$CellContext`m 2 1 -1 \$CellContext`v 2 b 1 2 -1 Binomial m m 2 -1 Binomial v v 2 -1 Erfi d 2 b z 2 b 1 2 -1 m 2 -1 m -1 v -1 Binomial v v 2 -1 1 -1 \$CellContext`v 2 1 2 k 0 m -1 2 -1 -1 k Binomial m k -1 d m -1 2 k p 2 -1 4 b a m -1 2 k e m -1 2 k q -1 m 2 -1 4 b a m -1 2 k -1 b a m -1 2 k 1 2 b -1 a m -1 2 k Erfi d m -1 2 k p 2 b a m -1 2 k z 2 b a m -1 2 k 1 2 -1 -1 d -1 m -1 2 k p 2 -1 4 b -1 a m -1 2 k e -1 m -1 2 k q m 2 -1 4 b -1 a m -1 2 k -1 b a m -1 2 k b -1 a m -1 2 k 1 2 Erfi d -1 m -1 2 k p 2 b z -1 2 a m -1 2 k z 2 b -1 a m -1 2 k 1 2 -1 b -1 a m -1 2 k b a m -1 2 k -1 m 2 -1 m -1 v -1 Binomial m m 2 -1 1 -1 \$CellContext`m 2 1 2 s 0 v -1 2 -1 Binomial v s -1 d f v -1 2 s 2 -1 4 b c v -1 2 s e g v -1 2 s 4 b c v -1 2 s -1 b c v -1 2 s 1 2 b -1 c v -1 2 s Erfi d f v -1 2 s 2 b c v -1 2 s z 2 b c v -1 2 s 1 2 -1 -1 d -1 f v -1 2 s 2 -1 4 b -1 c v -1 2 s e -1 g v -1 2 s 4 b -1 c v -1 2 s -1 b c v -1 2 s b -1 c v -1 2 s 1 2 Erfi d -1 f v -1 2 s 2 b z -1 2 c v -1 2 s z 2 b -1 c v -1 2 s 1 2 -1 b -1 c v -1 2 s b c v -1 2 s -1 m 2 -1 m -1 v -1 1 2 k 0 m -1 2 -1 -1 k Binomial m k s 0 v -1 2 -1 Binomial v s -1 d 2 k -1 m p f v -1 2 s 2 -1 4 b a 2 k -1 m c v -1 2 s e 2 k -1 m q g v -1 2 s m 2 -1 4 b a 2 k -1 m c v -1 2 s -1 b a 2 k -1 m c v -1 2 s 1 2 b -1 a 2 k -1 m -1 c v -1 2 s Erfi d 2 k -1 m p f v -1 2 s 2 b a 2 k -1 m c v -1 2 s z 2 b a 2 k -1 m c v -1 2 s 1 2 -1 -1 d -1 2 k -1 m p -1 f v -1 2 s 2 -1 4 b -1 a 2 k -1 m -1 c v -1 2 s e -1 2 k -1 m q -1 g v -1 2 s -1 m 2 -1 4 b -1 a 2 k -1 m -1 c v -1 2 s -1 b a 2 k -1 m c v -1 2 s b -1 a 2 k -1 m -1 c v -1 2 s 1 2 Erfi d -1 2 k -1 m p -1 f v -1 2 s 2 b z -1 2 a 2 k -1 m c v -1 2 s z 2 b -1 a 2 k -1 m -1 c v -1 2 s 1 2 -1 b -1 a 2 k -1 m -1 c v -1 2 s b a 2 k -1 m c v -1 2 s -1 -1 d m -1 2 k p f v -1 2 s 2 -1 4 b a m -1 2 k c v -1 2 s e m -1 2 k q g v -1 2 s -1 m 2 -1 4 b a m -1 2 k c v -1 2 s -1 b a m -1 2 k c v -1 2 s 1 2 b -1 a m -1 2 k -1 c v -1 2 s Erfi d m -1 2 k p f v -1 2 s 2 b a m -1 2 k c v -1 2 s z 2 b a m -1 2 k c v -1 2 s 1 2 -1 -1 d -1 m -1 2 k p -1 f v -1 2 s 2 -1 4 b -1 a m -1 2 k -1 c v -1 2 s e -1 m -1 2 k q -1 g v -1 2 s m 2 -1 4 b -1 a m -1 2 k -1 c v -1 2 s -1 b a m -1 2 k c v -1 2 s b -1 a m -1 2 k -1 c v -1 2 s 1 2 Erfi d -1 m -1 2 k p -1 f v -1 2 s 2 b z -1 2 a m -1 2 k c v -1 2 s z 2 b -1 a m -1 2 k -1 c v -1 2 s 1 2 -1 b -1 a m -1 2 k -1 c v -1 2 s b a m -1 2 k c v -1 2 s -1 m SuperPlus v SuperPlus [/itex]

 Rule Form

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

 2002-12-18