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 Coth

 http://functions.wolfram.com/01.22.21.0175.01

 Input Form

 Integrate[Sin[a z]^m Sinh[b z]^u Coth[c z], z] == (1/c) (I^u 2^(-m - u) Binomial[m, m/2] Binomial[u, u/2] Log[Sinh[c z]] (1 - Mod[m, 2]) (1 - Mod[u, 2])) - I^(u + 1) 2^(-m - u) Binomial[u, u/2] (1 - Mod[u, 2]) Sum[(-1)^k Binomial[m, k] ((E^((I m Pi)/2 - I a (-2 k + m) z) ((2 c - I a (-2 k + m)) Hypergeometric2F1[-((I a (-2 k + m))/(2 c)), 1, 1 - (I a (-2 k + m))/(2 c), E^(2 c z)] - I a E^(2 c z) (-2 k + m) Hypergeometric2F1[1 - (I a (-2 k + m))/(2 c), 1, 2 - (I a (-2 k + m))/(2 c), E^(2 c z)]))/ (a (-2 k + m) (2 c - I a (-2 k + m))) - (E^((-(1/2)) I m Pi + I a (-2 k + m) z) ((2 c + I a (-2 k + m)) Hypergeometric2F1[(I a (-2 k + m))/(2 c), 1, 1 + (I a (-2 k + m))/(2 c), E^(2 c z)] + I a E^(2 c z) (-2 k + m) Hypergeometric2F1[1 + (I a (-2 k + m))/(2 c), 1, 2 + (I a (-2 k + m))/(2 c), E^(2 c z)]))/ (a (-2 k + m) (2 c + I a (-2 k + m)))), {k, 0, Floor[(1/2) (-1 + m)]}] - 2^(-m - u) Binomial[m, m/2] (1 - Mod[m, 2]) Sum[(-1)^k Binomial[u, k] (-(((-1)^u ((2 c - b (-2 k + u)) Hypergeometric2F1[-((b (-2 k + u))/ (2 c)), 1, 1 - (b (-2 k + u))/(2 c), E^(2 c z)] - b E^(2 c z) (-2 k + u) Hypergeometric2F1[1 - (b (-2 k + u))/ (2 c), 1, 2 - (b (-2 k + u))/(2 c), E^(2 c z)]))/ E^(b (-2 k + u) z)/(b (-2 k + u) (2 c - b (-2 k + u)))) + (E^(b (-2 k + u) z) ((2 c + b (-2 k + u)) Hypergeometric2F1[ (b (-2 k + u))/(2 c), 1, 1 + (b (-2 k + u))/(2 c), E^(2 c z)] + b E^(2 c z) (-2 k + u) Hypergeometric2F1[1 + (b (-2 k + u))/(2 c), 1, 2 + (b (-2 k + u))/(2 c), E^(2 c z)]))/ (b (-2 k + u) (2 c + b (-2 k + u)))), {k, 0, Floor[(1/2) (-1 + u)]}] - 2^(-m - u) Sum[(-1)^(k + s) Binomial[m, k] Binomial[u, s] (((-1)^u E^((I m Pi)/2 + ((-I) a (-2 k + m) - b (-2 s + u)) z) ((2 c - I a (-2 k + m) - b (-2 s + u)) Hypergeometric2F1[ ((-I) a (-2 k + m) - b (-2 s + u))/(2 c), 1, 1 + ((-I) a (-2 k + m) - b (-2 s + u))/(2 c), E^(2 c z)] + E^(2 c z) ((-I) a (-2 k + m) - b (-2 s + u)) Hypergeometric2F1[ 1 + ((-I) a (-2 k + m) - b (-2 s + u))/(2 c), 1, 2 + ((-I) a (-2 k + m) - b (-2 s + u))/(2 c), E^(2 c z)]))/ (((-I) a (-2 k + m) - b (-2 s + u)) (2 c - I a (-2 k + m) - b (-2 s + u))) + ((-1)^u E^((-(1/2)) I m Pi + (I a (-2 k + m) - b (-2 s + u)) z) ((2 c + I a (-2 k + m) - b (-2 s + u)) Hypergeometric2F1[ (I a (-2 k + m) - b (-2 s + u))/(2 c), 1, 1 + (I a (-2 k + m) - b (-2 s + u))/(2 c), E^(2 c z)] + E^(2 c z) (I a (-2 k + m) - b (-2 s + u)) Hypergeometric2F1[ 1 + (I a (-2 k + m) - b (-2 s + u))/(2 c), 1, 2 + (I a (-2 k + m) - b (-2 s + u))/(2 c), E^(2 c z)]))/ ((I a (-2 k + m) - b (-2 s + u)) (2 c + I a (-2 k + m) - b (-2 s + u))) + (E^((I m Pi)/2 + ((-I) a (-2 k + m) + b (-2 s + u)) z) ((2 c - I a (-2 k + m) + b (-2 s + u)) Hypergeometric2F1[ ((-I) a (-2 k + m) + b (-2 s + u))/(2 c), 1, 1 + ((-I) a (-2 k + m) + b (-2 s + u))/(2 c), E^(2 c z)] + E^(2 c z) ((-I) a (-2 k + m) + b (-2 s + u)) Hypergeometric2F1[ 1 + ((-I) a (-2 k + m) + b (-2 s + u))/(2 c), 1, 2 + ((-I) a (-2 k + m) + b (-2 s + u))/(2 c), E^(2 c z)]))/ (((-I) a (-2 k + m) + b (-2 s + u)) (2 c - I a (-2 k + m) + b (-2 s + u))) + (E^((-(1/2)) I m Pi + (I a (-2 k + m) + b (-2 s + u)) z) ((2 c + I a (-2 k + m) + b (-2 s + u)) Hypergeometric2F1[ (I a (-2 k + m) + b (-2 s + u))/(2 c), 1, 1 + (I a (-2 k + m) + b (-2 s + u))/(2 c), E^(2 c z)] + E^(2 c z) (I a (-2 k + m) + b (-2 s + u)) Hypergeometric2F1[ 1 + (I a (-2 k + m) + b (-2 s + u))/(2 c), 1, 2 + (I a (-2 k + m) + b (-2 s + u))/(2 c), E^(2 c z)]))/ ((I a (-2 k + m) + b (-2 s + u)) (2 c + I a (-2 k + m) + b (-2 s + u)))), {s, 0, Floor[(1/2) (-1 + u)]}, {k, 0, Floor[(1/2) (-1 + m)]}] /; Element[m, Integers] && m > 0 && Element[u, Integers] && u > 0

 Standard Form

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 MathML Form

 sin m ( a z ) sinh u ( b z ) coth ( c z ) z u 2 - m - u log ( sinh ( c z ) ) ( 1 - m mod 2 \$CellContext`m 2 ) ( 1 - u mod 2 \$CellContext`u 2 ) c ( 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]]]]] ( u u 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["u", Identity]], List[TagBox[FractionBox["u", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] - u + 1 2 - m - u ( u u 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["u", Identity]], List[TagBox[FractionBox["u", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 1 - u mod 2 \$CellContext`u 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]]]]] ( ( m π 2 - a ( m - 2 k ) z ( ( 2 c - a ( m - 2 k ) ) 2 F 1 ( - a ( m - 2 k ) 2 c , 1 ; 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m + u + Condition z a z m b z u c z u 2 -1 m -1 u c z 1 -1 \$CellContext`m 2 1 -1 \$CellContext`u 2 c -1 Binomial m m 2 -1 Binomial u u 2 -1 -1 u 1 2 -1 m -1 u Binomial u u 2 -1 1 -1 \$CellContext`u 2 k 0 m -1 2 -1 -1 k Binomial m k m 2 -1 -1 a m -1 2 k z 2 c -1 a m -1 2 k Hypergeometric2F1 -1 a m -1 2 k 2 c -1 1 1 -1 a m -1 2 k 2 c -1 2 c z -1 a 2 c z m -1 2 k Hypergeometric2F1 1 -1 a m -1 2 k 2 c -1 1 2 -1 a m -1 2 k 2 c -1 2 c z a m -1 2 k 2 c -1 a m -1 2 k -1 -1 a m -1 2 k z -1 m 2 -1 2 c a m -1 2 k Hypergeometric2F1 a m -1 2 k 2 c -1 1 a m -1 2 k 2 c -1 1 2 c z a 2 c z m -1 2 k Hypergeometric2F1 a m -1 2 k 2 c -1 1 1 a m -1 2 k 2 c -1 2 2 c z a m -1 2 k 2 c a m -1 2 k -1 -1 2 -1 m -1 u Binomial m m 2 -1 1 -1 \$CellContext`m 2 k 0 u -1 2 -1 -1 k Binomial u k b u -1 2 k z 2 c b u -1 2 k Hypergeometric2F1 b u -1 2 k 2 c -1 1 b u -1 2 k 2 c -1 1 2 c z b 2 c z u -1 2 k Hypergeometric2F1 b u -1 2 k 2 c -1 1 1 b u -1 2 k 2 c -1 2 2 c z b u -1 2 k 2 c b u -1 2 k -1 -1 -1 u -1 b u -1 2 k z 2 c -1 b u -1 2 k Hypergeometric2F1 -1 b u -1 2 k 2 c -1 1 1 -1 b u -1 2 k 2 c -1 2 c z -1 b 2 c z u -1 2 k Hypergeometric2F1 1 -1 b u -1 2 k 2 c -1 1 2 -1 b u -1 2 k 2 c -1 2 c z b u -1 2 k 2 c -1 b u -1 2 k -1 -1 2 -1 m -1 u k 0 m -1 2 -1 s 0 u -1 2 -1 -1 k s Binomial m k Binomial u s m 2 -1 b u -1 2 s -1 a m -1 2 k z 2 c -1 a m -1 2 k b u -1 2 s Hypergeometric2F1 b u -1 2 s -1 a m -1 2 k 2 c -1 1 b u -1 2 s -1 a m -1 2 k 2 c -1 1 2 c z 2 c z b u -1 2 s -1 a m -1 2 k Hypergeometric2F1 b u -1 2 s -1 a m -1 2 k 2 c -1 1 1 b u -1 2 s -1 a m -1 2 k 2 c -1 2 2 c z b u -1 2 s -1 a m -1 2 k 2 c -1 a m -1 2 k b u -1 2 s -1 a m -1 2 k b u -1 2 s z -1 m 2 -1 2 c a m -1 2 k b u -1 2 s Hypergeometric2F1 a m -1 2 k b u -1 2 s 2 c -1 1 a m -1 2 k b u -1 2 s 2 c -1 1 2 c z 2 c z a m -1 2 k b u -1 2 s Hypergeometric2F1 a m -1 2 k b u -1 2 s 2 c -1 1 1 a m -1 2 k b u -1 2 s 2 c -1 2 2 c z a m -1 2 k b u -1 2 s 2 c a m -1 2 k b u -1 2 s -1 -1 u m 2 -1 -1 a m -1 2 k -1 b u -1 2 s z 2 c -1 a m -1 2 k -1 b u -1 2 s Hypergeometric2F1 -1 a m -1 2 k -1 b u -1 2 s 2 c -1 1 -1 a m -1 2 k -1 b u -1 2 s 2 c -1 1 2 c z 2 c z -1 a m -1 2 k -1 b u -1 2 s Hypergeometric2F1 -1 a m -1 2 k -1 b u -1 2 s 2 c -1 1 1 -1 a m -1 2 k -1 b u -1 2 s 2 c -1 2 2 c z -1 a m -1 2 k -1 b u -1 2 s 2 c -1 a m -1 2 k -1 b u -1 2 s -1 -1 u a m -1 2 k -1 b u -1 2 s z -1 m 2 -1 2 c a m -1 2 k -1 b u -1 2 s Hypergeometric2F1 a m -1 2 k -1 b u -1 2 s 2 c -1 1 a m -1 2 k -1 b u -1 2 s 2 c -1 1 2 c z 2 c z a m -1 2 k -1 b u -1 2 s Hypergeometric2F1 a m -1 2 k -1 b u -1 2 s 2 c -1 1 1 a m -1 2 k -1 b u -1 2 s 2 c -1 2 2 c z a m -1 2 k -1 b u -1 2 s 2 c a m -1 2 k -1 b u -1 2 s -1 m SuperPlus u SuperPlus [/itex]

 Rule Form

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

 2002-12-18

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