html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 Tanh

 http://functions.wolfram.com/01.21.21.0365.01

 Input Form

 Integrate[Cosh[c z]/Sqrt[a + b Tanh[c z]^2], z] == ((1 + Cosh[c z]) Sqrt[(a - b + (a + b) Cosh[2 c z])/(1 + Cosh[c z])^2] Sech[c z] (2 (a + 2 (b + Sqrt[b (a + b)])) EllipticE[ArcSin[Sqrt[-((a + 2 (b + Sqrt[b (a + b)]))/a)] Tanh[(c z)/2]], (a + 2 b - 2 Sqrt[b (a + b)])/ (a + 2 (b + Sqrt[b (a + b)]))] Sqrt[(1/a) ((-b + Sqrt[b (a + b)] + (a + b - Sqrt[b (a + b)]) Cosh[c z]) Sech[(c z)/2]^2)] Sqrt[(1/a) ((-b - Sqrt[b (a + b)] + (a + b + Sqrt[b (a + b)]) Cosh[c z]) Sech[(c z)/2]^2)] - 4 Sqrt[b (a + b)] EllipticF[ ArcSin[Sqrt[-((a + 2 (b + Sqrt[b (a + b)]))/a)] Tanh[(c z)/2]], (a + 2 b - 2 Sqrt[b (a + b)])/(a + 2 (b + Sqrt[b (a + b)]))] Sqrt[(1/a) ((-b + Sqrt[b (a + b)] + (a + b - Sqrt[b (a + b)]) Cosh[c z]) Sech[(c z)/2]^2)] Sqrt[(1/a) ((-b - Sqrt[b (a + b)] + (a + b + Sqrt[b (a + b)]) Cosh[c z]) Sech[(c z)/2]^2)] + Sqrt[-((a + 2 (b + Sqrt[b (a + b)]))/a)] (a - b + (a + b) Cosh[2 c z]) Sech[(c z)/2]^2 Tanh[(c z)/2]))/ (2 (a + b) Sqrt[-((a + 2 (b + Sqrt[b (a + b)]))/a)] c Sqrt[(a - b + (a + b) Cosh[2 c z]) Sech[(c z)/2]^4] Sqrt[a + b Tanh[c z]^2])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]], SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox[RowBox[List["Tanh", "[", RowBox[List["c", " ", "z"]], "]"]], "2"]]]]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]]]], ")"]], " ", SqrtBox[FractionBox[RowBox[List["a", "-", "b", "+", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]]]], ")"]], "2"]]], " ", RowBox[List["Sech", "[", RowBox[List["c", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["a", "+", RowBox[List["2", " ", 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 MathML Form

 cosh ( c z ) a + b tanh 2 ( c z ) z ( ( cosh ( c z ) + 1 ) a - b + ( a + b ) cosh ( 2 c z ) ( cosh ( c z ) + 1 ) 2 sech ( c z ) ( - a + 2 ( b + b ( a + b ) ) a ( a - b + ( a + b ) cosh ( 2 c z ) ) tanh ( c z 2 ) sech 2 ( c z 2 ) + 2 ( a + 2 ( b + b ( a + b ) ) ) E ( sin - 1 ( - a + 2 ( b + b ( a + b ) ) a tanh ( c z 2 ) ) a + 2 b - 2 b ( a + b ) a + 2 ( b + b ( a + b ) ) ) ( - b + ( a + b - b ( a + b ) ) cosh ( c z ) + b ( a + b ) ) sech 2 ( c z 2 ) a ( - b + ( a + b + b ( a + b ) ) cosh ( c z ) - b ( a + b ) ) sech 2 ( c z 2 ) a - 4 b ( a + b ) F ( sin - 1 ( - a + 2 ( b + b ( a + b ) ) a tanh ( c z 2 ) ) a + 2 b - 2 b ( a + b ) a + 2 ( b + b ( a + b ) ) ) ( - b + ( a + b - b ( a + b ) ) cosh ( c z ) + b ( a + b ) ) sech 2 ( c z 2 ) a ( - b + ( a + b + b ( a + b ) ) cosh ( c z ) - b ( a + b ) ) sech 2 ( c z 2 ) a ) ) / ( 2 ( a + b ) - a + 2 ( b + b ( a + b ) ) a c ( a - b + ( a + b ) cosh ( 2 c z ) ) sech 4 ( c z 2 ) b tanh 2 ( c z ) + a ) z c z a b c z 2 1 2 -1 c z 1 a -1 b a b 2 c z c z 1 2 -1 1 2 c z -1 a 2 b b a b 1 2 a -1 1 2 a -1 b a b 2 c z c z 2 -1 c z 2 -1 2 2 a 2 b b a b 1 2 EllipticE -1 a 2 b b a b 1 2 a -1 1 2 c z 2 -1 a 2 b -1 2 b a b 1 2 a 2 b b a b 1 2 -1 -1 b a b -1 b a b 1 2 c z b a b 1 2 c z 2 -1 2 a -1 1 2 -1 b a b b a b 1 2 c z -1 b a b 1 2 c z 2 -1 2 a -1 1 2 -1 4 b a b 1 2 EllipticF -1 a 2 b b a b 1 2 a -1 1 2 c z 2 -1 a 2 b -1 2 b a b 1 2 a 2 b b a b 1 2 -1 -1 b a b -1 b a b 1 2 c z b a b 1 2 c z 2 -1 2 a -1 1 2 -1 b a b b a b 1 2 c z -1 b a b 1 2 c z 2 -1 2 a -1 1 2 2 a b -1 a 2 b b a b 1 2 a -1 1 2 c a -1 b a b 2 c z c z 2 -1 4 1 2 b c z 2 a 1 2 -1 [/itex]

 Rule Form

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

 2002-12-18