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; }

 Cosh

 http://functions.wolfram.com/01.20.21.4406.01

 Input Form

 Integrate[Sqrt[a + b Sinh[e z] + c Cosh[e z]], z] == (b (b^2 - c^2) Sqrt[1 - c^2/b^2] AppellF1[-(1/2), -(1/2), -(1/2), 1/2, (a + c Cosh[e z] + b Sinh[e z])/(a + Sqrt[1 - b^2/c^2] c), (a + c Cosh[e z] + b Sinh[e z])/(a - Sqrt[1 - b^2/c^2] c)] Sinh[e z + ArcTanh[b/c]] + Sqrt[((b^2 - c (a Sqrt[1 - b^2/c^2] + c)) (1 + Cosh[e z + ArcTanh[b/c]]))/ (a^2 + b^2 - c^2)] Sqrt[(1/(a^2 + b^2 - c^2)) (b^2 + a Sqrt[1 - b^2/c^2] c - c^2 + c (-a + Sqrt[1 - b^2/c^2] c) Cosh[e z] - b (a - Sqrt[1 - b^2/c^2] c) Sinh[e z])] (Sqrt[1 - c^2/b^2] (2 Sqrt[1 - b^2/c^2] c^3 Cosh[e z] + 2 c (b^2 - c^2) Cosh[e z + ArcTanh[b/c]] + b (2 Sqrt[1 - b^2/c^2] c^2 Sinh[e z] + (-b^2 + c^2) Sinh[e z + ArcTanh[b/c]])) + 2 a Sqrt[1 - b^2/c^2] c AppellF1[1/2, 1/2, 1/2, 3/2, (a + c Cosh[e z] + b Sinh[e z])/ (a + I b Sqrt[1 - c^2/b^2]), (a + c Cosh[e z] + b Sinh[e z])/ (a - I b Sqrt[1 - c^2/b^2])] Sech[e z + ArcTanh[c/b]] (a + c Cosh[e z] + b Sinh[e z]) Sqrt[-((1/(a^2 + b^2 - c^2)) (-b^2 + c^2 - I a b Sqrt[1 - c^2/b^2] + c (a - I b Sqrt[1 - c^2/b^2]) Cosh[e z] + b (a - I b Sqrt[1 - c^2/b^2]) Sinh[e z]))] Sqrt[(1/(a^2 + b^2 - c^2)) ((b^2 - c^2 - I a b Sqrt[1 - c^2/b^2]) (Cosh[(1/2) (e z + ArcTanh[c/b])] - I Sinh[(1/2) (e z + ArcTanh[c/b])])^2)]))/ (b Sqrt[1 - b^2/c^2] c Sqrt[1 - c^2/b^2] e Sqrt[(Sqrt[1 - b^2/c^2] c (1 + Cosh[e z + ArcTanh[b/c]]))/ (-a + Sqrt[1 - b^2/c^2] c)] Sqrt[(Sqrt[1 - b^2/c^2] c - c Cosh[e z] - b Sinh[e z])/ (a + Sqrt[1 - b^2/c^2] c)] Sqrt[a + c Cosh[e z] + b Sinh[e z]])

 Standard Form

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"c"]], "-", RowBox[List["c", " ", RowBox[List["Cosh", "[", RowBox[List["e", " ", "z"]], "]"]]]], "-", RowBox[List["b", " ", RowBox[List["Sinh", "[", RowBox[List["e", " ", "z"]], "]"]]]]]], RowBox[List["a", "+", RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox[SuperscriptBox["b", "2"], SuperscriptBox["c", "2"]]]]], " ", "c"]]]]]], " ", SqrtBox[RowBox[List["a", "+", RowBox[List["c", " ", RowBox[List["Cosh", "[", RowBox[List["e", " ", "z"]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Sinh", "[", RowBox[List["e", " ", "z"]], "]"]]]]]]]]], ")"]]]]]]]]

 MathML Form

 a + b sinh ( e z ) + c cosh ( e z ) z ( b ( b 2 - c 2 ) 1 - c 2 b 2 F 1 AppellF1 ( - 1 2 ; - 1 2 , - 1 2 ; 1 2 ; a + c cosh ( e z ) + b sinh ( e z ) a + 1 - b 2 c 2 c , a + c cosh ( e z ) + b sinh ( e z ) a - 1 - b 2 c 2 c ) sinh ( e z + tanh - 1 ( b c ) ) + ( b 2 - c ( 1 - b 2 c 2 a + c ) ) ( cosh ( e z + tanh - 1 ( b c ) ) + 1 ) a 2 + b 2 - c 2 ( 1 a 2 + b 2 - c 2 ( b 2 - ( a - 1 - b 2 c 2 c ) sinh ( e z ) b - c 2 + a 1 - b 2 c 2 c + c ( 1 - b 2 c 2 c - a ) cosh ( e z ) ) ) ( 2 a 1 - b 2 c 2 c F 1 AppellF1 ( 1 2 ; 1 2 , 1 2 ; 3 2 ; a + c cosh ( e z ) + b sinh ( e z ) a + b 1 - c 2 b 2 , a + c cosh ( e z ) + b sinh ( e z ) a - b 1 - c 2 b 2 ) sech ( e z + tanh - 1 ( c b ) ) ( - 1 a 2 + b 2 - c 2 ( - b 2 + ( a - b 1 - c 2 b 2 ) sinh ( e z ) b - a 1 - c 2 b 2 b + c 2 + c ( a - b 1 - c 2 b 2 ) cosh ( e z ) ) ) ( 1 a 2 + b 2 - c 2 ( ( b 2 - a 1 - c 2 b 2 b - c 2 ) ( cosh ( 1 2 ( e z + tanh - 1 ( c b ) ) ) - sinh ( 1 2 ( e z + tanh - 1 ( c b ) ) ) ) 2 ) ) ( a + c cosh ( e z ) + b sinh ( e z ) ) + 1 - c 2 b 2 ( 2 1 - b 2 c 2 cosh ( e z ) c 3 + 2 ( b 2 - c 2 ) cosh ( e z + tanh - 1 ( b c ) ) c + b ( 2 1 - b 2 c 2 sinh ( e z ) c 2 + ( c 2 - b 2 ) sinh ( e z + tanh - 1 ( b c ) ) ) ) ) ) / ( b 1 - b 2 c 2 c 1 - c 2 b 2 e 1 - b 2 c 2 c ( cosh ( e z + tanh - 1 ( b c ) ) + 1 ) 1 - b 2 c 2 c - a - cosh ( e z ) c + 1 - b 2 c 2 c - b sinh ( e z ) a + 1 - b 2 c 2 c a + c cosh ( e z ) + b sinh ( e z ) ) z a b e z c e z 1 2 b b 2 -1 c 2 1 -1 c 2 b 2 -1 1 2 AppellF1 -1 1 2 -1 1 2 -1 1 2 1 2 a c e z b e z a 1 -1 b 2 c 2 -1 1 2 c -1 a c e z b e z a -1 1 -1 b 2 c 2 -1 1 2 c -1 e z b c -1 b 2 -1 c 1 -1 b 2 c 2 -1 1 2 a c e z b c -1 1 a 2 b 2 -1 c 2 -1 1 2 1 a 2 b 2 -1 c 2 -1 b 2 -1 a -1 1 -1 b 2 c 2 -1 1 2 c e z b -1 c 2 a 1 -1 b 2 c 2 -1 1 2 c c 1 -1 b 2 c 2 -1 1 2 c -1 a e z 2 a 1 -1 b 2 c 2 -1 1 2 c AppellF1 1 2 1 2 1 2 3 2 a c e z b e z a b 1 -1 c 2 b 2 -1 1 2 -1 a c e z b e z a -1 b 1 -1 c 2 b 2 -1 1 2 -1 e z c b -1 -1 1 a 2 b 2 -1 c 2 -1 -1 b 2 a -1 b 1 -1 c 2 b 2 -1 1 2 e z b -1 a 1 -1 c 2 b 2 -1 1 2 b c 2 c a -1 b 1 -1 c 2 b 2 -1 1 2 e z 1 a 2 b 2 -1 c 2 -1 b 2 -1 a 1 -1 c 2 b 2 -1 1 2 b -1 c 2 1 2 e z c b -1 -1 1 2 e z c b -1 2 a c e z b e z 1 -1 c 2 b 2 -1 1 2 2 1 -1 b 2 c 2 -1 1 2 e z c 3 2 b 2 -1 c 2 e z b c -1 c b 2 1 -1 b 2 c 2 -1 1 2 e z c 2 c 2 -1 b 2 e z b c -1 b 1 -1 b 2 c 2 -1 1 2 c 1 -1 c 2 b 2 -1 1 2 e 1 -1 b 2 c 2 -1 1 2 c e z b c -1 1 1 -1 b 2 c 2 -1 1 2 c -1 a -1 1 2 -1 e z c 1 -1 b 2 c 2 -1 1 2 c -1 b e z a 1 -1 b 2 c 2 -1 1 2 c -1 1 2 a c e z b e z 1 2 -1 [/itex]

 Rule Form

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

 2002-12-18