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 Cosh

 http://functions.wolfram.com/01.20.21.4408.01

 Input Form

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

 Standard Form

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 MathML Form

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