Sine: Integration (formula 01.06.21.0989)

Sin

$Mathematica Notation$

Elementary Functions

Sin[z]

Integration

Indefinite integration

Involving functions of the direct function

Involving algebraic functions of the direct function

Involving (a+b sin²(c z))^betaand rational function of sin(c z)

http://functions.wolfram.com/01.06.21.0989.01

Input Form

Integrate[1/((d + e Sin[c z]) Sqrt[a + b Sin[c z]^2]), z] == Sqrt[(2 a + b - b Cos[2 c z])/a] (((-(1/(Sqrt[1 - d^2/e^2] Sqrt[1 + (b d^2)/(a e^2)]))) ArcTanh[(Cos[c z]/(Sqrt[1 - d^2/e^2] Sqrt[(2 a + b - b Cos[2 c z])/a])) Sqrt[2 + (2 b d^2)/(a e^2)]] + (e/d) EllipticPi[e^2/d^2, c z, -(b/a)])/ (c e Sqrt[2 a + b - b Cos[2 c z]]))

Standard Form

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MathML Form

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2 </cn> <ci> a </ci> </apply> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> b </ci> <apply> <cos /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <ci> e </ci> <apply> <power /> <ci> d </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> EllipticPi </ci> <apply> <times /> <apply> <power /> <ci> e </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> d </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> 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Rule Form

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

2002-12-18