Sine: Integration (formula 01.06.21.0997)

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/01.06.21.0997.01

Input Form

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

Standard Form

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MathML Form

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

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

2002-12-18