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Sinh






Mathematica Notation

Traditional Notation









Elementary Functions > Sinh[z] > Integration > Indefinite integration > Involving one direct function and elementary functions > Involving trigonometric and exponential functions > Involving rational functions of sin, cos and exp > Involving ep zsinh(d z)(a sin2(e z)+b cos2(e z))-n





http://functions.wolfram.com/01.19.21.1333.01









  


  










Input Form





Integrate[(E^(p z) Sinh[d z])/(a Sin[e z]^2 + b Cos[e z]^2)^2, z] == (I ((1/(-d + 2 I e + p)) (E^((-d + 2 I e + p) z) ((-(Sqrt[-a] + I Sqrt[b])^2) (a + b) Hypergeometric2F1[ 1 + (I (d - p))/(2 e), 1, 2 + (I (d - p))/(2 e), ((-a + b) E^(2 I e z))/(Sqrt[-a] - I Sqrt[b])^2] + (Sqrt[-a] - I Sqrt[b])^2 (a + b) Hypergeometric2F1[ 1 + (I (d - p))/(2 e), 1, 2 + (I (d - p))/(2 e), ((-a + b) E^(2 I e z))/(Sqrt[-a] + I Sqrt[b])^2] - 2 I Sqrt[-a] Sqrt[b] ((Sqrt[-a] + I Sqrt[b])^2 Hypergeometric2F1[ 1 + (I (d - p))/(2 e), 2, 2 + (I (d - p))/(2 e), ((-a + b) E^(2 I e z))/(Sqrt[-a] - I Sqrt[b])^2] + (Sqrt[-a] - I Sqrt[b])^2 Hypergeometric2F1[1 + (I (d - p))/(2 e), 2, 2 + (I (d - p))/(2 e), ((-a + b) E^(2 I e z))/ (Sqrt[-a] + I Sqrt[b])^2]))) - (1/(d + 2 I e + p)) (E^((d + 2 I e + p) z) ((-(Sqrt[-a] + I Sqrt[b])^2) (a + b) Hypergeometric2F1[1 - (I (d + p))/(2 e), 1, 2 - (I (d + p))/(2 e), ((-a + b) E^(2 I e z))/(Sqrt[-a] - I Sqrt[b])^2] + (Sqrt[-a] - I Sqrt[b])^2 (a + b) Hypergeometric2F1[ 1 - (I (d + p))/(2 e), 1, 2 - (I (d + p))/(2 e), ((-a + b) E^(2 I e z))/(Sqrt[-a] + I Sqrt[b])^2] - 2 I Sqrt[-a] Sqrt[b] ((Sqrt[-a] + I Sqrt[b])^2 Hypergeometric2F1[ 1 - (I (d + p))/(2 e), 2, 2 - (I (d + p))/(2 e), ((-a + b) E^(2 I e z))/(Sqrt[-a] - I Sqrt[b])^2] + (Sqrt[-a] - I Sqrt[b])^2 Hypergeometric2F1[1 - (I (d + p))/(2 e), 2, 2 - (I (d + p))/(2 e), ((-a + b) E^(2 I e z))/ (Sqrt[-a] + I Sqrt[b])^2])))))/(4 (-a)^(3/2) b^(3/2) (-a + b))










Standard Form





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MathML Form







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InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], Hypergeometric2F1] </annotation> </semantics> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <msqrt> <mrow> <mo> - </mo> <mi> a </mi> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mi> b </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> + </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> e </mi> </mrow> </mfrac> </mrow> <mo> , </mo> <mn> 2 </mn> </mrow> <mo> ; </mo> <mrow> <mn> 2 </mn> <mo> - </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> + 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2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> 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</apply> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> <apply> <ci> Hypergeometric2F1 </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <plus /> <ci> d </ci> <ci> p </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> e </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 1 </cn> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 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Rule Form





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





2002-12-18