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Cosh






Mathematica Notation

Traditional Notation









Elementary Functions > Cosh[z] > Integration > Indefinite integration > Involving functions of the direct function > Involving algebraic functions of the direct function > Involving (a+b cosh(c z))betaand rational function of cosh(c z)





http://functions.wolfram.com/01.20.21.2339.01









  


  










Input Form





Integrate[Sqrt[a + b Cosh[c z]]/(d + e Cosh[c z])^2, z] == (-((I (4 a d - b e) Sqrt[(a + b Cosh[c z])/(a + b)])/ (2 c (d - e) (d + e)^2 Sqrt[a + b Cosh[c z]]))) EllipticPi[(2 e)/(d + e), (I c z)/2, (2 b)/(a + b)] + ((1/(b c (d^2 - e^2))) (I Sqrt[-(1/(a + b))] (-a^2 + b^2) Sqrt[(b (1 + Cosh[c z]))/(-a + b)] Sqrt[(b - b Cosh[c z])/(a + b)] Csch[c z])) EllipticE[I ArcSinh[Sqrt[-(1/(a + b))] Sqrt[a + b Cosh[c z]]], (a + b)/(a - b)] - ((I Sqrt[(-b - b Cosh[c z])/(a - b)] Sqrt[(b - b Cosh[c z])/(a + b)] Csch[c z])/(Sqrt[-(1/(a + b))] c e (d + e))) EllipticF[I ArcSinh[Sqrt[-(1/(a + b))] Sqrt[a + b Cosh[c z]]], (a + b)/(a - b)] - ((I b (2 d^2 + e^2) Sqrt[(b (1 + Cosh[c z]))/(-a + b)] Sqrt[(b - b Cosh[c z])/(a + b)] Csch[c z])/(2 Sqrt[-(1/(a + b))] c e (b d - a e) (-d^2 + e^2))) EllipticPi[((a + b) e)/((-b) d + a e), I ArcSinh[Sqrt[-(1/(a + b))] Sqrt[a + b Cosh[c z]]], (a + b)/(a - b)] - (e Sqrt[a + b Cosh[c z]] Sinh[c z])/(c (d - e) (d + e) (d + e Cosh[c z]))










Standard Form





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MathML Form







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<mi> b </mi> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <mrow> <mi> csch </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> F </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> sinh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <msqrt> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> </mfrac> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#10072; </mo> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <msqrt> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> </mfrac> </mrow> </msqrt> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> e </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> + </mo> <mi> e </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> - </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> a </mi> <mo> &#8290; </mo> <mi> d </mi> </mrow> <mo> - </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> e </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msqrt> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mi> a </mi> <mo> + </mo> <mi> b 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Date Added to functions.wolfram.com (modification date)





2002-12-18