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Sinh






Mathematica Notation

Traditional Notation









Elementary Functions > Sinh[z] > Integration > Indefinite integration > Involving functions of the direct function and exponential function > Involving products of the direct function and exponential function > Involving products of two direct functions and exponential function > Involving eb zr+d z+e sinh(a zr+p z+q) sinh(c zr+f z+g)





http://functions.wolfram.com/01.19.21.2191.01









  


  










Input Form





Integrate[E^(b Sqrt[z] + d z + e) Sinh[a Sqrt[z] + p z + q] Sinh[c Sqrt[z] + f z + g], z] == (1/4) E^(e - g - q) (E^((-(a - b + c)) Sqrt[z] + (d - f - p) z)/ (d - f - p) - E^(2 g + (-a + b + c) Sqrt[z] + (d + f - p) z)/ (d + f - p) - E^(2 q + (a + b - c) Sqrt[z] + (d - f + p) z)/ (d - f + p) + E^(2 g + 2 q + (a + b + c) Sqrt[z] + (d + f + p) z)/ (d + f + p)) + (1/(8 (d + f - p)^(3/2))) (((a - b - c) Sqrt[Pi] Erfi[(a - b - c - 2 d Sqrt[z] - 2 f Sqrt[z] + 2 p Sqrt[z])/(2 Sqrt[d + f - p])])/ E^((a^2 + b^2 + 2 b c + c^2 - 2 a (b + c) - 4 d e - 4 e f - 4 d g - 4 f g + 4 e p + 4 g p + 4 d q + 4 f q - 4 p q)/(4 (d + f - p)))) + (1/(8 (d - f + p)^(3/2))) (((a + b - c) Sqrt[Pi] Erfi[(a + b - c + 2 d Sqrt[z] - 2 f Sqrt[z] + 2 p Sqrt[z])/ (2 Sqrt[d - f + p])])/E^((a^2 + b^2 + 2 a (b - c) - 2 b c + c^2 - 4 d e + 4 e f + 4 d g - 4 f g - 4 e p + 4 g p - 4 d q + 4 f q - 4 p q)/ (4 (d - f + p)))) - (1/(8 (d - f - p)^(3/2))) ((a - b + c) E^((a^2 - 2 a b + b^2 + 2 a c - 2 b c + c^2 - 4 d e + 4 e f + 4 d g - 4 f g + 4 e p - 4 g p + 4 d q - 4 f q - 4 p q)/ (4 (-d + f + p))) Sqrt[Pi] Erfi[(a - b + c - 2 d Sqrt[z] + 2 f Sqrt[z] + 2 p Sqrt[z])/ (2 Sqrt[d - f - p])]) - (1/(8 (d + f + p)^(3/2))) (((a + b + c) Sqrt[Pi] Erfi[(a + b + c + 2 d Sqrt[z] + 2 f Sqrt[z] + 2 p Sqrt[z])/(2 Sqrt[d + f + p])])/ E^((a^2 + b^2 + 2 b c + c^2 + 2 a (b + c) - 4 d e - 4 e f - 4 d g - 4 f g - 4 e p - 4 g p - 4 d q - 4 f q - 4 p q)/(4 (d + f + p))))










Standard Form





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MathML Form







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</msup> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mo> - </mo> <mfrac> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> + </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> c </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> b </mi> <mo> &#8290; </mo> <mi> c </mi> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> d </mi> <mo> &#8290; </mo> <mi> e </mi> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> e </mi> <mo> &#8290; </mo> <mi> f </mi> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> d </mi> <mo> &#8290; </mo> <mi> g </mi> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> f </mi> <mo> &#8290; </mo> <mi> g </mi> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> e </mi> <mo> &#8290; </mo> <mi> p </mi> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> g </mi> <mo> &#8290; </mo> <mi> p </mi> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> d </mi> <mo> &#8290; </mo> <mi> q </mi> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> f </mi> <mo> &#8290; </mo> <mi> q </mi> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> p </mi> <mo> &#8290; </mo> <mi> q </mi> </mrow> </mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> + </mo> <mi> f </mi> <mo> - </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <msqrt> <mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mi> erfi </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> <mo> - </mo> <mi> c </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> d </mi> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> f </mi> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> p </mi> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <mi> d </mi> <mo> + </mo> <mi> f </mi> <mo> - </mo> <mi> p </mi> </mrow> </msqrt> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> - </mo> <mi> f </mi> <mo> + </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mo> - </mo> <mfrac> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> + </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> c </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> b </mi> <mo> &#8290; </mo> <mi> c </mi> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> d </mi> <mo> &#8290; </mo> <mi> e </mi> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> e </mi> <mo> &#8290; </mo> <mi> f </mi> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> d </mi> <mo> &#8290; </mo> <mi> g </mi> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> f </mi> <mo> &#8290; </mo> <mi> g </mi> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> e </mi> <mo> &#8290; </mo> <mi> p </mi> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> g </mi> <mo> &#8290; </mo> <mi> p </mi> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> d </mi> <mo> &#8290; </mo> <mi> q </mi> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> f </mi> <mo> &#8290; </mo> <mi> q </mi> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> p </mi> <mo> &#8290; </mo> <mi> q </mi> </mrow> </mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> - </mo> <mi> f </mi> <mo> + </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <msqrt> <mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mi> erfi </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> - </mo> <mi> c </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> d </mi> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> f </mi> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> p </mi> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <mi> d </mi> <mo> - </mo> <mi> f </mi> <mo> + </mo> <mi> p </mi> </mrow> </msqrt> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> + </mo> <mi> f </mi> <mo> + </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> + </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mo> - </mo> <mfrac> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> + </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> c </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> b </mi> <mo> &#8290; </mo> <mi> c </mi> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> d </mi> <mo> &#8290; </mo> <mi> e </mi> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> e </mi> <mo> &#8290; </mo> <mi> f </mi> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> d </mi> <mo> &#8290; </mo> <mi> g </mi> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> f </mi> <mo> &#8290; </mo> <mi> g </mi> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> e </mi> <mo> &#8290; </mo> <mi> p </mi> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> g </mi> <mo> &#8290; </mo> <mi> p </mi> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> d </mi> <mo> &#8290; </mo> <mi> q </mi> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> f </mi> <mo> &#8290; </mo> <mi> q </mi> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> p </mi> <mo> &#8290; </mo> <mi> q </mi> </mrow> </mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> + </mo> <mi> f </mi> <mo> + </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <msqrt> <mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mi> erfi </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> + </mo> <mi> c </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> d </mi> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> f </mi> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> p </mi> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <mi> d </mi> <mo> + </mo> <mi> f </mi> <mo> + </mo> <mi> p </mi> </mrow> </msqrt> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> b </ci> </apply> <apply> <times /> <ci> d </ci> <ci> z </ci> </apply> <ci> e </ci> </apply> </apply> <apply> <sinh /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> a </ci> </apply> <apply> <times /> <ci> p </ci> <ci> z </ci> </apply> <ci> q </ci> </apply> </apply> <apply> <sinh /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> c </ci> </apply> <apply> <times /> <ci> f </ci> <ci> z </ci> </apply> <ci> g </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <power /> <exponentiale /> <apply> <plus /> <ci> e </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> g </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> q </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <ci> b </ci> <ci> c </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> g </ci> </apply> <apply> <times /> <apply> <plus /> <ci> d </ci> <ci> f </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> p </ci> </apply> </apply> <ci> z </ci> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> d </ci> <ci> f </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> p </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <ci> a </ci> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> q </ci> </apply> <apply> <times /> <apply> <plus /> <ci> d </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> f </ci> </apply> <ci> p </ci> </apply> <ci> z </ci> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> d </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> f </ci> </apply> <ci> p </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <ci> a </ci> <ci> b </ci> <ci> c </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> g </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> q </ci> </apply> <apply> <times /> <apply> <plus /> <ci> d </ci> <ci> f </ci> <ci> p </ci> </apply> <ci> z </ci> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> d </ci> <ci> f </ci> <ci> p </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> d </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> f </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> p </ci> </apply> </apply> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <ci> c </ci> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> d </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> f </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> p </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <apply> <plus /> <ci> d </ci> <apply> <times /> <cn 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Rule Form





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





2002-12-18