Hyperbolic sine: Integration (formula 01.19.21.1335)

Sinh

$Mathematica Notation$

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 e^{p z}sinh(d z)(a+b sin²(e z)+c cos²(e z))^-n

http://functions.wolfram.com/01.19.21.1335.01

Input Form

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

Standard Form

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MathML Form

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- </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox[RowBox[List["d", "+", "p"]], RowBox[List["2", " ", "e"]]], "+", "1"]], Hypergeometric2F1], ",", TagBox["2", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox[RowBox[List["d", "+", "p"]], RowBox[List["2", " ", "e"]]], "+", "2"]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", 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<mo> + </mo> <mn> 2 </mn> </mrow> <mo> ; </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> </mrow> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> + </mo> <mi> b </mi> <mo> - </mo> <mi> c </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", 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Rule Form

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

2002-12-18