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http://functions.wolfram.com/01.20.21.4228.01
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Integrate[Sinh[d z]/(a + b Cosh[c z])^2, z] ==
(1/(2 b (a^2 - b^2)^(3/2)))
((1/(c + d)) (E^((c + d) z) (a (a + Sqrt[a^2 - b^2])
Hypergeometric2F1[(c + d)/c, 1, 2 + d/c, (b E^(c z))/
(-a + Sqrt[a^2 - b^2])] + a (-a + Sqrt[a^2 - b^2])
Hypergeometric2F1[(c + d)/c, 1, 2 + d/c,
-((b E^(c z))/(a + Sqrt[a^2 - b^2]))] -
a^2 Hypergeometric2F1[(c + d)/c, 2, 2 + d/c,
(b E^(c z))/(-a + Sqrt[a^2 - b^2])] +
b^2 Hypergeometric2F1[(c + d)/c, 2, 2 + d/c,
(b E^(c z))/(-a + Sqrt[a^2 - b^2])] - a Sqrt[a^2 - b^2]
Hypergeometric2F1[(c + d)/c, 2, 2 + d/c, (b E^(c z))/
(-a + Sqrt[a^2 - b^2])] + a^2 Hypergeometric2F1[(c + d)/c, 2,
2 + d/c, -((b E^(c z))/(a + Sqrt[a^2 - b^2]))] -
b^2 Hypergeometric2F1[(c + d)/c, 2, 2 + d/c,
-((b E^(c z))/(a + Sqrt[a^2 - b^2]))] - a Sqrt[a^2 - b^2]
Hypergeometric2F1[(c + d)/c, 2, 2 + d/c,
-((b E^(c z))/(a + Sqrt[a^2 - b^2]))])) +
(1/(c - d)) (E^((c - d) z) ((-a) (a + Sqrt[a^2 - b^2])
Hypergeometric2F1[1 - d/c, 1, 2 - d/c, (b E^(c z))/
(-a + Sqrt[a^2 - b^2])] + a (a - Sqrt[a^2 - b^2])
Hypergeometric2F1[1 - d/c, 1, 2 - d/c,
-((b E^(c z))/(a + Sqrt[a^2 - b^2]))] +
a^2 Hypergeometric2F1[1 - d/c, 2, 2 - d/c, (b E^(c z))/
(-a + Sqrt[a^2 - b^2])] - b^2 Hypergeometric2F1[1 - d/c, 2, 2 - d/c,
(b E^(c z))/(-a + Sqrt[a^2 - b^2])] + a Sqrt[a^2 - b^2]
Hypergeometric2F1[1 - d/c, 2, 2 - d/c, (b E^(c z))/
(-a + Sqrt[a^2 - b^2])] - a^2 Hypergeometric2F1[1 - d/c, 2, 2 - d/c,
-((b E^(c z))/(a + Sqrt[a^2 - b^2]))] +
b^2 Hypergeometric2F1[1 - d/c, 2, 2 - d/c,
-((b E^(c z))/(a + Sqrt[a^2 - b^2]))] + a Sqrt[a^2 - b^2]
Hypergeometric2F1[1 - d/c, 2, 2 - d/c,
-((b E^(c z))/(a + Sqrt[a^2 - b^2]))])))
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mo> ∫ </mo> <mrow> <mfrac> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mi> c </mi> <mo> - </mo> <mi> d </mi> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> - </mo> <mi> d </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mi> d </mi> <mi> c </mi> </mfrac> </mrow> <mo> , </mo> <mn> 2 </mn> </mrow> <mo> ; </mo> <mrow> <mn> 2 </mn> <mo> - </mo> <mfrac> <mi> d </mi> <mi> c </mi> </mfrac> </mrow> <mo> ; </mo> <mfrac> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> </mrow> <mrow> <msqrt> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mo> - </mo> <mi> a </mi> </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["1", "-", FractionBox["d", "c"]]], Hypergeometric2F1], ",", TagBox["2", Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List["2", "-", FractionBox["d", "c"]]], Hypergeometric2F1], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]]]], RowBox[List[SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]], "-", "a"]]], 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Date Added to functions.wolfram.com (modification date)
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