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http://functions.wolfram.com/01.20.21.4774.01
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Integrate[(Cos[e z] Sinh[d z])/(a + b Cosh[c z]^2), z] ==
(1/4) ((E^((2 c - d - I e) z) ((-2 a - b - 2 Sqrt[a] Sqrt[a + b])
Hypergeometric2F1[1 + (-d - I e)/(2 c), 1, 2 + (-d - I e)/(2 c),
-((b E^(2 c z))/(2 a + b - 2 Sqrt[a] Sqrt[a + b]))] +
(2 a + b - 2 Sqrt[a] Sqrt[a + b]) Hypergeometric2F1[
1 + (-d - I e)/(2 c), 1, 2 + (-d - I e)/(2 c),
-((b E^(2 c z))/(2 a + b + 2 Sqrt[a] Sqrt[a + b]))]))/
(Sqrt[a] b Sqrt[a + b] (2 c - d - I e)) -
(E^((2 c + d - I e) z) ((-2 a - b - 2 Sqrt[a] Sqrt[a + b])
Hypergeometric2F1[1 + (d - I e)/(2 c), 1, 2 + (d - I e)/(2 c),
-((b E^(2 c z))/(2 a + b - 2 Sqrt[a] Sqrt[a + b]))] +
(2 a + b - 2 Sqrt[a] Sqrt[a + b]) Hypergeometric2F1[
1 + (d - I e)/(2 c), 1, 2 + (d - I e)/(2 c),
-((b E^(2 c z))/(2 a + b + 2 Sqrt[a] Sqrt[a + b]))]))/
(Sqrt[a] b Sqrt[a + b] (2 c + d - I e)) +
(E^((2 c - d + I e) z) ((-2 a - b - 2 Sqrt[a] Sqrt[a + b])
Hypergeometric2F1[1 + (-d + I e)/(2 c), 1, 2 + (-d + I e)/(2 c),
-((b E^(2 c z))/(2 a + b - 2 Sqrt[a] Sqrt[a + b]))] +
(2 a + b - 2 Sqrt[a] Sqrt[a + b]) Hypergeometric2F1[
1 + (-d + I e)/(2 c), 1, 2 + (-d + I e)/(2 c),
-((b E^(2 c z))/(2 a + b + 2 Sqrt[a] Sqrt[a + b]))]))/
(Sqrt[a] b Sqrt[a + b] (2 c - d + I e)) -
(E^((2 c + d + I e) z) ((-2 a - b - 2 Sqrt[a] Sqrt[a + b])
Hypergeometric2F1[1 + (d + I e)/(2 c), 1, 2 + (d + I e)/(2 c),
-((b E^(2 c z))/(2 a + b - 2 Sqrt[a] Sqrt[a + b]))] +
(2 a + b - 2 Sqrt[a] Sqrt[a + b]) Hypergeometric2F1[
1 + (d + I e)/(2 c), 1, 2 + (d + I e)/(2 c),
-((b E^(2 c z))/(2 a + b + 2 Sqrt[a] Sqrt[a + b]))]))/
(Sqrt[a] b Sqrt[a + b] (2 c + d + I e)))
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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> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> cosh </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mrow> <msqrt> <mi> a </mi> </msqrt> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <msqrt> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> - </mo> <mi> d </mi> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> e </mi> </mrow> </mrow> <mtext> </mtext> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> - </mo> <mi> d </mi> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> e </mi> </mrow> </mrow> <mtext> </mtext> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mi> a </mi> </msqrt> </mrow> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <semantics> 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Date Added to functions.wolfram.com (modification date)
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