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http://functions.wolfram.com/01.19.21.2796.01
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Integrate[(Cos[e z] Sinh[d z])/(a + b Sinh[c z]^2), z] ==
(1/(4 Sqrt[a] Sqrt[a - b] b))
((1/(-2 c + d - I e)) (E^((-2 c + d - I e) z)
((-2 a + 2 Sqrt[a] Sqrt[a - b] + b) Hypergeometric2F1[
1 - (d - I e)/(2 c), 1, 2 - (d - I e)/(2 c),
b/(E^(2 c z) (-2 a - 2 Sqrt[a] Sqrt[a - b] + b))] +
(2 a + 2 Sqrt[a] Sqrt[a - b] - b) Hypergeometric2F1[
1 - (d - I e)/(2 c), 1, 2 - (d - I e)/(2 c),
b/(E^(2 c z) (-2 a + 2 Sqrt[a] Sqrt[a - b] + b))])) +
(1/(-2 c + d + I e)) (E^((-2 c + d + I e) z)
((-2 a + 2 Sqrt[a] Sqrt[a - b] + b) Hypergeometric2F1[
1 - (d + I e)/(2 c), 1, 2 - (d + I e)/(2 c),
b/(E^(2 c z) (-2 a - 2 Sqrt[a] Sqrt[a - b] + b))] +
(2 a + 2 Sqrt[a] Sqrt[a - b] - b) Hypergeometric2F1[
1 - (d + I e)/(2 c), 1, 2 - (d + I e)/(2 c),
b/(E^(2 c z) (-2 a + 2 Sqrt[a] Sqrt[a - b] + b))])) -
(1/(-2 c - d + I e)) (((-2 a + 2 Sqrt[a] Sqrt[a - b] + b)
Hypergeometric2F1[(2 c + d - I e)/(2 c), 1, (4 c + d - I e)/(2 c),
b/(E^(2 c z) (-2 a - 2 Sqrt[a] Sqrt[a - b] + b))] +
(2 a + 2 Sqrt[a] Sqrt[a - b] - b) Hypergeometric2F1[
(2 c + d - I e)/(2 c), 1, (4 c + d - I e)/(2 c),
b/(E^(2 c z) (-2 a + 2 Sqrt[a] Sqrt[a - b] + b))])/
E^((2 c + d - I e) z)) + (1/(2 c + d + I e))
(((-2 a + 2 Sqrt[a] Sqrt[a - b] + b) Hypergeometric2F1[
(2 c + d + I e)/(2 c), 1, (4 c + d + I e)/(2 c),
b/(E^(2 c z) (-2 a - 2 Sqrt[a] Sqrt[a - b] + b))] +
(2 a + 2 Sqrt[a] Sqrt[a - b] - b) Hypergeometric2F1[
(2 c + d + I e)/(2 c), 1, (4 c + d + I e)/(2 c),
b/(E^(2 c z) (-2 a + 2 Sqrt[a] Sqrt[a - b] + b))])/
E^((2 c + d + I e) z)))
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