I'm bored (Part 2)

Calculus Level 4

The integral \[\int _{ 1}^{ 5 } { \frac { dx }{ x^4 + 4x^3 + 9x^2 + 10x } } \] can be expressed as \(\frac { \alpha }{ \beta } \pi -\frac { \gamma }\delta { \arctan { \varepsilon } }+\frac { \zeta }{ \eta } \ln { \frac { \theta }{ \iota } } \) where \(\alpha\), \(\beta\), \(\gamma\), \(\delta\), \(\varepsilon\), \(\zeta\), \(\eta\), \(\theta\), and \(\iota\) are all positive integers. Also \(gcd(\alpha, \beta) = gcd(\gamma, \delta) = gcd(\zeta, \eta) = gcd(\theta, \iota) = 1\).

Find \(\alpha +\beta +\gamma +\delta +\varepsilon +\zeta +\eta +\theta +\iota\)

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