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 $$\alpha +\beta +\gamma +\delta +\varepsilon +\zeta +\eta +\theta +\iota$$

×