\[ \large \int \frac{\tan^{-1}(x)}{x^4} \, dx \]

If the indefinite integral above equals to \[ - \frac{\tan^{-1}(x)}{a\cdot x^a} + \frac bc \ln \left( \frac{x^d+b}{x^d} \right) - \frac b{c\cdot x^d} + C \]

for positive integer constants \(a,b,c\) and \(d\) with \(b,c\) coprime and arbitrary constant \(C\), find the value of \(a+b+c+d\).

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