Let's do some calculus! (45)

Calculus Level 5

\[\large \displaystyle \int \dfrac{\sin^4 x + \cos^4 x}{\sin^3 x + \cos^3 x} \, dx \]

If the above integral can be represented as

\[\sin x - \cos x + \frac{\sqrt{a}}{b} \cdot \operatorname{arctanh} \left( \frac{ \tan \left( \frac x2 \right) - 1 }{\sqrt{a}} \right) + \frac ab \cdot \arctan \left( \cos x - \sin x \right) + C, \]

where \(a\) and \(b\) are coprime positive integers with \(a\) square free, then evaluate \(a+b\).

Clarification: \(C\) denotes the arbitrary constant of integration.

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