# 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

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

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


Clarification: $$C$$ denotes the arbitrary constant of integration.