$\int_0^\pi\frac{x\,\cos\frac x3}{\sqrt[3]{\sin x}}dx =\frac{\pi\sqrt[a]b}{c}\big(d\pi\sqrt a -e\ln a\big)$ If the above integral is true for positive integers $$a,b,c,d,e$$, where $$a,b$$ are prime
Evaluate $$a+b+c+d+e$$.