Compute $\int \frac{\sin x+\cos x}{\sqrt[3]{\sin x-\cos x}}\,\mathrm dx$

Question

Compute $$\int \frac{\sin x+\cos x}{\sqrt[3]{\sin x-\cos x}}\,\mathrm dx$$

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Let $t=\tan\frac x2$, then

$$\sin x=\frac{2t}{1+t^2},\; \cos x=\frac{1-t^2}{1+t^2},\; \mathrm dx =\frac{2\mathrm dt}{1+t^2}.$$

then, ..

Any suggestions are welcomed. Thanks

Answer 1

Hint: If $u=\sin x-\cos x$, then $du = (\cos x+\sin x)dx$.