I just thought of this integral: $$I(a,b,x)=\int_0^x\sin^a(t)\cos^b(t)dt$$ I know that $$I(a,b,\pi/2)=\frac{\Gamma(\frac{a+1}2)\Gamma(\frac{b+1}2)}{2\Gamma(\frac{a+b}2+1)}$$ And I know how to reach this result myself, but I do not know how to even start on evaluating $I(a,b,x)$, so that's why I have no attempts to show you.
Is there any way to find a general form for $I(a,b,x)$ (potentially in terms of special functions)?