If ∫(sinx)2−11(cosx)2−5dx= −q1p1(cotx)29−q2p2(cotx)25−q3p3(cotx)21+q4p4(cotx)2−3+C, where pi and qi are positive integers with gcd(pi,qi)=1 for i=1,2,3,4 and C is the constant of integration, then q1q2q3q415p1p2p3p4 is equal to
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Let $y = y(x)$ be the solution of the differential equation $x\sin\left(\dfrac{y}{x}\right)dy = \left(y\sin\left(\dfrac{y}{x}\right) - x\right)dx$, $y(1) = \dfrac{\pi}{2}$ and let $\alpha = \cos\left(\dfrac{y(e^{12})}{e^{12}}\right)$. Then the number of integral values of $p$, for which the equation $x^2 + y^2 - 2px + 2py + \alpha + 2 = 0$ represents a circle of radius $r \leq 6$, is __________.
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