Step 1: Simplify the expression inside the denominator before cubing.
The denominator term is $\sin^2 x + \sin^{-3} x \cos^5 x$. Rewrite $\sin^{-3} x \cos^5 x = \frac{\cos^5 x}{\sin^3 x}$. Factor out $\sin^2 x$: \[\sin^2 x + \frac{\cos^5 x}{\sin^3 x} = \sin^2 x\!\left(1 + \frac{\cos^5 x}{\sin^5 x}\right) = \sin^2 x(1 + \cot^5 x)\]
Step 2: Raise the simplified denominator to the power of 3.
\[\left(\sin^2 x + \sin^{-3} x \cos^5 x\right)^3 = \sin^6 x \cdot (1 + \cot^5 x)^3\]
Step 3: Rewrite the integral in a clean form.
The integral becomes $\int \frac{\cos^4 x}{\sin^6 x (1 + \cot^5 x)^3}\, dx$. Note $\frac{\cos^4 x}{\sin^6 x} = \cot^4 x \cdot \cosec^2 x$. So: \[\int \frac{\cot^4 x \cosec^2 x}{(1 + \cot^5 x)^3}\, dx\]
Step 4: Apply the substitution u = 1 + cot^5(x).
Let $u = 1 + \cot^5 x$. Then $du = 5\cot^4 x \cdot (-\cosec^2 x)\, dx$, so $\cot^4 x \cosec^2 x\, dx = -\frac{du}{5}$. The integral becomes: \[-\frac{1}{5}\int u^{-3}\, du\]
Step 5: Integrate using the power rule.
\[-\frac{1}{5} \cdot \frac{u^{-2}}{-2} + C = \frac{1}{10} u^{-2} + C\]
Step 6: Substitute back and state the answer.
Replacing $u = 1 + \cot^5 x$: \[\boxed{\dfrac{1}{10}(1 + \cot^5 x)^{-2} + C}\]