Integrate the functions`(x^3sin(tan^(-1)x^4))/(1+x^8)`

Explanation

To solve the integral \( \int \frac{x^3 \sin(\tan^{-1}(x^4))}{1 + x^8} \, dx \), we will use substitution and integration techniques. Here’s a step-by-step solution:

### Step 1: Substitution
Let \( t = \tan^{-1}(x^4) \). Then, we differentiate \( t \) with respect to \( x \):

\[ \frac{dt}{dx} = \frac{4x^3}{1 + (x^4)^2} = \frac{4x^3}{1 + x^8} \]
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