1. **State the problem:** We need to evaluate the integral $$\int x^2(1+\sin x \cos x)^2 (x^2+1)^3 (\sin x + \cos x)^4 \ln(x^2+1) \, dx.$$\n\n2. **Analyze the integral:** This integral is a product of polynomial, trigonometric, and logarithmic functions, making it quite complex. There is no straightforward elementary antiderivative.\n\n3. **Consider substitution or simplification:**\n- Note that $\sin x \cos x = \frac{1}{2} \sin(2x)$, so $1 + \sin x \cos x = 1 + \frac{1}{2} \sin(2x)$.\n- Also, $(\sin x + \cos x)^2 = \sin^2 x + 2 \sin x \cos x + \cos^2 x = 1 + \sin(2x)$, so $(\sin x + \cos x)^4 = (1 + \sin(2x))^2$.\n\n4. **Rewrite the integral:**\n$$\int x^2 \left(1 + \frac{1}{2} \sin(2x)\right)^2 (x^2+1)^3 (1 + \sin(2x))^2 \ln(x^2+1) \, dx.$$\n\n5. **No elementary antiderivative:** Given the complexity and mixture of functions, this integral does not have a closed-form elementary antiderivative. It is best approached with numerical methods or special functions if a definite integral is given.\n\n**Final answer:** The integral $$\int x^2(1+\sin x \cos x)^2 (x^2+1)^3 (\sin x + \cos x)^4 \ln(x^2+1) \, dx$$ has no elementary closed-form antiderivative.