-(5) \( \int \frac{3 x^{3}+x}{1+x^{4}} \mathrm{~d} x \);

Calculate the integral \( \int \frac{3x^3+x}{1+x^4} \, dx \). Evaluate the integral by following steps: - step0: Evaluate using substitution: \(\int \frac{3x^{3}+x}{1+x^{4}} dx\) - step1: Rewrite the fraction: \(\int \left(\frac{3x^{3}}{1+x^{4}}+\frac{x}{1+x^{4}}\right) dx\) - step2: Use properties of integrals: \(\int \frac{3x^{3}}{1+x^{4}} dx+\int \frac{x}{1+x^{4}} dx\) - step3: Evaluate the integral: \(\frac{3}{4}\ln{\left(x^{4}+1\right)}+\int \frac{x}{1+x^{4}} dx\) - step4: Evaluate the integral: \(\frac{3}{4}\ln{\left(x^{4}+1\right)}+\frac{1}{2}\arctan\left(x^{2}\right)\) - step5: Add the constant of integral C: \(\frac{3}{4}\ln{\left(x^{4}+1\right)}+\frac{1}{2}\arctan\left(x^{2}\right) + C, C \in \mathbb{R}\) The integral of \( \frac{3x^{3}+x}{1+x^{4}} \) with respect to \( x \) is \( \frac{3}{4}\ln{(x^{4}+1)}+\frac{1}{2}\arctan(x^{2}) + C \), where \( C \) is an arbitrary constant.