<s>[INST] Calculation of improper integral: $\int_{0}^{\infty}\frac{x^3\ln x}{(x^4+1)^3} \,dx$ One of an exam's task was to calculate the following integral: $$I=\int_{0}^{\infty}\frac{x^3\ln x}{(x^4+1)^3} \,dx$$
I tried integration by parts:
$$I=\frac{1}{4}\int_{0}^{\infty}\ln x \cdot (x^4+1)^{-3} \,d(x^4+1)$$
but then things got messy and I ultimately concluded that the improper integral diverges, which is wrong since the answer is:
$$I=-\frac{1}{32}$$
What should I do?
 [/INST] Integrating by parts,
$$
\int\frac{x^3\log x}{(1+x^4)^3}\,dx=-\frac{\log x}{8(1+x^4)^2}+\int \frac{1}{8(1+x^4)^2x}\,dx
$$
Doing partial fraction decomposition,
$$
\frac{1}{8(1+x^4)^2x}=\frac{1}{8x}-\frac{x^3}{8(1+x^4)}-\frac{x^3}{8(1+x^4)^2}.
$$
Can you take it from here?
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