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this (almost) looks like u^-1 du

This smells of parts, but I'm having trouble with \[\int\limits \frac{1}{x^2+1}dx\]

that is arctangent

Although that might not help

can you enlighten us with the problem?

Do a substitution \(u=x^2, du=2xdx\)

\[0.5 \ln(u+1)\]

Yes. Now make the inverse substitution.

hmm, can we try reduction of order with \[y=v(ix)\]