Integral de ln|1+x^2| dx

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 |                           | x*im (x)*--(im(x))*sign\1 + x /         | x*re (x)*--(re(x))*sign\1 + x /         | x*--(re(x))*re(x)*sign\1 + x /         | x*im (x)*--(re(x))*re(x)*sign\1 + x /         | x*re (x)*--(im(x))*im(x)*sign\1 + x /         | x*--(im(x))*im(x)*sign\1 + x /                     
 |    /|     2|\             |          dx                             |          dx                             |   dx                                   |          dx                                   |          dx                                   |   dx                                     /|     2|\
 | log\|1 + x |/ dx = C - 2* | ------------------------------- dx - 2* | ------------------------------- dx - 2* | ------------------------------ dx - 2* | ------------------------------------- dx - 2* | ------------------------------------- dx + 2* | ------------------------------ dx + x*log\|1 + x |/
 |                           |        /     2\ |     2|                |        /     2\ |     2|                |       /     2\ |     2|                |           /     2\ |     2|                   |           /     2\ |     2|                   |       /     2\ |     2|                            
/                            |        \1 + x /*|1 + x |                |        \1 + x /*|1 + x |                |       \1 + x /*|1 + x |                |           \1 + x /*|1 + x |                   |           \1 + x /*|1 + x |                   |       \1 + x /*|1 + x |                            
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$$\int \log{\left(\left|{x^{2} + 1}\right| \right)}\, dx = C + x \log{\left(\left|{x^{2} + 1}\right| \right)} - 2 \int \frac{x \operatorname{re}{\left(x\right)} \operatorname{sign}{\left(x^{2} + 1 \right)} \frac{d}{d x} \operatorname{re}{\left(x\right)}}{\left(x^{2} + 1\right) \left|{x^{2} + 1}\right|}\, dx - 2 \int \frac{x \left(\operatorname{re}{\left(x\right)}\right)^{3} \operatorname{sign}{\left(x^{2} + 1 \right)} \frac{d}{d x} \operatorname{re}{\left(x\right)}}{\left(x^{2} + 1\right) \left|{x^{2} + 1}\right|}\, dx + 2 \int \frac{x \operatorname{im}{\left(x\right)} \operatorname{sign}{\left(x^{2} + 1 \right)} \frac{d}{d x} \operatorname{im}{\left(x\right)}}{\left(x^{2} + 1\right) \left|{x^{2} + 1}\right|}\, dx - 2 \int \frac{x \left(\operatorname{im}{\left(x\right)}\right)^{3} \operatorname{sign}{\left(x^{2} + 1 \right)} \frac{d}{d x} \operatorname{im}{\left(x\right)}}{\left(x^{2} + 1\right) \left|{x^{2} + 1}\right|}\, dx - 2 \int \frac{x \operatorname{re}{\left(x\right)} \left(\operatorname{im}{\left(x\right)}\right)^{2} \operatorname{sign}{\left(x^{2} + 1 \right)} \frac{d}{d x} \operatorname{re}{\left(x\right)}}{\left(x^{2} + 1\right) \left|{x^{2} + 1}\right|}\, dx - 2 \int \frac{x \left(\operatorname{re}{\left(x\right)}\right)^{2} \operatorname{im}{\left(x\right)} \operatorname{sign}{\left(x^{2} + 1 \right)} \frac{d}{d x} \operatorname{im}{\left(x\right)}}{\left(x^{2} + 1\right) \left|{x^{2} + 1}\right|}\, dx$$

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