Evaluate the following integral:

Solution:

Let

$I=\int \frac{1}{x\left(x^{4}+1\right)} d x$

$\frac{1}{x\left(x^{4}+1\right)}=\frac{A}{x}+\frac{B x^{3}+C x^{2}+D x+E}{x^{4}+1}$

$I=A\left(x^{4}+1\right)+\left(B x^{3}+C x^{2}+D x+E\right)(x)$

Equating constants

$A=1$

Equating coefficients of $x^{4}$

$0=A+B$

$0=1+B$

$B=-1$

Equating coefficients of $x^{2}$

$D=0$

Equating coefficients of $x$

$E=0$

Thus,

$I=\int \frac{d x}{x}+\int-\frac{x^{2} d x}{x^{4}+1}$

$=\log |x|-\frac{1}{4} \log \left|x^{4}+1\right|+C$

$=\frac{4}{4} \log |x|-\frac{1}{4} \log \left|x^{4}+1\right|+C$

$=\frac{1}{4} \log \left|x^{4}\right|-\frac{1}{4} \log \left|x^{4}+1\right|+C$

$\frac{1}{4} \log \left|\frac{x^{4}}{x^{4}+1}\right|+C$