I'm assuming the integral is
[tex]\displaystyle \int \frac{dx}{x (\ln(x^2))^5}[/tex]
We have
[tex]\ln(x^2) = 2 \ln|x| \implies (\ln(x^2))^5 = 32 (\ln|x|)^5[/tex]
Then substituting [tex]y=\ln|x|[/tex] and [tex]dy=\frac{dx}x[/tex], the integral transforms and reduces to
[tex]\displaystyle \int \frac{dx}{x(\ln(x^2))^5} = \frac1{32} \int \frac{dy}{y^5} \\\\ ~~~~~~~~ = \frac1{32} \left(-\frac1{4y^4}\right) + C \\\\ ~~~~~~~~ = -\frac1{128(\ln|x|)^4} + C[/tex]
which we can rewrite as
[tex]128 (\ln|x|)^4 = 8\cdot2^4(\ln|x|)^4 = 8 (2\ln|x|)^4 = 8 (\ln(x^2))^4[/tex]
and so
[tex]\displaystyle \int \frac{dx}{x (\ln(x^2))^5} = \boxed{-\frac1{8(\ln(x^2))^4} + C}[/tex]
