\[ y'(x)=\frac {1}{16} x (y(x)+1)^2 (-\log (y(x)-1)+\log (y(x)+1)+2 \log (x))^2 \] ✓ Mathematica : cpu = 4.8318 (sec), leaf count = 1391
DSolve[Derivative[1][y][x] == (x*(2*Log[x] - Log[-1 + y[x]] + Log[1 + y[x]])^2*(1 + y[x])^2)/16,y[x],x]
\[\text {Solve}\left [\int _1^x-\frac {K[1] (2 \log (K[1])-\log (y(x)-1)+\log (y(x)+1))^2 (y(x)+1)}{4 \log ^2(K[1]) K[1]^2+\log ^2(y(x)-1) K[1]^2+\log ^2(y(x)+1) K[1]^2-4 \log (K[1]) \log (y(x)-1) K[1]^2+4 \log (K[1]) \log (y(x)+1) K[1]^2-2 \log (y(x)-1) \log (y(x)+1) K[1]^2+4 \log ^2(K[1]) y(x) K[1]^2+\log ^2(y(x)-1) y(x) K[1]^2+\log ^2(y(x)+1) y(x) K[1]^2-4 \log (K[1]) \log (y(x)-1) y(x) K[1]^2+4 \log (K[1]) \log (y(x)+1) y(x) K[1]^2-2 \log (y(x)-1) \log (y(x)+1) y(x) K[1]^2-16 y(x)+16}dK[1]+\int _1^{y(x)}\left (\frac {-4 \log ^2(x) x^2-\log ^2(K[2]-1) x^2-\log ^2(K[2]+1) x^2+4 \log (x) \log (K[2]-1) x^2-4 \log (x) \log (K[2]+1) x^2+2 \log (K[2]-1) \log (K[2]+1) x^2+16}{2 \left (4 \log ^2(x) x^2+\log ^2(K[2]-1) x^2+\log ^2(K[2]+1) x^2-4 \log (x) \log (K[2]-1) x^2+4 \log (x) \log (K[2]+1) x^2-2 \log (K[2]-1) \log (K[2]+1) x^2+K[2] \left (4 \log ^2(x) x^2+\log ^2(K[2]-1) x^2+\log ^2(K[2]+1) x^2-4 \log (x) \log (K[2]-1) x^2+4 \log (x) \log (K[2]+1) x^2-2 \log (K[2]-1) \log (K[2]+1) x^2-16\right )+16\right )}-\int _1^x\left (-\frac {K[1] (2 \log (K[1])-\log (K[2]-1)+\log (K[2]+1))^2}{4 K[2] \log ^2(K[1]) K[1]^2+4 \log ^2(K[1]) K[1]^2+K[2] \log ^2(K[2]-1) K[1]^2+\log ^2(K[2]-1) K[1]^2+K[2] \log ^2(K[2]+1) K[1]^2+\log ^2(K[2]+1) K[1]^2-4 K[2] \log (K[1]) \log (K[2]-1) K[1]^2-4 \log (K[1]) \log (K[2]-1) K[1]^2+4 K[2] \log (K[1]) \log (K[2]+1) K[1]^2+4 \log (K[1]) \log (K[2]+1) K[1]^2-2 K[2] \log (K[2]-1) \log (K[2]+1) K[1]^2-2 \log (K[2]-1) \log (K[2]+1) K[1]^2-16 K[2]+16}+\frac {K[1] (K[2]+1) \left (4 \log ^2(K[1]) K[1]^2+\log ^2(K[2]-1) K[1]^2+\log ^2(K[2]+1) K[1]^2-\frac {4 K[2] \log (K[1]) K[1]^2}{K[2]-1}-\frac {4 \log (K[1]) K[1]^2}{K[2]-1}+\frac {4 K[2] \log (K[1]) K[1]^2}{K[2]+1}+\frac {4 \log (K[1]) K[1]^2}{K[2]+1}+\frac {2 K[2] \log (K[2]-1) K[1]^2}{K[2]-1}-4 \log (K[1]) \log (K[2]-1) K[1]^2+\frac {2 \log (K[2]-1) K[1]^2}{K[2]-1}-\frac {2 K[2] \log (K[2]-1) K[1]^2}{K[2]+1}-\frac {2 \log (K[2]-1) K[1]^2}{K[2]+1}-\frac {2 K[2] \log (K[2]+1) K[1]^2}{K[2]-1}+4 \log (K[1]) \log (K[2]+1) K[1]^2-2 \log (K[2]-1) \log (K[2]+1) K[1]^2-\frac {2 \log (K[2]+1) K[1]^2}{K[2]-1}+\frac {2 K[2] \log (K[2]+1) K[1]^2}{K[2]+1}+\frac {2 \log (K[2]+1) K[1]^2}{K[2]+1}-16\right ) (2 \log (K[1])-\log (K[2]-1)+\log (K[2]+1))^2}{\left (4 K[2] \log ^2(K[1]) K[1]^2+4 \log ^2(K[1]) K[1]^2+K[2] \log ^2(K[2]-1) K[1]^2+\log ^2(K[2]-1) K[1]^2+K[2] \log ^2(K[2]+1) K[1]^2+\log ^2(K[2]+1) K[1]^2-4 K[2] \log (K[1]) \log (K[2]-1) K[1]^2-4 \log (K[1]) \log (K[2]-1) K[1]^2+4 K[2] \log (K[1]) \log (K[2]+1) K[1]^2+4 \log (K[1]) \log (K[2]+1) K[1]^2-2 K[2] \log (K[2]-1) \log (K[2]+1) K[1]^2-2 \log (K[2]-1) \log (K[2]+1) K[1]^2-16 K[2]+16\right )^2}-\frac {2 K[1] (K[2]+1) \left (\frac {1}{K[2]+1}-\frac {1}{K[2]-1}\right ) (2 \log (K[1])-\log (K[2]-1)+\log (K[2]+1))}{4 K[2] \log ^2(K[1]) K[1]^2+4 \log ^2(K[1]) K[1]^2+K[2] \log ^2(K[2]-1) K[1]^2+\log ^2(K[2]-1) K[1]^2+K[2] \log ^2(K[2]+1) K[1]^2+\log ^2(K[2]+1) K[1]^2-4 K[2] \log (K[1]) \log (K[2]-1) K[1]^2-4 \log (K[1]) \log (K[2]-1) K[1]^2+4 K[2] \log (K[1]) \log (K[2]+1) K[1]^2+4 \log (K[1]) \log (K[2]+1) K[1]^2-2 K[2] \log (K[2]-1) \log (K[2]+1) K[1]^2-2 \log (K[2]-1) \log (K[2]+1) K[1]^2-16 K[2]+16}\right )dK[1]+\frac {1}{2 (K[2]+1)}\right )dK[2]=c_1,y(x)\right ]\] ✓ Maple : cpu = 0.61 (sec), leaf count = 105
dsolve(diff(y(x),x) = 1/16*(-ln(-1+y(x))+ln(1+y(x))+2*ln(x))^2*x*(1+y(x))^2,y(x))
\[\int _{\textit {\_b}}^{y \left (x \right )}\frac {1}{4 \left (1+\textit {\_a} \right ) \left (\frac {x^{2} \left (1+\textit {\_a} \right ) \ln \left (1+\textit {\_a} \right )^{2}}{4}+\left (-\frac {\ln \left (\textit {\_a} -1\right )}{2}+\ln \left (x \right )\right ) x^{2} \left (1+\textit {\_a} \right ) \ln \left (1+\textit {\_a} \right )+\frac {x^{2} \left (1+\textit {\_a} \right ) \ln \left (\textit {\_a} -1\right )^{2}}{4}-x^{2} \ln \left (x \right ) \left (1+\textit {\_a} \right ) \ln \left (\textit {\_a} -1\right )+x^{2} \left (1+\textit {\_a} \right ) \ln \left (x \right )^{2}-4 \textit {\_a} +4\right )}d \textit {\_a} -\frac {\ln \left (x \right )}{16}-c_{1} = 0\]