#### 2.707   ODE No. 707

$y'(x)=\frac {1}{16} x (y(x)+1)^2 (-\log (y(x)-1)+\log (y(x)+1)+2 \log (x))^2$ Mathematica : cpu = 4.95341 (sec), leaf count = 1391

$\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 = 0.368 (sec), leaf count = 105

$\left \{ \int _{{\it \_b}}^{y \left ( x \right ) }\!{\frac {1}{4\,{\it \_a}+4} \left ( {\frac {{x}^{2} \left ( {\it \_a}+1 \right ) \left ( \ln \left ( {\it \_a}+1 \right ) \right ) ^{2}}{4}}+ \left ( -{\frac {\ln \left ( {\it \_a}-1 \right ) }{2}}+\ln \left ( x \right ) \right ) {x}^{2} \left ( {\it \_a}+1 \right ) \ln \left ( {\it \_a}+1 \right ) +{\frac {{x}^{2} \left ( {\it \_a}+1 \right ) \left ( \ln \left ( {\it \_a}-1 \right ) \right ) ^{2}}{4}}-\ln \left ( x \right ) {x}^{2} \left ( {\it \_a}+1 \right ) \ln \left ( {\it \_a}-1 \right ) +{x}^{2} \left ( {\it \_a}+1 \right ) \left ( \ln \left ( x \right ) \right ) ^{2}-4\,{\it \_a}+4 \right ) ^{-1}}\,{\rm d}{\it \_a}-{\frac {\ln \left ( x \right ) }{16}}-{\it \_C1}=0 \right \}$