\[ \boxed { {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) ={\frac {2\,{x}^{2}\cosh \left ( \left ( x-1 \right ) ^{-1} \right ) -2\,x\cosh \left ( \left ( x-1 \right ) ^{-1} \right ) -1+ \left ( y \left ( x \right ) \right ) ^{2}-2\,{x}^{2}y \left ( x \right ) +{x}^{4}-x+x \left ( y \left ( x \right ) \right ) ^{2}-2\,{x}^{3}y \left ( x \right ) +{x}^{5}}{ \left ( x-1 \right ) \cosh \left ( \left ( x-1 \right ) ^{-1} \right ) }}=0} \]
Mathematica: cpu = 1670.398114 (sec), leaf count = 87 \[ \left \{\left \{y(x)\to \frac {\exp \left (\int _1^x \frac {2 (K[5]+1) \text {sech}\left (\frac {1}{K[5]-1}\right )}{K[5]-1} \, dK[5]\right )}{c_1-\frac {1}{2} \exp \left (\text {Integrate}\left [\frac {2 (K[5]+1) \text {sech}\left (\frac {1}{K[5]-1}\right )}{K[5]-1},\{K[5],1,x\},\text {Assumptions}\to \text {True}\right ]\right )}+\frac {x^3+x^2}{x+1}+1\right \}\right \} \]
Maple: cpu = 12.699 (sec), leaf count = 634 \[ \left \{ y \left ( x \right ) ={1 \left ( {{x}^{2} \left ( {{\rm e}^{{ \frac {{\it \_C1}}{ \left ( {{\rm e}^{ \left ( x-1 \right ) ^{-1}}} \right ) ^{2}+1}{{\rm e}^{2\, \left ( x-1 \right ) ^{-1}}}}}} \right ) ^{ 4} \left ( {{\rm e}^{{\frac {{\it \_C1}}{ \left ( {{\rm e}^{ \left ( x-1 \right ) ^{-1}}} \right ) ^{2}+1}}}} \right ) ^{4} \left ( {{\rm e}^{{ \frac {1}{ \left ( {{\rm e}^{ \left ( x-1 \right ) ^{-1}}} \right ) ^{2}+1 }\int \! \left ( {\frac {{{\rm e}^{ \left ( x-1 \right ) ^{-1}}}x}{1+x}}+ {\frac {x}{{{\rm e}^{ \left ( x-1 \right ) ^{-1}}} \left ( 1+x \right ) }} -{\frac {{{\rm e}^{ \left ( x-1 \right ) ^{-1}}}}{1+x}}-{\frac {1}{{ {\rm e}^{ \left ( x-1 \right ) ^{-1}}} \left ( 1+x \right ) }} \right ) ^{- 1}\,{\rm d}x{{\rm e}^{2\, \left ( x-1 \right ) ^{-1}}}}}} \right ) ^{-4} \left ( {{\rm e}^{{\frac {1}{ \left ( {{\rm e}^{ \left ( x-1 \right ) ^{- 1}}} \right ) ^{2}+1}\int \! \left ( {\frac {{{\rm e}^{ \left ( x-1 \right ) ^{-1}}}x}{1+x}}+{\frac {x}{{{\rm e}^{ \left ( x-1 \right ) ^{-1 }}} \left ( 1+x \right ) }}-{\frac {{{\rm e}^{ \left ( x-1 \right ) ^{-1}} }}{1+x}}-{\frac {1}{{{\rm e}^{ \left ( x-1 \right ) ^{-1}}} \left ( 1+x \right ) }} \right ) ^{-1}\,{\rm d}x}}} \right ) ^{-4}}-{x}^{2}+{1 \left ( {{\rm e}^{{\frac {{\it \_C1}}{ \left ( {{\rm e}^{ \left ( x-1 \right ) ^{-1}}} \right ) ^{2}+1}{{\rm e}^{2\, \left ( x-1 \right ) ^{-1} }}}}} \right ) ^{4} \left ( {{\rm e}^{{\frac {{\it \_C1}}{ \left ( { {\rm e}^{ \left ( x-1 \right ) ^{-1}}} \right ) ^{2}+1}}}} \right ) ^{4} \left ( {{\rm e}^{{\frac {1}{ \left ( {{\rm e}^{ \left ( x-1 \right ) ^{- 1}}} \right ) ^{2}+1}\int \! \left ( {\frac {{{\rm e}^{ \left ( x-1 \right ) ^{-1}}}x}{1+x}}+{\frac {x}{{{\rm e}^{ \left ( x-1 \right ) ^{-1 }}} \left ( 1+x \right ) }}-{\frac {{{\rm e}^{ \left ( x-1 \right ) ^{-1}} }}{1+x}}-{\frac {1}{{{\rm e}^{ \left ( x-1 \right ) ^{-1}}} \left ( 1+x \right ) }} \right ) ^{-1}\,{\rm d}x{{\rm e}^{2\, \left ( x-1 \right ) ^{ -1}}}}}} \right ) ^{-4} \left ( {{\rm e}^{{\frac {1}{ \left ( {{\rm e}^{ \left ( x-1 \right ) ^{-1}}} \right ) ^{2}+1}\int \! \left ( {\frac {{ {\rm e}^{ \left ( x-1 \right ) ^{-1}}}x}{1+x}}+{\frac {x}{{{\rm e}^{ \left ( x-1 \right ) ^{-1}}} \left ( 1+x \right ) }}-{\frac {{{\rm e}^{ \left ( x-1 \right ) ^{-1}}}}{1+x}}-{\frac {1}{{{\rm e}^{ \left ( x-1 \right ) ^{-1}}} \left ( 1+x \right ) }} \right ) ^{-1}\,{\rm d}x}}} \right ) ^{-4}}+1 \right ) \left ( -1+{1 \left ( {{\rm e}^{{\frac {{\it \_C1}}{ \left ( {{\rm e}^{ \left ( x-1 \right ) ^{-1}}} \right ) ^{2}+1}{ {\rm e}^{2\, \left ( x-1 \right ) ^{-1}}}}}} \right ) ^{4} \left ( { {\rm e}^{{\frac {{\it \_C1}}{ \left ( {{\rm e}^{ \left ( x-1 \right ) ^{- 1}}} \right ) ^{2}+1}}}} \right ) ^{4} \left ( {{\rm e}^{{\frac {1}{ \left ( {{\rm e}^{ \left ( x-1 \right ) ^{-1}}} \right ) ^{2}+1}\int \! \left ( {\frac {{{\rm e}^{ \left ( x-1 \right ) ^{-1}}}x}{1+x}}+{\frac { x}{{{\rm e}^{ \left ( x-1 \right ) ^{-1}}} \left ( 1+x \right ) }}-{\frac {{{\rm e}^{ \left ( x-1 \right ) ^{-1}}}}{1+x}}-{\frac {1}{{{\rm e}^{ \left ( x-1 \right ) ^{-1}}} \left ( 1+x \right ) }} \right ) ^{-1} \,{\rm d}x{{\rm e}^{2\, \left ( x-1 \right ) ^{-1}}}}}} \right ) ^{-4} \left ( {{\rm e}^{{\frac {1}{ \left ( {{\rm e}^{ \left ( x-1 \right ) ^{- 1}}} \right ) ^{2}+1}\int \! \left ( {\frac {{{\rm e}^{ \left ( x-1 \right ) ^{-1}}}x}{1+x}}+{\frac {x}{{{\rm e}^{ \left ( x-1 \right ) ^{-1 }}} \left ( 1+x \right ) }}-{\frac {{{\rm e}^{ \left ( x-1 \right ) ^{-1}} }}{1+x}}-{\frac {1}{{{\rm e}^{ \left ( x-1 \right ) ^{-1}}} \left ( 1+x \right ) }} \right ) ^{-1}\,{\rm d}x}}} \right ) ^{-4}} \right ) ^{-1}} \right \} \]