ODE
\[ y'(x)=\sec ^2(x) \text {Cosy}(y(x)) \cot (y(x)) \] ODE Classification
[_separable]
Book solution method
Separable ODE, Neither variable missing
Mathematica ✓
cpu = 0.427983 (sec), leaf count = 28
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}} \frac {\tan (K[1])}{\text {Cosy}(K[1])} \, dK[1]\& \right ]\left [c_1+\tan (x)\right ]\right \}\right \}\]
Maple ✓
cpu = 0.049 (sec), leaf count = 33
\[ \left \{ {\frac {1}{\cos \left ( x \right ) } \left ( -\int ^{y \left ( x \right ) }\!{\frac {1}{\cot \left ( {\it \_a} \right ) {\it Cosy} \left ( {\it \_a} \right ) }}{d{\it \_a}}\cos \left ( x \right ) +{\it \_C1}\,\cos \left ( x \right ) +\sin \left ( x \right ) \right ) }=0 \right \} \] Mathematica raw input
DSolve[y'[x] == Cosy[y[x]]*Cot[y[x]]*Sec[x]^2,y[x],x]
Mathematica raw output
{{y[x] -> InverseFunction[Integrate[Tan[K[1]]/Cosy[K[1]], {K[1], 1, #1}] & ][C[1] + Tan[x]]}}
Maple raw input
dsolve(diff(y(x),x) = sec(x)^2*cot(y(x))*Cosy(y(x)), y(x),'implicit')
Maple raw output
(-Intat(1/cot(_a)/Cosy(_a),_a = y(x))*cos(x)+_C1*cos(x)+sin(x))/cos(x) = 0