ODE
\[ y'(x)=\tan (y(x)) (\cos (x) \sin (y(x))+1) \] ODE Classification
[`y=_G(x,y')`]
Book solution method
Change of Variable, new dependent variable
Mathematica ✓
cpu = 0.802158 (sec), leaf count = 53
\[\left \{\left \{y(x)\to \csc ^{-1}\left (\frac {1}{2} \left (-\sin (x)-\cos (x)-2 c_1 e^{-x}\right )\right )\right \},\left \{y(x)\to -\csc ^{-1}\left (\frac {1}{2} \left (\sin (x)+\cos (x)+2 c_1 e^{-x}\right )\right )\right \}\right \}\]
Maple ✗
cpu = 5.142 (sec), leaf count = 0 , could not solve
dsolve(diff(y(x),x) = (1+cos(x)*sin(y(x)))*tan(y(x)), y(x))
Mathematica raw input
DSolve[y'[x] == (1 + Cos[x]*Sin[y[x]])*Tan[y[x]],y[x],x]
Mathematica raw output
{{y[x] -> ArcCsc[((-2*C[1])/E^x - Cos[x] - Sin[x])/2]}, {y[x] -> -ArcCsc[((2*C[1])/E^x + Cos[x] + Sin[x])/2]}}
Maple raw input
dsolve(diff(y(x),x) = (1+cos(x)*sin(y(x)))*tan(y(x)), y(x))
Maple raw output
dsolve(diff(y(x),x) = (1+cos(x)*sin(y(x)))*tan(y(x)), y(x))