\[ \boxed { \begin {array}{rl} x_1'(t) \sin (x_2(t)) &= x_4(t) \sin (x_3(t))+x_5(t) \cos (x_3(t))\\ x_2'(t) &= x_4(t) \cos (x_3(t))-x_5(t) \sin (x_3(t))\\ x_1'(t) \cos (x_2(t))+x_3'(t) &= a\\ x_4'(t)-a (1-\lambda ) x_5(t) &= -m \sin (x_2(t)) \cos (x_3(t))\\ a (1-\lambda ) x_4(t)+x_5'(t) &= m \sin (x_2(t)) \sin (x_3(t)) \end {array}} \]
Mathematica: cpu = 0.007001 (sec), leaf count = 118 \[ \text {DSolve}\left [\left \{\text {x1}'(t) \sin (\text {x2}(t))=\text {x4}(t) \sin (\text {x3}(t))+\text {x5}(t) \cos (\text {x3}(t)),\text {x2}'(t)=\text {x4}(t) \cos (\text {x3}(t))-\text {x5}(t) \sin (\text {x3}(t)),\text {x1}'(t) \cos (\text {x2}(t))+\text {x3}'(t)=a,\text {x4}'(t)-a (1-\lambda ) \text {x5}(t)=-m \sin (\text {x2}(t)) \cos (\text {x3}(t)),a (1-\lambda ) \text {x4}(t)+\text {x5}'(t)=m \sin (\text {x2}(t)) \sin (\text {x3}(t))\right \},\{\text {x1}(t),\text {x2}(t),\text {x3}(t),\text {x4}(t),\text {x5}(t)\},t\right ] \]
Maple: cpu = 0 (sec), leaf count = 0 \[ \text {hanged} \]