ODE No. 1917

$$\{x^{\prime }(t)=y(t{)}^2-\cos (x(t)),y^{\prime }(t)=y(t)(-\sin (x(t)))\}$$ ✓ Mathematica : cpu = 202.221 (sec), leaf count = 3406

DSolve[{Derivative[1][x][t] == -Cos[x[t]] + y[t]^2, Derivative[1][y][t] == -(Sin[x[t]]*y[t])},{x[t], y[t]},t]
 

$$\{\{y(t)\to \frac{3\sqrt[3]{2}\cos \left(\text{InverseFunction}[{\int }_1^{\mathrm{\#}\text{1}}\frac{(3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[1])}){}^{2/3}}{22^{2/3}{\cos }^2(K[1])+2(3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[1])}){}^{2/3}\cos (K[1])+3\sqrt[3]{2}{c}_1\sqrt[3]{3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[1])}}+\sqrt[3]{2}\sqrt{9c_1{}^2-4{\cos }^3(K[1])}\sqrt[3]{3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[1])}}}dK[1]\mathrm{\&}][\frac{t}{2}+c_2]\right)}{\sqrt[3]{81c_1+\sqrt{6561c_1{}^2-2916{\cos }^3\left(\text{InverseFunction}[{\int }_1^{\mathrm{\#}\text{1}}\frac{(3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[1])}){}^{2/3}}{22^{2/3}{\cos }^2(K[1])+2(3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[1])}){}^{2/3}\cos (K[1])+3\sqrt[3]{2}{c}_1\sqrt[3]{3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[1])}}+\sqrt[3]{2}\sqrt{9c_1{}^2-4{\cos }^3(K[1])}\sqrt[3]{3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[1])}}}dK[1]\mathrm{\&}][\frac{t}{2}+c_2]\right)}}}+\frac{\sqrt[3]{81c_1+\sqrt{6561c_1{}^2-2916{\cos }^3\left(\text{InverseFunction}[{\int }_1^{\mathrm{\#}\text{1}}\frac{(3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[1])}){}^{2/3}}{22^{2/3}{\cos }^2(K[1])+2(3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[1])}){}^{2/3}\cos (K[1])+3\sqrt[3]{2}{c}_1\sqrt[3]{3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[1])}}+\sqrt[3]{2}\sqrt{9c_1{}^2-4{\cos }^3(K[1])}\sqrt[3]{3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[1])}}}dK[1]\mathrm{\&}][\frac{t}{2}+c_2]\right)}}}{3\sqrt[3]{2}},x(t)\to \text{InverseFunction}[{\int }_1^{\mathrm{\#}\text{1}}\frac{(3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[1])}){}^{2/3}}{22^{2/3}{\cos }^2(K[1])+2(3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[1])}){}^{2/3}\cos (K[1])+3\sqrt[3]{2}{c}_1\sqrt[3]{3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[1])}}+\sqrt[3]{2}\sqrt{9c_1{}^2-4{\cos }^3(K[1])}\sqrt[3]{3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[1])}}}dK[1]\mathrm{\&}][\frac{t}{2}+c_2]\},\{y(t)\to -\frac{3(1+i\sqrt{3})\cos \left(\text{InverseFunction}[{\int }_1^{\mathrm{\#}\text{1}}\frac{(3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[2])}){}^{2/3}}{2i{2}^{2/3}\sqrt{3}{\cos }^2(K[2])-22^{2/3}{\cos }^2(K[2])+4(3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[2])}){}^{2/3}\cos (K[2])-3i\sqrt[3]{2}\sqrt{3}{c}_1\sqrt[3]{3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[2])}}-3\sqrt[3]{2}{c}_1\sqrt[3]{3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[2])}}-i\sqrt[3]{2}\sqrt{3}\sqrt{9c_1{}^2-4{\cos }^3(K[2])}\sqrt[3]{3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[2])}}-\sqrt[3]{2}\sqrt{9c_1{}^2-4{\cos }^3(K[2])}\sqrt[3]{3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[2])}}}dK[2]\mathrm{\&}][\frac{t}{4}+c_2]\right)}{2^{2/3}\sqrt[3]{81c_1+\sqrt{6561c_1{}^2-2916{\cos }^3\left(\text{InverseFunction}[{\int }_1^{\mathrm{\#}\text{1}}\frac{(3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[2])}){}^{2/3}}{2i{2}^{2/3}\sqrt{3}{\cos }^2(K[2])-22^{2/3}{\cos }^2(K[2])+4(3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[2])}){}^{2/3}\cos (K[2])-3i\sqrt[3]{2}\sqrt{3}{c}_1\sqrt[3]{3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[2])}}-3\sqrt[3]{2}{c}_1\sqrt[3]{3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[2])}}-i\sqrt[3]{2}\sqrt{3}\sqrt{9c_1{}^2-4{\cos }^3(K[2])}\sqrt[3]{3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[2])}}-\sqrt[3]{2}\sqrt{9c_1{}^2-4{\cos }^3(K[2])}\sqrt[3]{3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[2])}}}dK[2]\mathrm{\&}][\frac{t}{4}+c_2]\right)}}}-\frac{(1-i\sqrt{3})\sqrt[3]{81c_1+\sqrt{6561c_1{}^2-2916{\cos }^3\left(\text{InverseFunction}[{\int }_1^{\mathrm{\#}\text{1}}\frac{(3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[2])}){}^{2/3}}{2i{2}^{2/3}\sqrt{3}{\cos }^2(K[2])-22^{2/3}{\cos }^2(K[2])+4(3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[2])}){}^{2/3}\cos (K[2])-3i\sqrt[3]{2}\sqrt{3}{c}_1\sqrt[3]{3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[2])}}-3\sqrt[3]{2}{c}_1\sqrt[3]{3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[2])}}-i\sqrt[3]{2}\sqrt{3}\sqrt{9c_1{}^2-4{\cos }^3(K[2])}\sqrt[3]{3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[2])}}-\sqrt[3]{2}\sqrt{9c_1{}^2-4{\cos }^3(K[2])}\sqrt[3]{3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[2])}}}dK[2]\mathrm{\&}][\frac{t}{4}+c_2]\right)}}}{6\sqrt[3]{2}},x(t)\to \text{InverseFunction}[{\int }_1^{\mathrm{\#}\text{1}}\frac{(3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[2])}){}^{2/3}}{2i{2}^{2/3}\sqrt{3}{\cos }^2(K[2])-22^{2/3}{\cos }^2(K[2])+4(3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[2])}){}^{2/3}\cos (K[2])-3i\sqrt[3]{2}\sqrt{3}{c}_1\sqrt[3]{3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[2])}}-3\sqrt[3]{2}{c}_1\sqrt[3]{3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[2])}}-i\sqrt[3]{2}\sqrt{3}\sqrt{9c_1{}^2-4{\cos }^3(K[2])}\sqrt[3]{3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[2])}}-\sqrt[3]{2}\sqrt{9c_1{}^2-4{\cos }^3(K[2])}\sqrt[3]{3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[2])}}}dK[2]\mathrm{\&}][\frac{t}{4}+c_2]\},\{y(t)\to -\frac{3(1-i\sqrt{3})\cos \left(\text{InverseFunction}[{\int }_1^{\mathrm{\#}\text{1}}\frac{(3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[3])}){}^{2/3}}{-2i{2}^{2/3}\sqrt{3}{\cos }^2(K[3])-22^{2/3}{\cos }^2(K[3])+4(3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[3])}){}^{2/3}\cos (K[3])+3i\sqrt[3]{2}\sqrt{3}{c}_1\sqrt[3]{3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[3])}}-3\sqrt[3]{2}{c}_1\sqrt[3]{3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[3])}}+i\sqrt[3]{2}\sqrt{3}\sqrt{9c_1{}^2-4{\cos }^3(K[3])}\sqrt[3]{3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[3])}}-\sqrt[3]{2}\sqrt{9c_1{}^2-4{\cos }^3(K[3])}\sqrt[3]{3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[3])}}}dK[3]\mathrm{\&}][\frac{t}{4}+c_2]\right)}{2^{2/3}\sqrt[3]{81c_1+\sqrt{6561c_1{}^2-2916{\cos }^3\left(\text{InverseFunction}[{\int }_1^{\mathrm{\#}\text{1}}\frac{(3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[3])}){}^{2/3}}{-2i{2}^{2/3}\sqrt{3}{\cos }^2(K[3])-22^{2/3}{\cos }^2(K[3])+4(3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[3])}){}^{2/3}\cos (K[3])+3i\sqrt[3]{2}\sqrt{3}{c}_1\sqrt[3]{3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[3])}}-3\sqrt[3]{2}{c}_1\sqrt[3]{3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[3])}}+i\sqrt[3]{2}\sqrt{3}\sqrt{9c_1{}^2-4{\cos }^3(K[3])}\sqrt[3]{3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[3])}}-\sqrt[3]{2}\sqrt{9c_1{}^2-4{\cos }^3(K[3])}\sqrt[3]{3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[3])}}}dK[3]\mathrm{\&}][\frac{t}{4}+c_2]\right)}}}-\frac{(1+i\sqrt{3})\sqrt[3]{81c_1+\sqrt{6561c_1{}^2-2916{\cos }^3\left(\text{InverseFunction}[{\int }_1^{\mathrm{\#}\text{1}}\frac{(3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[3])}){}^{2/3}}{-2i{2}^{2/3}\sqrt{3}{\cos }^2(K[3])-22^{2/3}{\cos }^2(K[3])+4(3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[3])}){}^{2/3}\cos (K[3])+3i\sqrt[3]{2}\sqrt{3}{c}_1\sqrt[3]{3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[3])}}-3\sqrt[3]{2}{c}_1\sqrt[3]{3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[3])}}+i\sqrt[3]{2}\sqrt{3}\sqrt{9c_1{}^2-4{\cos }^3(K[3])}\sqrt[3]{3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[3])}}-\sqrt[3]{2}\sqrt{9c_1{}^2-4{\cos }^3(K[3])}\sqrt[3]{3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[3])}}}dK[3]\mathrm{\&}][\frac{t}{4}+c_2]\right)}}}{6\sqrt[3]{2}},x(t)\to \text{InverseFunction}[{\int }_1^{\mathrm{\#}\text{1}}\frac{(3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[3])}){}^{2/3}}{-2i{2}^{2/3}\sqrt{3}{\cos }^2(K[3])-22^{2/3}{\cos }^2(K[3])+4(3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[3])}){}^{2/3}\cos (K[3])+3i\sqrt[3]{2}\sqrt{3}{c}_1\sqrt[3]{3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[3])}}-3\sqrt[3]{2}{c}_1\sqrt[3]{3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[3])}}+i\sqrt[3]{2}\sqrt{3}\sqrt{9c_1{}^2-4{\cos }^3(K[3])}\sqrt[3]{3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[3])}}-\sqrt[3]{2}\sqrt{9c_1{}^2-4{\cos }^3(K[3])}\sqrt[3]{3c_1+\sqrt{9c_1{}^2-4{\cos }^3(K[3])}}}dK[3]\mathrm{\&}][\frac{t}{4}+c_2]\}\}$$ ✓ Maple : cpu = 1.166 (sec), leaf count = 108

dsolve({diff(x(t),t) = y(t)^2-cos(x(t)), diff(y(t),t) = -y(t)*sin(x(t))})
 

$$[\{x\left(t\right)=\mathrm{RootOf}(-2\left({\int }_{}^{\mathit{\text{\_Z}}}\frac{1}{-\tan \left(\mathrm{RootOf}(-3\sqrt{-({\cos }^2\left(\mathit{\text{\_f}}\right))}\ln \left(\frac{9({\cos }^2\left(\mathit{\text{\_f}}\right))}{4\cos {\left(\mathit{\text{\_Z}}\right)}^2}\right)+3c_1\sqrt{-({\cos }^2\left(\mathit{\text{\_f}}\right))}+2\mathit{\text{\_Z}}\cos \left(\mathit{\text{\_f}}\right))\right)\sqrt{-4\cos \left(2\mathit{\text{\_f}}\right)-4-({\cos }^2\left(\mathit{\text{\_f}}\right))}+\cos \left(\mathit{\text{\_f}}\right)}d\mathit{\text{\_f}}\right)+t+c_2)\},\{y\left(t\right)=\sqrt{\frac{d}{dt}x\left(t\right)+\cos \left(x\left(t\right)\right)},y\left(t\right)=-\sqrt{\frac{d}{dt}x\left(t\right)+\cos \left(x\left(t\right)\right)}\}]$$