2.338   ODE No. 338

1. Problem in Latex
2. Mathematica input
3. Maple input

$$y^{\prime }(x)(\sin (\alpha )(y(x{)}^2-x^2)-2x\cos (\alpha )y(x)+\sqrt{x^2+y(x{)}^2}y(x))+\cos (\alpha )(y(x{)}^2-x^2)+2x\sin (\alpha )y(x)+x\sqrt{x^2+y(x{)}^2}=0$$ ✓ Mathematica : cpu = 100.276 (sec), leaf count = 17681

$$\{\{y(x)\to -\frac{1}{2}{e}^{c_1}\sin (\alpha )-\frac{1}{2}\sqrt{e^{2c_1}{\sin }^2(\alpha )+\frac{1}{6}(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+\frac{1}{2}(-4x^2+4e^{c_1}\cos (\alpha )x+e^{2c_1}+e^{2c_1}\cos (2\alpha ))+\frac{\sqrt[3]{2(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^3-144x^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+72e^{2c_1}x\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+432e^{2c_1}{x}^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha )){\sin }^2(\alpha )+216e^{2c_1}{x}^2(e^{c_1}\sin (2\alpha )-2x\sin (\alpha )){}^2+\sqrt{(2(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^3-144x^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+72e^{2c_1}x\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+432e^{2c_1}{x}^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha )){\sin }^2(\alpha )+216e^{2c_1}{x}^2(e^{c_1}\sin (2\alpha )-2x\sin (\alpha )){}^2){}^2-4(24(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))x^2+24e^{2c_1}\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))x+(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^2){}^3}}}{6\sqrt[3]{2}}+\frac{128x^4-256e^{c_1}\cos (\alpha )x^3-96e^{2c_1}{x}^2+96e^{2c_1}\cos (2\alpha )x^2+48e^{3c_1}\cos (\alpha )x-16e^{3c_1}\cos (3\alpha )x+3e^{4c_1}+4e^{4c_1}\cos (2\alpha )+e^{4c_1}\cos (4\alpha )}{62^{2/3}\sqrt[3]{2(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^3-144x^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+72e^{2c_1}x\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+432e^{2c_1}{x}^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha )){\sin }^2(\alpha )+216e^{2c_1}{x}^2(e^{c_1}\sin (2\alpha )-2x\sin (\alpha )){}^2+\sqrt{(2(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^3-144x^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+72e^{2c_1}x\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+432e^{2c_1}{x}^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha )){\sin }^2(\alpha )+216e^{2c_1}{x}^2(e^{c_1}\sin (2\alpha )-2x\sin (\alpha )){}^2){}^2-4(24(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))x^2+24e^{2c_1}\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))x+(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^2){}^3}}}}-\frac{1}{2}\sqrt{2e^{2c_1}{\sin }^2(\alpha )+\frac{2}{3}(-4x^2+4e^{c_1}\cos (\alpha )x+e^{2c_1}+e^{2c_1}\cos (2\alpha ))-\frac{\sqrt[3]{2(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^3-144x^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+72e^{2c_1}x\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+432e^{2c_1}{x}^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha )){\sin }^2(\alpha )+216e^{2c_1}{x}^2(e^{c_1}\sin (2\alpha )-2x\sin (\alpha )){}^2+\sqrt{(2(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^3-144x^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+72e^{2c_1}x\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+432e^{2c_1}{x}^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha )){\sin }^2(\alpha )+216e^{2c_1}{x}^2(e^{c_1}\sin (2\alpha )-2x\sin (\alpha )){}^2){}^2-4(24(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))x^2+24e^{2c_1}\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))x+(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^2){}^3}}}{6\sqrt[3]{2}}-\frac{128x^4-256e^{c_1}\cos (\alpha )x^3-96e^{2c_1}{x}^2+96e^{2c_1}\cos (2\alpha )x^2+48e^{3c_1}\cos (\alpha )x-16e^{3c_1}\cos (3\alpha )x+3e^{4c_1}+4e^{4c_1}\cos (2\alpha )+e^{4c_1}\cos (4\alpha )}{62^{2/3}\sqrt[3]{2(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^3-144x^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+72e^{2c_1}x\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+432e^{2c_1}{x}^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha )){\sin }^2(\alpha )+216e^{2c_1}{x}^2(e^{c_1}\sin (2\alpha )-2x\sin (\alpha )){}^2+\sqrt{(2(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^3-144x^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+72e^{2c_1}x\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+432e^{2c_1}{x}^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha )){\sin }^2(\alpha )+216e^{2c_1}{x}^2(e^{c_1}\sin (2\alpha )-2x\sin (\alpha )){}^2){}^2-4(24(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))x^2+24e^{2c_1}\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))x+(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^2){}^3}}}-\frac{-8e^{3c_1}{\sin }^3(\alpha )+4e^{c_1}(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))\sin (\alpha )+8e^{c_1}x(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))}{4\sqrt{e^{2c_1}{\sin }^2(\alpha )+\frac{1}{6}(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+\frac{1}{2}(-4x^2+4e^{c_1}\cos (\alpha )x+e^{2c_1}+e^{2c_1}\cos (2\alpha ))+\frac{\sqrt[3]{2(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^3-144x^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+72e^{2c_1}x\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+432e^{2c_1}{x}^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha )){\sin }^2(\alpha )+216e^{2c_1}{x}^2(e^{c_1}\sin (2\alpha )-2x\sin (\alpha )){}^2+\sqrt{(2(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^3-144x^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+72e^{2c_1}x\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+432e^{2c_1}{x}^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha )){\sin }^2(\alpha )+216e^{2c_1}{x}^2(e^{c_1}\sin (2\alpha )-2x\sin (\alpha )){}^2){}^2-4(24(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))x^2+24e^{2c_1}\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))x+(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^2){}^3}}}{6\sqrt[3]{2}}+\frac{128x^4-256e^{c_1}\cos (\alpha )x^3-96e^{2c_1}{x}^2+96e^{2c_1}\cos (2\alpha )x^2+48e^{3c_1}\cos (\alpha )x-16e^{3c_1}\cos (3\alpha )x+3e^{4c_1}+4e^{4c_1}\cos (2\alpha )+e^{4c_1}\cos (4\alpha )}{62^{2/3}\sqrt[3]{2(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^3-144x^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+72e^{2c_1}x\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+432e^{2c_1}{x}^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha )){\sin }^2(\alpha )+216e^{2c_1}{x}^2(e^{c_1}\sin (2\alpha )-2x\sin (\alpha )){}^2+\sqrt{(2(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^3-144x^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+72e^{2c_1}x\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+432e^{2c_1}{x}^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha )){\sin }^2(\alpha )+216e^{2c_1}{x}^2(e^{c_1}\sin (2\alpha )-2x\sin (\alpha )){}^2){}^2-4(24(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))x^2+24e^{2c_1}\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))x+(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^2){}^3}}}}}}\},\{y(x)\to -\frac{1}{2}{e}^{c_1}\sin (\alpha )-\frac{1}{2}\sqrt{e^{2c_1}{\sin }^2(\alpha )+\frac{1}{6}(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+\frac{1}{2}(-4x^2+4e^{c_1}\cos (\alpha )x+e^{2c_1}+e^{2c_1}\cos (2\alpha ))+\frac{\sqrt[3]{2(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^3-144x^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+72e^{2c_1}x\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+432e^{2c_1}{x}^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha )){\sin }^2(\alpha )+216e^{2c_1}{x}^2(e^{c_1}\sin (2\alpha )-2x\sin (\alpha )){}^2+\sqrt{(2(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^3-144x^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+72e^{2c_1}x\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+432e^{2c_1}{x}^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha )){\sin }^2(\alpha )+216e^{2c_1}{x}^2(e^{c_1}\sin (2\alpha )-2x\sin (\alpha )){}^2){}^2-4(24(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))x^2+24e^{2c_1}\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))x+(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^2){}^3}}}{6\sqrt[3]{2}}+\frac{128x^4-256e^{c_1}\cos (\alpha )x^3-96e^{2c_1}{x}^2+96e^{2c_1}\cos (2\alpha )x^2+48e^{3c_1}\cos (\alpha )x-16e^{3c_1}\cos (3\alpha )x+3e^{4c_1}+4e^{4c_1}\cos (2\alpha )+e^{4c_1}\cos (4\alpha )}{62^{2/3}\sqrt[3]{2(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^3-144x^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+72e^{2c_1}x\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+432e^{2c_1}{x}^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha )){\sin }^2(\alpha )+216e^{2c_1}{x}^2(e^{c_1}\sin (2\alpha )-2x\sin (\alpha )){}^2+\sqrt{(2(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^3-144x^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+72e^{2c_1}x\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+432e^{2c_1}{x}^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha )){\sin }^2(\alpha )+216e^{2c_1}{x}^2(e^{c_1}\sin (2\alpha )-2x\sin (\alpha )){}^2){}^2-4(24(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))x^2+24e^{2c_1}\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))x+(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^2){}^3}}}}+\frac{1}{2}\sqrt{2e^{2c_1}{\sin }^2(\alpha )+\frac{2}{3}(-4x^2+4e^{c_1}\cos (\alpha )x+e^{2c_1}+e^{2c_1}\cos (2\alpha ))-\frac{\sqrt[3]{2(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^3-144x^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+72e^{2c_1}x\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+432e^{2c_1}{x}^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha )){\sin }^2(\alpha )+216e^{2c_1}{x}^2(e^{c_1}\sin (2\alpha )-2x\sin (\alpha )){}^2+\sqrt{(2(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^3-144x^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+72e^{2c_1}x\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+432e^{2c_1}{x}^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha )){\sin }^2(\alpha )+216e^{2c_1}{x}^2(e^{c_1}\sin (2\alpha )-2x\sin (\alpha )){}^2){}^2-4(24(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))x^2+24e^{2c_1}\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))x+(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^2){}^3}}}{6\sqrt[3]{2}}-\frac{128x^4-256e^{c_1}\cos (\alpha )x^3-96e^{2c_1}{x}^2+96e^{2c_1}\cos (2\alpha )x^2+48e^{3c_1}\cos (\alpha )x-16e^{3c_1}\cos (3\alpha )x+3e^{4c_1}+4e^{4c_1}\cos (2\alpha )+e^{4c_1}\cos (4\alpha )}{62^{2/3}\sqrt[3]{2(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^3-144x^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+72e^{2c_1}x\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+432e^{2c_1}{x}^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha )){\sin }^2(\alpha )+216e^{2c_1}{x}^2(e^{c_1}\sin (2\alpha )-2x\sin (\alpha )){}^2+\sqrt{(2(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^3-144x^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+72e^{2c_1}x\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+432e^{2c_1}{x}^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha )){\sin }^2(\alpha )+216e^{2c_1}{x}^2(e^{c_1}\sin (2\alpha )-2x\sin (\alpha )){}^2){}^2-4(24(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))x^2+24e^{2c_1}\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))x+(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^2){}^3}}}-\frac{-8e^{3c_1}{\sin }^3(\alpha )+4e^{c_1}(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))\sin (\alpha )+8e^{c_1}x(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))}{4\sqrt{e^{2c_1}{\sin }^2(\alpha )+\frac{1}{6}(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+\frac{1}{2}(-4x^2+4e^{c_1}\cos (\alpha )x+e^{2c_1}+e^{2c_1}\cos (2\alpha ))+\frac{\sqrt[3]{2(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^3-144x^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+72e^{2c_1}x\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+432e^{2c_1}{x}^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha )){\sin }^2(\alpha )+216e^{2c_1}{x}^2(e^{c_1}\sin (2\alpha )-2x\sin (\alpha )){}^2+\sqrt{(2(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^3-144x^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+72e^{2c_1}x\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+432e^{2c_1}{x}^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha )){\sin }^2(\alpha )+216e^{2c_1}{x}^2(e^{c_1}\sin (2\alpha )-2x\sin (\alpha )){}^2){}^2-4(24(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))x^2+24e^{2c_1}\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))x+(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^2){}^3}}}{6\sqrt[3]{2}}+\frac{128x^4-256e^{c_1}\cos (\alpha )x^3-96e^{2c_1}{x}^2+96e^{2c_1}\cos (2\alpha )x^2+48e^{3c_1}\cos (\alpha )x-16e^{3c_1}\cos (3\alpha )x+3e^{4c_1}+4e^{4c_1}\cos (2\alpha )+e^{4c_1}\cos (4\alpha )}{62^{2/3}\sqrt[3]{2(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^3-144x^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+72e^{2c_1}x\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+432e^{2c_1}{x}^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha )){\sin }^2(\alpha )+216e^{2c_1}{x}^2(e^{c_1}\sin (2\alpha )-2x\sin (\alpha )){}^2+\sqrt{(2(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^3-144x^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+72e^{2c_1}x\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+432e^{2c_1}{x}^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha )){\sin }^2(\alpha )+216e^{2c_1}{x}^2(e^{c_1}\sin (2\alpha )-2x\sin (\alpha )){}^2){}^2-4(24(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))x^2+24e^{2c_1}\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))x+(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^2){}^3}}}}}}\},\{y(x)\to -\frac{1}{2}{e}^{c_1}\sin (\alpha )+\frac{1}{2}\sqrt{e^{2c_1}{\sin }^2(\alpha )+\frac{1}{6}(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+\frac{1}{2}(-4x^2+4e^{c_1}\cos (\alpha )x+e^{2c_1}+e^{2c_1}\cos (2\alpha ))+\frac{\sqrt[3]{2(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^3-144x^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+72e^{2c_1}x\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+432e^{2c_1}{x}^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha )){\sin }^2(\alpha )+216e^{2c_1}{x}^2(e^{c_1}\sin (2\alpha )-2x\sin (\alpha )){}^2+\sqrt{(2(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^3-144x^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+72e^{2c_1}x\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+432e^{2c_1}{x}^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha )){\sin }^2(\alpha )+216e^{2c_1}{x}^2(e^{c_1}\sin (2\alpha )-2x\sin (\alpha )){}^2){}^2-4(24(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))x^2+24e^{2c_1}\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))x+(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^2){}^3}}}{6\sqrt[3]{2}}+\frac{128x^4-256e^{c_1}\cos (\alpha )x^3-96e^{2c_1}{x}^2+96e^{2c_1}\cos (2\alpha )x^2+48e^{3c_1}\cos (\alpha )x-16e^{3c_1}\cos (3\alpha )x+3e^{4c_1}+4e^{4c_1}\cos (2\alpha )+e^{4c_1}\cos (4\alpha )}{62^{2/3}\sqrt[3]{2(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^3-144x^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+72e^{2c_1}x\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+432e^{2c_1}{x}^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha )){\sin }^2(\alpha )+216e^{2c_1}{x}^2(e^{c_1}\sin (2\alpha )-2x\sin (\alpha )){}^2+\sqrt{(2(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^3-144x^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+72e^{2c_1}x\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+432e^{2c_1}{x}^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha )){\sin }^2(\alpha )+216e^{2c_1}{x}^2(e^{c_1}\sin (2\alpha )-2x\sin (\alpha )){}^2){}^2-4(24(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))x^2+24e^{2c_1}\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))x+(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^2){}^3}}}}-\frac{1}{2}\sqrt{2e^{2c_1}{\sin }^2(\alpha )+\frac{2}{3}(-4x^2+4e^{c_1}\cos (\alpha )x+e^{2c_1}+e^{2c_1}\cos (2\alpha ))-\frac{\sqrt[3]{2(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^3-144x^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+72e^{2c_1}x\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+432e^{2c_1}{x}^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha )){\sin }^2(\alpha )+216e^{2c_1}{x}^2(e^{c_1}\sin (2\alpha )-2x\sin (\alpha )){}^2+\sqrt{(2(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^3-144x^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+72e^{2c_1}x\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+432e^{2c_1}{x}^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha )){\sin }^2(\alpha )+216e^{2c_1}{x}^2(e^{c_1}\sin (2\alpha )-2x\sin (\alpha )){}^2){}^2-4(24(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))x^2+24e^{2c_1}\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))x+(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^2){}^3}}}{6\sqrt[3]{2}}-\frac{128x^4-256e^{c_1}\cos (\alpha )x^3-96e^{2c_1}{x}^2+96e^{2c_1}\cos (2\alpha )x^2+48e^{3c_1}\cos (\alpha )x-16e^{3c_1}\cos (3\alpha )x+3e^{4c_1}+4e^{4c_1}\cos (2\alpha )+e^{4c_1}\cos (4\alpha )}{62^{2/3}\sqrt[3]{2(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^3-144x^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+72e^{2c_1}x\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+432e^{2c_1}{x}^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha )){\sin }^2(\alpha )+216e^{2c_1}{x}^2(e^{c_1}\sin (2\alpha )-2x\sin (\alpha )){}^2+\sqrt{(2(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^3-144x^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+72e^{2c_1}x\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+432e^{2c_1}{x}^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha )){\sin }^2(\alpha )+216e^{2c_1}{x}^2(e^{c_1}\sin (2\alpha )-2x\sin (\alpha )){}^2){}^2-4(24(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))x^2+24e^{2c_1}\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))x+(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^2){}^3}}}+\frac{-8e^{3c_1}{\sin }^3(\alpha )+4e^{c_1}(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))\sin (\alpha )+8e^{c_1}x(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))}{4\sqrt{e^{2c_1}{\sin }^2(\alpha )+\frac{1}{6}(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+\frac{1}{2}(-4x^2+4e^{c_1}\cos (\alpha )x+e^{2c_1}+e^{2c_1}\cos (2\alpha ))+\frac{\sqrt[3]{2(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^3-144x^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+72e^{2c_1}x\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+432e^{2c_1}{x}^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha )){\sin }^2(\alpha )+216e^{2c_1}{x}^2(e^{c_1}\sin (2\alpha )-2x\sin (\alpha )){}^2+\sqrt{(2(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^3-144x^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+72e^{2c_1}x\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+432e^{2c_1}{x}^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha )){\sin }^2(\alpha )+216e^{2c_1}{x}^2(e^{c_1}\sin (2\alpha )-2x\sin (\alpha )){}^2){}^2-4(24(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))x^2+24e^{2c_1}\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))x+(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^2){}^3}}}{6\sqrt[3]{2}}+\frac{128x^4-256e^{c_1}\cos (\alpha )x^3-96e^{2c_1}{x}^2+96e^{2c_1}\cos (2\alpha )x^2+48e^{3c_1}\cos (\alpha )x-16e^{3c_1}\cos (3\alpha )x+3e^{4c_1}+4e^{4c_1}\cos (2\alpha )+e^{4c_1}\cos (4\alpha )}{62^{2/3}\sqrt[3]{2(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^3-144x^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+72e^{2c_1}x\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+432e^{2c_1}{x}^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha )){\sin }^2(\alpha )+216e^{2c_1}{x}^2(e^{c_1}\sin (2\alpha )-2x\sin (\alpha )){}^2+\sqrt{(2(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^3-144x^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+72e^{2c_1}x\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+432e^{2c_1}{x}^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha )){\sin }^2(\alpha )+216e^{2c_1}{x}^2(e^{c_1}\sin (2\alpha )-2x\sin (\alpha )){}^2){}^2-4(24(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))x^2+24e^{2c_1}\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))x+(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^2){}^3}}}}}}\},\{y(x)\to -\frac{1}{2}{e}^{c_1}\sin (\alpha )+\frac{1}{2}\sqrt{e^{2c_1}{\sin }^2(\alpha )+\frac{1}{6}(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+\frac{1}{2}(-4x^2+4e^{c_1}\cos (\alpha )x+e^{2c_1}+e^{2c_1}\cos (2\alpha ))+\frac{\sqrt[3]{2(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^3-144x^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+72e^{2c_1}x\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+432e^{2c_1}{x}^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha )){\sin }^2(\alpha )+216e^{2c_1}{x}^2(e^{c_1}\sin (2\alpha )-2x\sin (\alpha )){}^2+\sqrt{(2(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^3-144x^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+72e^{2c_1}x\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+432e^{2c_1}{x}^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha )){\sin }^2(\alpha )+216e^{2c_1}{x}^2(e^{c_1}\sin (2\alpha )-2x\sin (\alpha )){}^2){}^2-4(24(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))x^2+24e^{2c_1}\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))x+(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^2){}^3}}}{6\sqrt[3]{2}}+\frac{128x^4-256e^{c_1}\cos (\alpha )x^3-96e^{2c_1}{x}^2+96e^{2c_1}\cos (2\alpha )x^2+48e^{3c_1}\cos (\alpha )x-16e^{3c_1}\cos (3\alpha )x+3e^{4c_1}+4e^{4c_1}\cos (2\alpha )+e^{4c_1}\cos (4\alpha )}{62^{2/3}\sqrt[3]{2(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^3-144x^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+72e^{2c_1}x\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+432e^{2c_1}{x}^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha )){\sin }^2(\alpha )+216e^{2c_1}{x}^2(e^{c_1}\sin (2\alpha )-2x\sin (\alpha )){}^2+\sqrt{(2(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^3-144x^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+72e^{2c_1}x\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+432e^{2c_1}{x}^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha )){\sin }^2(\alpha 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)){}^3-144x^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+72e^{2c_1}x\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+432e^{2c_1}{x}^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha )){\sin }^2(\alpha )+216e^{2c_1}{x}^2(e^{c_1}\sin (2\alpha )-2x\sin (\alpha )){}^2){}^2-4(24(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))x^2+24e^{2c_1}\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))x+(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^2){}^3}}}{6\sqrt[3]{2}}-\frac{128x^4-256e^{c_1}\cos (\alpha )x^3-96e^{2c_1}{x}^2+96e^{2c_1}\cos (2\alpha )x^2+48e^{3c_1}\cos (\alpha )x-16e^{3c_1}\cos (3\alpha )x+3e^{4c_1}+4e^{4c_1}\cos (2\alpha )+e^{4c_1}\cos (4\alpha )}{62^{2/3}\sqrt[3]{2(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^3-144x^2(2x^2-4e^{c_1}\cos (\alpha 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)(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))x+(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^2){}^3}}}+\frac{-8e^{3c_1}{\sin }^3(\alpha )+4e^{c_1}(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))\sin (\alpha )+8e^{c_1}x(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))}{4\sqrt{e^{2c_1}{\sin }^2(\alpha )+\frac{1}{6}(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+\frac{1}{2}(-4x^2+4e^{c_1}\cos (\alpha )x+e^{2c_1}+e^{2c_1}\cos (2\alpha ))+\frac{\sqrt[3]{2(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^3-144x^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+72e^{2c_1}x\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+432e^{2c_1}{x}^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha )){\sin }^2(\alpha )+216e^{2c_1}{x}^2(e^{c_1}\sin (2\alpha )-2x\sin (\alpha )){}^2+\sqrt{(2(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^3-144x^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+72e^{2c_1}x\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+432e^{2c_1}{x}^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha )){\sin }^2(\alpha )+216e^{2c_1}{x}^2(e^{c_1}\sin (2\alpha )-2x\sin (\alpha )){}^2){}^2-4(24(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))x^2+24e^{2c_1}\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))x+(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^2){}^3}}}{6\sqrt[3]{2}}+\frac{128x^4-256e^{c_1}\cos (\alpha )x^3-96e^{2c_1}{x}^2+96e^{2c_1}\cos (2\alpha )x^2+48e^{3c_1}\cos (\alpha )x-16e^{3c_1}\cos (3\alpha )x+3e^{4c_1}+4e^{4c_1}\cos (2\alpha )+e^{4c_1}\cos (4\alpha )}{62^{2/3}\sqrt[3]{2(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^3-144x^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+72e^{2c_1}x\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+432e^{2c_1}{x}^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha )){\sin }^2(\alpha )+216e^{2c_1}{x}^2(e^{c_1}\sin (2\alpha )-2x\sin (\alpha )){}^2+\sqrt{(2(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^3-144x^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+72e^{2c_1}x\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha ))+432e^{2c_1}{x}^2(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha )){\sin }^2(\alpha )+216e^{2c_1}{x}^2(e^{c_1}\sin (2\alpha )-2x\sin (\alpha )){}^2){}^2-4(24(2x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}+e^{2c_1}\cos (2\alpha ))x^2+24e^{2c_1}\sin (\alpha )(e^{c_1}\sin (2\alpha )-2x\sin (\alpha ))x+(4x^2-4e^{c_1}\cos (\alpha )x-e^{2c_1}-e^{2c_1}\cos (2\alpha )){}^2){}^3}}}}}}\}\}$$

✓ Maple : cpu = 0.743 (sec), leaf count = 136

$$\{y\left(x\right)=\mathit{R}\mathit{o}\mathit{o}\mathit{t}\mathit{O}\mathit{f}(-\ln \left(x\right)+{\int }^{\mathit{\_}\mathit{Z}}-\frac{1}{({\mathit{\_}\mathit{a}}^2+1)(\cos \left(2\alpha \right){\mathit{\_}\mathit{a}}^2+2\mathit{\_}\mathit{a}\sin \left(2\alpha \right)+{\mathit{\_}\mathit{a}}^2-\cos \left(2\alpha \right)+1)}({\mathit{\_}\mathit{a}}^3\cos \left(2\alpha \right)+3{\mathit{\_}\mathit{a}}^2\sin \left(2\alpha \right)+{\mathit{\_}\mathit{a}}^3-3\mathit{\_}\mathit{a}\cos \left(2\alpha \right)-\sin \left(2\alpha \right)-\sqrt{2\cos \left(2\alpha \right){\mathit{\_}\mathit{a}}^4+4\sin \left(2\alpha \right){\mathit{\_}\mathit{a}}^3+2{\mathit{\_}\mathit{a}}^4+4\mathit{\_}\mathit{a}\sin \left(2\alpha \right)+4{\mathit{\_}\mathit{a}}^2-2\cos \left(2\alpha \right)+2}+\mathit{\_}\mathit{a})d\mathit{\_}\mathit{a}+\mathit{\_}\mathit{C}\mathit{1})x\}$$