4.3.25 $y^{\prime }(x)={\sec }^2(x){\sec }^3(y(x))$

ODE
$$y^{\prime }(x)={\sec }^2(x){\sec }^3(y(x))$$ ODE Classification

[_separable]

Book solution method
Separable ODE, Neither variable missing

Mathematica ✓
cpu = 0.389479 (sec), leaf count = 2959

$$\{\{y(x)\to -{\cos }^{-1}(-\frac{\sqrt{2^{2/3}\sqrt[3]{-9c_1^2-18\tan (x)c_1-9{\tan }^2(x)+3\sqrt{(c_1+\tan (x)){}^2(9c_1^2+18\tan (x)c_1+9{\tan }^2(x)-4)}+2}-2+\frac{2\sqrt[3]{2}}{\sqrt[3]{-9c_1^2-18\tan (x)c_1-9{\tan }^2(x)+3\sqrt{(c_1+\tan (x)){}^2(9c_1^2+18\tan (x)c_1+9{\tan }^2(x)-4)}+2}}}}{\sqrt{2}})\},\{y(x)\to {\cos }^{-1}(-\frac{\sqrt{2^{2/3}\sqrt[3]{-9c_1^2-18\tan (x)c_1-9{\tan }^2(x)+3\sqrt{(c_1+\tan (x)){}^2(9c_1^2+18\tan (x)c_1+9{\tan }^2(x)-4)}+2}-2+\frac{2\sqrt[3]{2}}{\sqrt[3]{-9c_1^2-18\tan (x)c_1-9{\tan }^2(x)+3\sqrt{(c_1+\tan (x)){}^2(9c_1^2+18\tan (x)c_1+9{\tan }^2(x)-4)}+2}}}}{\sqrt{2}})\},\{y(x)\to -{\cos }^{-1}\left(\frac{\sqrt{2^{2/3}\sqrt[3]{-9c_1^2-18\tan (x)c_1-9{\tan }^2(x)+3\sqrt{(c_1+\tan (x)){}^2(9c_1^2+18\tan (x)c_1+9{\tan }^2(x)-4)}+2}-2+\frac{2\sqrt[3]{2}}{\sqrt[3]{-9c_1^2-18\tan (x)c_1-9{\tan }^2(x)+3\sqrt{(c_1+\tan (x)){}^2(9c_1^2+18\tan (x)c_1+9{\tan }^2(x)-4)}+2}}}}{\sqrt{2}}\right)\},\{y(x)\to {\cos }^{-1}\left(\frac{\sqrt{2^{2/3}\sqrt[3]{-9c_1^2-18\tan (x)c_1-9{\tan }^2(x)+3\sqrt{(c_1+\tan (x)){}^2(9c_1^2+18\tan (x)c_1+9{\tan }^2(x)-4)}+2}-2+\frac{2\sqrt[3]{2}}{\sqrt[3]{-9c_1^2-18\tan (x)c_1-9{\tan }^2(x)+3\sqrt{(c_1+\tan (x)){}^2(9c_1^2+18\tan (x)c_1+9{\tan }^2(x)-4)}+2}}}}{\sqrt{2}}\right)\},\{y(x)\to -{\cos }^{-1}(-\frac{1}{2}\sqrt{-\frac{i(2^{2/3}\sqrt{3}(-9c_1^2-18\tan (x)c_1-9{\tan }^2(x)+3\sqrt{(c_1+\tan (x)){}^2(9c_1^2+18\tan (x)c_1+9{\tan }^2(x)-4)}+2){}^{2/3}-i{2}^{2/3}(-9c_1^2-18\tan (x)c_1-9{\tan }^2(x)+3\sqrt{(c_1+\tan (x)){}^2(9c_1^2+18\tan (x)c_1+9{\tan }^2(x)-4)}+2){}^{2/3}-4i\sqrt[3]{-9c_1^2-18\tan (x)c_1-9{\tan }^2(x)+3\sqrt{(c_1+\tan (x)){}^2(9c_1^2+18\tan (x)c_1+9{\tan }^2(x)-4)}+2}-2\sqrt[3]{2}\sqrt{3}-2i\sqrt[3]{2})}{\sqrt[3]{-9c_1^2-18\tan (x)c_1-9{\tan }^2(x)+3\sqrt{(c_1+\tan (x)){}^2(9c_1^2+18\tan (x)c_1+9{\tan }^2(x)-4)}+2}}})\},\{y(x)\to {\cos }^{-1}(-\frac{1}{2}\sqrt{-\frac{i(2^{2/3}\sqrt{3}(-9c_1^2-18\tan (x)c_1-9{\tan }^2(x)+3\sqrt{(c_1+\tan (x)){}^2(9c_1^2+18\tan (x)c_1+9{\tan }^2(x)-4)}+2){}^{2/3}-i{2}^{2/3}(-9c_1^2-18\tan (x)c_1-9{\tan }^2(x)+3\sqrt{(c_1+\tan (x)){}^2(9c_1^2+18\tan (x)c_1+9{\tan }^2(x)-4)}+2){}^{2/3}-4i\sqrt[3]{-9c_1^2-18\tan (x)c_1-9{\tan }^2(x)+3\sqrt{(c_1+\tan (x)){}^2(9c_1^2+18\tan (x)c_1+9{\tan }^2(x)-4)}+2}-2\sqrt[3]{2}\sqrt{3}-2i\sqrt[3]{2})}{\sqrt[3]{-9c_1^2-18\tan (x)c_1-9{\tan }^2(x)+3\sqrt{(c_1+\tan (x)){}^2(9c_1^2+18\tan (x)c_1+9{\tan }^2(x)-4)}+2}}})\},\{y(x)\to -{\cos }^{-1}\left(\frac{1}{2}\sqrt{-\frac{i(2^{2/3}\sqrt{3}(-9c_1^2-18\tan (x)c_1-9{\tan }^2(x)+3\sqrt{(c_1+\tan (x)){}^2(9c_1^2+18\tan (x)c_1+9{\tan }^2(x)-4)}+2){}^{2/3}-i{2}^{2/3}(-9c_1^2-18\tan (x)c_1-9{\tan }^2(x)+3\sqrt{(c_1+\tan (x)){}^2(9c_1^2+18\tan (x)c_1+9{\tan }^2(x)-4)}+2){}^{2/3}-4i\sqrt[3]{-9c_1^2-18\tan (x)c_1-9{\tan }^2(x)+3\sqrt{(c_1+\tan (x)){}^2(9c_1^2+18\tan (x)c_1+9{\tan }^2(x)-4)}+2}-2\sqrt[3]{2}\sqrt{3}-2i\sqrt[3]{2})}{\sqrt[3]{-9c_1^2-18\tan (x)c_1-9{\tan }^2(x)+3\sqrt{(c_1+\tan (x)){}^2(9c_1^2+18\tan (x)c_1+9{\tan }^2(x)-4)}+2}}}\right)\},\{y(x)\to {\cos }^{-1}\left(\frac{1}{2}\sqrt{-\frac{i(2^{2/3}\sqrt{3}(-9c_1^2-18\tan (x)c_1-9{\tan }^2(x)+3\sqrt{(c_1+\tan (x)){}^2(9c_1^2+18\tan (x)c_1+9{\tan }^2(x)-4)}+2){}^{2/3}-i{2}^{2/3}(-9c_1^2-18\tan (x)c_1-9{\tan }^2(x)+3\sqrt{(c_1+\tan (x)){}^2(9c_1^2+18\tan (x)c_1+9{\tan }^2(x)-4)}+2){}^{2/3}-4i\sqrt[3]{-9c_1^2-18\tan (x)c_1-9{\tan }^2(x)+3\sqrt{(c_1+\tan (x)){}^2(9c_1^2+18\tan (x)c_1+9{\tan }^2(x)-4)}+2}-2\sqrt[3]{2}\sqrt{3}-2i\sqrt[3]{2})}{\sqrt[3]{-9c_1^2-18\tan (x)c_1-9{\tan }^2(x)+3\sqrt{(c_1+\tan (x)){}^2(9c_1^2+18\tan (x)c_1+9{\tan }^2(x)-4)}+2}}}\right)\},\{y(x)\to -{\cos }^{-1}(-\frac{1}{2}\sqrt{\frac{i(2^{2/3}\sqrt{3}(-9c_1^2-18\tan (x)c_1-9{\tan }^2(x)+3\sqrt{(c_1+\tan (x)){}^2(9c_1^2+18\tan (x)c_1+9{\tan }^2(x)-4)}+2){}^{2/3}+i{2}^{2/3}(-9c_1^2-18\tan (x)c_1-9{\tan }^2(x)+3\sqrt{(c_1+\tan (x)){}^2(9c_1^2+18\tan (x)c_1+9{\tan }^2(x)-4)}+2){}^{2/3}+4i\sqrt[3]{-9c_1^2-18\tan (x)c_1-9{\tan }^2(x)+3\sqrt{(c_1+\tan (x)){}^2(9c_1^2+18\tan (x)c_1+9{\tan }^2(x)-4)}+2}-2\sqrt[3]{2}\sqrt{3}+2i\sqrt[3]{2})}{\sqrt[3]{-9c_1^2-18\tan (x)c_1-9{\tan }^2(x)+3\sqrt{(c_1+\tan (x)){}^2(9c_1^2+18\tan (x)c_1+9{\tan }^2(x)-4)}+2}}})\},\{y(x)\to {\cos }^{-1}(-\frac{1}{2}\sqrt{\frac{i(2^{2/3}\sqrt{3}(-9c_1^2-18\tan (x)c_1-9{\tan }^2(x)+3\sqrt{(c_1+\tan (x)){}^2(9c_1^2+18\tan (x)c_1+9{\tan }^2(x)-4)}+2){}^{2/3}+i{2}^{2/3}(-9c_1^2-18\tan (x)c_1-9{\tan }^2(x)+3\sqrt{(c_1+\tan (x)){}^2(9c_1^2+18\tan (x)c_1+9{\tan }^2(x)-4)}+2){}^{2/3}+4i\sqrt[3]{-9c_1^2-18\tan (x)c_1-9{\tan }^2(x)+3\sqrt{(c_1+\tan (x)){}^2(9c_1^2+18\tan (x)c_1+9{\tan }^2(x)-4)}+2}-2\sqrt[3]{2}\sqrt{3}+2i\sqrt[3]{2})}{\sqrt[3]{-9c_1^2-18\tan (x)c_1-9{\tan }^2(x)+3\sqrt{(c_1+\tan (x)){}^2(9c_1^2+18\tan (x)c_1+9{\tan }^2(x)-4)}+2}}})\},\{y(x)\to -{\cos }^{-1}\left(\frac{1}{2}\sqrt{\frac{i(2^{2/3}\sqrt{3}(-9c_1^2-18\tan (x)c_1-9{\tan }^2(x)+3\sqrt{(c_1+\tan (x)){}^2(9c_1^2+18\tan (x)c_1+9{\tan }^2(x)-4)}+2){}^{2/3}+i{2}^{2/3}(-9c_1^2-18\tan (x)c_1-9{\tan }^2(x)+3\sqrt{(c_1+\tan (x)){}^2(9c_1^2+18\tan (x)c_1+9{\tan }^2(x)-4)}+2){}^{2/3}+4i\sqrt[3]{-9c_1^2-18\tan (x)c_1-9{\tan }^2(x)+3\sqrt{(c_1+\tan (x)){}^2(9c_1^2+18\tan (x)c_1+9{\tan }^2(x)-4)}+2}-2\sqrt[3]{2}\sqrt{3}+2i\sqrt[3]{2})}{\sqrt[3]{-9c_1^2-18\tan (x)c_1-9{\tan }^2(x)+3\sqrt{(c_1+\tan (x)){}^2(9c_1^2+18\tan (x)c_1+9{\tan }^2(x)-4)}+2}}}\right)\},\{y(x)\to {\cos }^{-1}\left(\frac{1}{2}\sqrt{\frac{i(2^{2/3}\sqrt{3}(-9c_1^2-18\tan (x)c_1-9{\tan }^2(x)+3\sqrt{(c_1+\tan (x)){}^2(9c_1^2+18\tan (x)c_1+9{\tan }^2(x)-4)}+2){}^{2/3}+i{2}^{2/3}(-9c_1^2-18\tan (x)c_1-9{\tan }^2(x)+3\sqrt{(c_1+\tan (x)){}^2(9c_1^2+18\tan (x)c_1+9{\tan }^2(x)-4)}+2){}^{2/3}+4i\sqrt[3]{-9c_1^2-18\tan (x)c_1-9{\tan }^2(x)+3\sqrt{(c_1+\tan (x)){}^2(9c_1^2+18\tan (x)c_1+9{\tan }^2(x)-4)}+2}-2\sqrt[3]{2}\sqrt{3}+2i\sqrt[3]{2})}{\sqrt[3]{-9c_1^2-18\tan (x)c_1-9{\tan }^2(x)+3\sqrt{(c_1+\tan (x)){}^2(9c_1^2+18\tan (x)c_1+9{\tan }^2(x)-4)}+2}}}\right)\}\}$$

Maple ✓
cpu = 0.038 (sec), leaf count = 51

$$\{\frac{24\mathit{\_}\mathit{C}\mathit{1}\cos \left(x\right)+24\sin \left(x\right)-9\sin (x+y\left(x\right))+9\sin (x-y\left(x\right))-\sin (3y\left(x\right)+x)+\sin (x-3y\left(x\right))}{24\cos \left(x\right)}=0\}$$ Mathematica raw input

DSolve[y'[x] == Sec[x]^2*Sec[y[x]]^3,y[x],x]

Mathematica raw output

{{y[x] -> -ArcCos[-(Sqrt[-2 + (2*2^(1/3))/(2 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9*Tan
[x]^2 + 3*Sqrt[(C[1] + Tan[x])^2*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^2)])
^(1/3) + 2^(2/3)*(2 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1] + Ta
n[x])^2*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^2)])^(1/3)]/Sqrt[2])]}, {y[x]
 -> ArcCos[-(Sqrt[-2 + (2*2^(1/3))/(2 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9*Tan[x]^2 +
 3*Sqrt[(C[1] + Tan[x])^2*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^2)])^(1/3) 
+ 2^(2/3)*(2 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1] + Tan[x])^2
*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^2)])^(1/3)]/Sqrt[2])]}, {y[x] -> -Ar
cCos[Sqrt[-2 + (2*2^(1/3))/(2 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[
(C[1] + Tan[x])^2*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^2)])^(1/3) + 2^(2/3
)*(2 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1] + Tan[x])^2*(-4 + 9
*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^2)])^(1/3)]/Sqrt[2]]}, {y[x] -> ArcCos[Sqrt[
-2 + (2*2^(1/3))/(2 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1] + Ta
n[x])^2*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^2)])^(1/3) + 2^(2/3)*(2 - 9*C
[1]^2 - 18*C[1]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1] + Tan[x])^2*(-4 + 9*C[1]^2 + 
18*C[1]*Tan[x] + 9*Tan[x]^2)])^(1/3)]/Sqrt[2]]}, {y[x] -> -ArcCos[-Sqrt[((-I)*((
-2*I)*2^(1/3) - 2*2^(1/3)*Sqrt[3] - (4*I)*(2 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9*Tan
[x]^2 + 3*Sqrt[(C[1] + Tan[x])^2*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^2)])
^(1/3) - I*2^(2/3)*(2 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1] + 
Tan[x])^2*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^2)])^(2/3) + 2^(2/3)*Sqrt[3
]*(2 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1] + Tan[x])^2*(-4 + 9
*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^2)])^(2/3)))/(2 - 9*C[1]^2 - 18*C[1]*Tan[x] 
- 9*Tan[x]^2 + 3*Sqrt[(C[1] + Tan[x])^2*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[
x]^2)])^(1/3)]/2]}, {y[x] -> ArcCos[-Sqrt[((-I)*((-2*I)*2^(1/3) - 2*2^(1/3)*Sqrt
[3] - (4*I)*(2 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1] + Tan[x])
^2*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^2)])^(1/3) - I*2^(2/3)*(2 - 9*C[1]
^2 - 18*C[1]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1] + Tan[x])^2*(-4 + 9*C[1]^2 + 18*
C[1]*Tan[x] + 9*Tan[x]^2)])^(2/3) + 2^(2/3)*Sqrt[3]*(2 - 9*C[1]^2 - 18*C[1]*Tan[
x] - 9*Tan[x]^2 + 3*Sqrt[(C[1] + Tan[x])^2*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9*T
an[x]^2)])^(2/3)))/(2 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1] + 
Tan[x])^2*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^2)])^(1/3)]/2]}, {y[x] -> -
ArcCos[Sqrt[((-I)*((-2*I)*2^(1/3) - 2*2^(1/3)*Sqrt[3] - (4*I)*(2 - 9*C[1]^2 - 18
*C[1]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1] + Tan[x])^2*(-4 + 9*C[1]^2 + 18*C[1]*Ta
n[x] + 9*Tan[x]^2)])^(1/3) - I*2^(2/3)*(2 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9*Tan[x]
^2 + 3*Sqrt[(C[1] + Tan[x])^2*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^2)])^(2
/3) + 2^(2/3)*Sqrt[3]*(2 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1]
 + Tan[x])^2*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^2)])^(2/3)))/(2 - 9*C[1]
^2 - 18*C[1]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1] + Tan[x])^2*(-4 + 9*C[1]^2 + 18*
C[1]*Tan[x] + 9*Tan[x]^2)])^(1/3)]/2]}, {y[x] -> ArcCos[Sqrt[((-I)*((-2*I)*2^(1/
3) - 2*2^(1/3)*Sqrt[3] - (4*I)*(2 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9*Tan[x]^2 + 3*S
qrt[(C[1] + Tan[x])^2*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^2)])^(1/3) - I*
2^(2/3)*(2 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1] + Tan[x])^2*(
-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^2)])^(2/3) + 2^(2/3)*Sqrt[3]*(2 - 9*C[
1]^2 - 18*C[1]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1] + Tan[x])^2*(-4 + 9*C[1]^2 + 1
8*C[1]*Tan[x] + 9*Tan[x]^2)])^(2/3)))/(2 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9*Tan[x]^
2 + 3*Sqrt[(C[1] + Tan[x])^2*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^2)])^(1/
3)]/2]}, {y[x] -> -ArcCos[-Sqrt[(I*((2*I)*2^(1/3) - 2*2^(1/3)*Sqrt[3] + (4*I)*(2
 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1] + Tan[x])^2*(-4 + 9*C[1
]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^2)])^(1/3) + I*2^(2/3)*(2 - 9*C[1]^2 - 18*C[1]*T
an[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1] + Tan[x])^2*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 
9*Tan[x]^2)])^(2/3) + 2^(2/3)*Sqrt[3]*(2 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9*Tan[x]^
2 + 3*Sqrt[(C[1] + Tan[x])^2*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^2)])^(2/
3)))/(2 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1] + Tan[x])^2*(-4 
+ 9*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^2)])^(1/3)]/2]}, {y[x] -> ArcCos[-Sqrt[(I
*((2*I)*2^(1/3) - 2*2^(1/3)*Sqrt[3] + (4*I)*(2 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9*T
an[x]^2 + 3*Sqrt[(C[1] + Tan[x])^2*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^2)
])^(1/3) + I*2^(2/3)*(2 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1] 
+ Tan[x])^2*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^2)])^(2/3) + 2^(2/3)*Sqrt
[3]*(2 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1] + Tan[x])^2*(-4 +
 9*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^2)])^(2/3)))/(2 - 9*C[1]^2 - 18*C[1]*Tan[x
] - 9*Tan[x]^2 + 3*Sqrt[(C[1] + Tan[x])^2*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9*Ta
n[x]^2)])^(1/3)]/2]}, {y[x] -> -ArcCos[Sqrt[(I*((2*I)*2^(1/3) - 2*2^(1/3)*Sqrt[3
] + (4*I)*(2 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1] + Tan[x])^2
*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^2)])^(1/3) + I*2^(2/3)*(2 - 9*C[1]^2
 - 18*C[1]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1] + Tan[x])^2*(-4 + 9*C[1]^2 + 18*C[
1]*Tan[x] + 9*Tan[x]^2)])^(2/3) + 2^(2/3)*Sqrt[3]*(2 - 9*C[1]^2 - 18*C[1]*Tan[x]
 - 9*Tan[x]^2 + 3*Sqrt[(C[1] + Tan[x])^2*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan
[x]^2)])^(2/3)))/(2 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1] + Ta
n[x])^2*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^2)])^(1/3)]/2]}, {y[x] -> Arc
Cos[Sqrt[(I*((2*I)*2^(1/3) - 2*2^(1/3)*Sqrt[3] + (4*I)*(2 - 9*C[1]^2 - 18*C[1]*T
an[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1] + Tan[x])^2*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 
9*Tan[x]^2)])^(1/3) + I*2^(2/3)*(2 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9*Tan[x]^2 + 3*
Sqrt[(C[1] + Tan[x])^2*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^2)])^(2/3) + 2
^(2/3)*Sqrt[3]*(2 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1] + Tan[
x])^2*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^2)])^(2/3)))/(2 - 9*C[1]^2 - 18
*C[1]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1] + Tan[x])^2*(-4 + 9*C[1]^2 + 18*C[1]*Ta
n[x] + 9*Tan[x]^2)])^(1/3)]/2]}}

Maple raw input

dsolve(diff(y(x),x) = sec(x)^2*sec(y(x))^3, y(x),'implicit')

Maple raw output

1/24*(24*_C1*cos(x)+24*sin(x)-9*sin(x+y(x))+9*sin(x-y(x))-sin(3*y(x)+x)+sin(x-3*
y(x)))/cos(x) = 0