4.3.25 \(y'(x)=\sec ^2(x) \sec ^3(y(x))\)

ODE
\[ y'(x)=\sec ^2(x) \sec ^3(y(x)) \] ODE Classification

[_separable]

Book solution method
Separable ODE, Neither variable missing

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

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

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

\[ \left \{ {\frac {24\,{\it \_C1}\,\cos \relax (x ) +24\,\sin \relax (x ) -9\,\sin \left (x+y \relax (x ) \right ) +9\,\sin \left (x-y \relax (x ) \right ) -\sin \left (3\,y \relax (x ) +x \right ) +\sin \left (x-3\,y \relax (x ) \right ) }{24\,\cos \relax (x ) }}=0 \right \} \] 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