##### 4.2.44 $$y'(x)=y(x) \left (y(x)^3 \sec (x)+\tan (x)\right )$$

ODE
$y'(x)=y(x) \left (y(x)^3 \sec (x)+\tan (x)\right )$ ODE Classiﬁcation

[_Bernoulli]

Book solution method
The Bernoulli ODE

Mathematica
cpu = 0.266836 (sec), leaf count = 98

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

Maple
cpu = 0.133 (sec), leaf count = 364

$\left [y \left (x \right ) = \frac {\left (\cos \left (x \right ) \left (\textit {\_C1} \left (\sin ^{4}\left (x \right )\right )+2 \cos \left (x \right ) \left (\sin ^{3}\left (x \right )\right )-2 \textit {\_C1} \left (\sin ^{2}\left (x \right )\right )-3 \cos \left (x \right ) \sin \left (x \right )+\textit {\_C1} \right )^{2}\right )^{\frac {1}{3}}}{\textit {\_C1} \left (\sin ^{4}\left (x \right )\right )+2 \cos \left (x \right ) \left (\sin ^{3}\left (x \right )\right )-2 \textit {\_C1} \left (\sin ^{2}\left (x \right )\right )-3 \cos \left (x \right ) \sin \left (x \right )+\textit {\_C1}}, y \left (x \right ) = -\frac {\left (\cos \left (x \right ) \left (\textit {\_C1} \left (\sin ^{4}\left (x \right )\right )+2 \cos \left (x \right ) \left (\sin ^{3}\left (x \right )\right )-2 \textit {\_C1} \left (\sin ^{2}\left (x \right )\right )-3 \cos \left (x \right ) \sin \left (x \right )+\textit {\_C1} \right )^{2}\right )^{\frac {1}{3}}}{2 \left (\textit {\_C1} \left (\sin ^{4}\left (x \right )\right )+2 \cos \left (x \right ) \left (\sin ^{3}\left (x \right )\right )-2 \textit {\_C1} \left (\sin ^{2}\left (x \right )\right )-3 \cos \left (x \right ) \sin \left (x \right )+\textit {\_C1} \right )}-\frac {i \sqrt {3}\, \left (\cos \left (x \right ) \left (\textit {\_C1} \left (\sin ^{4}\left (x \right )\right )+2 \cos \left (x \right ) \left (\sin ^{3}\left (x \right )\right )-2 \textit {\_C1} \left (\sin ^{2}\left (x \right )\right )-3 \cos \left (x \right ) \sin \left (x \right )+\textit {\_C1} \right )^{2}\right )^{\frac {1}{3}}}{2 \left (\textit {\_C1} \left (\sin ^{4}\left (x \right )\right )+2 \cos \left (x \right ) \left (\sin ^{3}\left (x \right )\right )-2 \textit {\_C1} \left (\sin ^{2}\left (x \right )\right )-3 \cos \left (x \right ) \sin \left (x \right )+\textit {\_C1} \right )}, y \left (x \right ) = -\frac {\left (\cos \left (x \right ) \left (\textit {\_C1} \left (\sin ^{4}\left (x \right )\right )+2 \cos \left (x \right ) \left (\sin ^{3}\left (x \right )\right )-2 \textit {\_C1} \left (\sin ^{2}\left (x \right )\right )-3 \cos \left (x \right ) \sin \left (x \right )+\textit {\_C1} \right )^{2}\right )^{\frac {1}{3}}}{2 \left (\textit {\_C1} \left (\sin ^{4}\left (x \right )\right )+2 \cos \left (x \right ) \left (\sin ^{3}\left (x \right )\right )-2 \textit {\_C1} \left (\sin ^{2}\left (x \right )\right )-3 \cos \left (x \right ) \sin \left (x \right )+\textit {\_C1} \right )}+\frac {i \sqrt {3}\, \left (\cos \left (x \right ) \left (\textit {\_C1} \left (\sin ^{4}\left (x \right )\right )+2 \cos \left (x \right ) \left (\sin ^{3}\left (x \right )\right )-2 \textit {\_C1} \left (\sin ^{2}\left (x \right )\right )-3 \cos \left (x \right ) \sin \left (x \right )+\textit {\_C1} \right )^{2}\right )^{\frac {1}{3}}}{2 \textit {\_C1} \left (\sin ^{4}\left (x \right )\right )+4 \cos \left (x \right ) \left (\sin ^{3}\left (x \right )\right )-4 \textit {\_C1} \left (\sin ^{2}\left (x \right )\right )-6 \cos \left (x \right ) \sin \left (x \right )+2 \textit {\_C1}}\right ]$ Mathematica raw input

DSolve[y'[x] == y[x]*(Tan[x] + Sec[x]*y[x]^3),y[x],x]

Mathematica raw output

{{y[x] -> (C[1]*Cos[x]^3 - Sin[x] - 2*Cos[x]^2*Sin[x])^(-1/3)}, {y[x] -> -((-1)^
(1/3)/(C[1]*Cos[x]^3 - Sin[x] - 2*Cos[x]^2*Sin[x])^(1/3))}, {y[x] -> (-1)^(2/3)/
(C[1]*Cos[x]^3 - Sin[x] - 2*Cos[x]^2*Sin[x])^(1/3)}}

Maple raw input

dsolve(diff(y(x),x) = (tan(x)+y(x)^3*sec(x))*y(x), y(x))

Maple raw output

[y(x) = 1/(_C1*sin(x)^4+2*cos(x)*sin(x)^3-2*_C1*sin(x)^2-3*cos(x)*sin(x)+_C1)*(c
os(x)*(_C1*sin(x)^4+2*cos(x)*sin(x)^3-2*_C1*sin(x)^2-3*cos(x)*sin(x)+_C1)^2)^(1/
3), y(x) = -1/2/(_C1*sin(x)^4+2*cos(x)*sin(x)^3-2*_C1*sin(x)^2-3*cos(x)*sin(x)+_
C1)*(cos(x)*(_C1*sin(x)^4+2*cos(x)*sin(x)^3-2*_C1*sin(x)^2-3*cos(x)*sin(x)+_C1)^
2)^(1/3)-1/2*I*3^(1/2)/(_C1*sin(x)^4+2*cos(x)*sin(x)^3-2*_C1*sin(x)^2-3*cos(x)*s
in(x)+_C1)*(cos(x)*(_C1*sin(x)^4+2*cos(x)*sin(x)^3-2*_C1*sin(x)^2-3*cos(x)*sin(x
)+_C1)^2)^(1/3), y(x) = -1/2/(_C1*sin(x)^4+2*cos(x)*sin(x)^3-2*_C1*sin(x)^2-3*co
s(x)*sin(x)+_C1)*(cos(x)*(_C1*sin(x)^4+2*cos(x)*sin(x)^3-2*_C1*sin(x)^2-3*cos(x)
*sin(x)+_C1)^2)^(1/3)+1/2*I*3^(1/2)/(_C1*sin(x)^4+2*cos(x)*sin(x)^3-2*_C1*sin(x)
^2-3*cos(x)*sin(x)+_C1)*(cos(x)*(_C1*sin(x)^4+2*cos(x)*sin(x)^3-2*_C1*sin(x)^2-3
*cos(x)*sin(x)+_C1)^2)^(1/3)]