\(\int \frac {\sqrt [3]{\tan (x)}}{(\cos (x)+\sin (x))^2} \, dx\) [318]

Optimal result

Integrand size = 14, antiderivative size = 61 \[ \int \frac {\sqrt [3]{\tan (x)}}{(\cos (x)+\sin (x))^2} \, dx=-\frac {\arctan \left (\frac {1-2 \sqrt [3]{\tan (x)}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} \log \left (1+\sqrt [3]{\tan (x)}\right )-\frac {1}{6} \log (1+\tan (x))-\frac {\sqrt [3]{\tan (x)}}{1+\tan (x)} \]

[Out]

-1/3*arctan(1/3*(1-2*tan(x)^(1/3))*3^(1/2))*3^(1/2)+1/2*ln(1+tan(x)^(1/3))-1/6*ln(1+tan(x))-tan(x)^(1/3)/(1+ta
n(x))

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {43, 60, 632, 210, 31} \[ \int \frac {\sqrt [3]{\tan (x)}}{(\cos (x)+\sin (x))^2} \, dx=-\frac {\arctan \left (\frac {1-2 \sqrt [3]{\tan (x)}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\sqrt [3]{\tan (x)}}{\tan (x)+1}+\frac {1}{2} \log \left (\sqrt [3]{\tan (x)}+1\right )-\frac {1}{6} \log (\tan (x)+1) \]

[In]

Int[Tan[x]^(1/3)/(Cos[x] + Sin[x])^2,x]

[Out]

-(ArcTan[(1 - 2*Tan[x]^(1/3))/Sqrt[3]]/Sqrt[3]) + Log[1 + Tan[x]^(1/3)]/2 - Log[1 + Tan[x]]/6 - Tan[x]^(1/3)/(
1 + Tan[x])

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + 1))), x] - Dist[d*(n/(b*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

Rule 60

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[-(b*c - a*d)/b, 3]}, Simp[-
Log[RemoveContent[a + b*x, x]]/(2*b*q^2), x] + (Dist[3/(2*b*q), Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d*x
)^(1/3)], x] + Dist[3/(2*b*q^2), Subst[Int[1/(q + x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x]
&& NegQ[(b*c - a*d)/b]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {\sqrt [3]{x}}{(1+x)^2} \, dx,x,\tan (x)\right ) \\ & = -\frac {\sqrt [3]{\tan (x)}}{1+\tan (x)}+\frac {1}{3} \text {Subst}\left (\int \frac {1}{x^{2/3} (1+x)} \, dx,x,\tan (x)\right ) \\ & = -\frac {1}{6} \log (1+\tan (x))-\frac {\sqrt [3]{\tan (x)}}{1+\tan (x)}+\frac {1}{2} \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sqrt [3]{\tan (x)}\right )+\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt [3]{\tan (x)}\right ) \\ & = \frac {1}{2} \log \left (1+\sqrt [3]{\tan (x)}\right )-\frac {1}{6} \log (1+\tan (x))-\frac {\sqrt [3]{\tan (x)}}{1+\tan (x)}-\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \sqrt [3]{\tan (x)}\right ) \\ & = \frac {\arctan \left (\frac {-1+2 \sqrt [3]{\tan (x)}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} \log \left (1+\sqrt [3]{\tan (x)}\right )-\frac {1}{6} \log (1+\tan (x))-\frac {\sqrt [3]{\tan (x)}}{1+\tan (x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.26 \[ \int \frac {\sqrt [3]{\tan (x)}}{(\cos (x)+\sin (x))^2} \, dx=\frac {\arctan \left (\frac {-1+2 \sqrt [3]{\tan (x)}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (1+\sqrt [3]{\tan (x)}\right )-\frac {1}{6} \log \left (1-\sqrt [3]{\tan (x)}+\tan ^{\frac {2}{3}}(x)\right )+\left (-1+\frac {\sin (x)}{\cos (x)+\sin (x)}\right ) \sqrt [3]{\tan (x)} \]

[In]

Integrate[Tan[x]^(1/3)/(Cos[x] + Sin[x])^2,x]

[Out]

ArcTan[(-1 + 2*Tan[x]^(1/3))/Sqrt[3]]/Sqrt[3] + Log[1 + Tan[x]^(1/3)]/3 - Log[1 - Tan[x]^(1/3) + Tan[x]^(2/3)]
/6 + (-1 + Sin[x]/(Cos[x] + Sin[x]))*Tan[x]^(1/3)

Maple [F]

\[\int \frac {\tan \left (x \right )^{\frac {1}{3}}}{\left (\cos \left (x \right )+\sin \left (x \right )\right )^{2}}d x\]

[In]

int(tan(x)^(1/3)/(cos(x)+sin(x))^2,x)

[Out]

int(tan(x)^(1/3)/(cos(x)+sin(x))^2,x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 108 vs. \(2 (48) = 96\).

Time = 0.25 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.77 \[ \int \frac {\sqrt [3]{\tan (x)}}{(\cos (x)+\sin (x))^2} \, dx=\frac {2 \, {\left (\sqrt {3} \cos \left (x\right ) + \sqrt {3} \sin \left (x\right )\right )} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (\frac {\sin \left (x\right )}{\cos \left (x\right )}\right )^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) - {\left (\cos \left (x\right ) + \sin \left (x\right )\right )} \log \left (\left (\frac {\sin \left (x\right )}{\cos \left (x\right )}\right )^{\frac {2}{3}} - \left (\frac {\sin \left (x\right )}{\cos \left (x\right )}\right )^{\frac {1}{3}} + 1\right ) + 2 \, {\left (\cos \left (x\right ) + \sin \left (x\right )\right )} \log \left (\left (\frac {\sin \left (x\right )}{\cos \left (x\right )}\right )^{\frac {1}{3}} + 1\right ) - 6 \, \left (\frac {\sin \left (x\right )}{\cos \left (x\right )}\right )^{\frac {1}{3}} \cos \left (x\right )}{6 \, {\left (\cos \left (x\right ) + \sin \left (x\right )\right )}} \]

[In]

integrate(tan(x)^(1/3)/(cos(x)+sin(x))^2,x, algorithm="fricas")

[Out]

1/6*(2*(sqrt(3)*cos(x) + sqrt(3)*sin(x))*arctan(2/3*sqrt(3)*(sin(x)/cos(x))^(1/3) - 1/3*sqrt(3)) - (cos(x) + s
in(x))*log((sin(x)/cos(x))^(2/3) - (sin(x)/cos(x))^(1/3) + 1) + 2*(cos(x) + sin(x))*log((sin(x)/cos(x))^(1/3)
+ 1) - 6*(sin(x)/cos(x))^(1/3)*cos(x))/(cos(x) + sin(x))

Sympy [F]

\[ \int \frac {\sqrt [3]{\tan (x)}}{(\cos (x)+\sin (x))^2} \, dx=\int \frac {\sqrt [3]{\tan {\left (x \right )}}}{\left (\sin {\left (x \right )} + \cos {\left (x \right )}\right )^{2}}\, dx \]

[In]

integrate(tan(x)**(1/3)/(cos(x)+sin(x))**2,x)

[Out]

Integral(tan(x)**(1/3)/(sin(x) + cos(x))**2, x)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt [3]{\tan (x)}}{(\cos (x)+\sin (x))^2} \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \tan \left (x\right )^{\frac {1}{3}} - 1\right )}\right ) - \frac {\tan \left (x\right )^{\frac {1}{3}}}{\tan \left (x\right ) + 1} - \frac {1}{6} \, \log \left (\tan \left (x\right )^{\frac {2}{3}} - \tan \left (x\right )^{\frac {1}{3}} + 1\right ) + \frac {1}{3} \, \log \left (\tan \left (x\right )^{\frac {1}{3}} + 1\right ) \]

[In]

integrate(tan(x)^(1/3)/(cos(x)+sin(x))^2,x, algorithm="maxima")

[Out]

1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*tan(x)^(1/3) - 1)) - tan(x)^(1/3)/(tan(x) + 1) - 1/6*log(tan(x)^(2/3) - tan(
x)^(1/3) + 1) + 1/3*log(tan(x)^(1/3) + 1)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt [3]{\tan (x)}}{(\cos (x)+\sin (x))^2} \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \tan \left (x\right )^{\frac {1}{3}} - 1\right )}\right ) - \frac {\tan \left (x\right )^{\frac {1}{3}}}{\tan \left (x\right ) + 1} - \frac {1}{6} \, \log \left (\tan \left (x\right )^{\frac {2}{3}} - \tan \left (x\right )^{\frac {1}{3}} + 1\right ) + \frac {1}{3} \, \log \left ({\left | \tan \left (x\right )^{\frac {1}{3}} + 1 \right |}\right ) \]

[In]

integrate(tan(x)^(1/3)/(cos(x)+sin(x))^2,x, algorithm="giac")

[Out]

1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*tan(x)^(1/3) - 1)) - tan(x)^(1/3)/(tan(x) + 1) - 1/6*log(tan(x)^(2/3) - tan(
x)^(1/3) + 1) + 1/3*log(abs(tan(x)^(1/3) + 1))

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt [3]{\tan (x)}}{(\cos (x)+\sin (x))^2} \, dx=\int \frac {{\mathrm {tan}\left (x\right )}^{1/3}}{{\left (\cos \left (x\right )+\sin \left (x\right )\right )}^2} \,d x \]

[In]

int(tan(x)^(1/3)/(cos(x) + sin(x))^2,x)

[Out]

int(tan(x)^(1/3)/(cos(x) + sin(x))^2, x)