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

Optimal result

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

-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+tan(x))
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.26 \[ \int \frac {\sqrt [3]{\tan (x)}}{(\cos (x)+\sin (x))^2} \, dx=\frac {\arctan \left (\frac {-1+2 \sqrt [3]{\tan (x)}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (1+\sqrt [3]{\tan (x)}\right )-\frac {1}{6} \log \left (1-\sqrt [3]{\tan (x)}+\tan ^{\frac {2}{3}}(x)\right )+\left (-1+\frac {\sin (x)}{\cos (x)+\sin (x)}\right ) \sqrt [3]{\tan (x)} \] Input:

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

Output:

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]))*T 
an[x]^(1/3)
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.07, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 4889, 51, 70, 16, 1083, 217}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4889

\(\displaystyle \int \frac {\sqrt [3]{\tan (x)}}{(\tan (x)+1)^2}d\tan (x)\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {1}{3} \int \frac {1}{\tan ^{\frac {2}{3}}(x) (\tan (x)+1)}d\tan (x)-\frac {\sqrt [3]{\tan (x)}}{\tan (x)+1}\)

\(\Big \downarrow \) 70

\(\displaystyle \frac {1}{3} \left (\frac {3}{2} \int \frac {1}{\tan ^{\frac {2}{3}}(x)-\sqrt [3]{\tan (x)}+1}d\sqrt [3]{\tan (x)}+\frac {3}{2} \int \frac {1}{\sqrt [3]{\tan (x)}+1}d\sqrt [3]{\tan (x)}-\frac {1}{2} \log (\tan (x)+1)\right )-\frac {\sqrt [3]{\tan (x)}}{\tan (x)+1}\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {1}{3} \left (\frac {3}{2} \int \frac {1}{\tan ^{\frac {2}{3}}(x)-\sqrt [3]{\tan (x)}+1}d\sqrt [3]{\tan (x)}+\frac {3}{2} \log \left (\sqrt [3]{\tan (x)}+1\right )-\frac {1}{2} \log (\tan (x)+1)\right )-\frac {\sqrt [3]{\tan (x)}}{\tan (x)+1}\)

\(\Big \downarrow \) 1083

\(\displaystyle \frac {1}{3} \left (-3 \int \frac {1}{-\tan ^{\frac {2}{3}}(x)-3}d\left (2 \sqrt [3]{\tan (x)}-1\right )+\frac {3}{2} \log \left (\sqrt [3]{\tan (x)}+1\right )-\frac {1}{2} \log (\tan (x)+1)\right )-\frac {\sqrt [3]{\tan (x)}}{\tan (x)+1}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{3} \left (\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{\tan (x)}-1}{\sqrt {3}}\right )+\frac {3}{2} \log \left (\sqrt [3]{\tan (x)}+1\right )-\frac {1}{2} \log (\tan (x)+1)\right )-\frac {\sqrt [3]{\tan (x)}}{\tan (x)+1}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 16

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

rule 51

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] - Simp[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 
] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
 

rule 70

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] + (Simp[3/(2*b*q)   Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d*x)^(1 
/3)], x] + Simp[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 217

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 1083

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

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4889

Int[u_, x_Symbol] :> With[{v = FunctionOfTrig[u, x]}, With[{d = FreeFactors 
[Tan[v], x]}, Simp[d/Coefficient[v, x, 1]   Subst[Int[SubstFor[1/(1 + d^2*x 
^2), Tan[v]/d, u, x], x], x, Tan[v]/d], x]] /;  !FalseQ[v] && FunctionOfQ[N 
onfreeFactors[Tan[v], x], u, x]] /; InverseFunctionFreeQ[u, x] &&  !MatchQ[ 
u, (v_.)*((c_.)*tan[w_]^(n_.)*tan[z_]^(n_.))^(p_.) /; FreeQ[{c, p}, x] && I 
ntegerQ[n] && LinearQ[w, x] && EqQ[z, 2*w]]
 

Maple [F]

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

Input:

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

Output:

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 = 0.10 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.77 \[ \int \frac {\sqrt [3]{\tan (x)}}{(\cos (x)+\sin (x))^2} \, dx=\frac {2 \, {\left (\sqrt {3} \cos \left (x\right ) + \sqrt {3} \sin \left (x\right )\right )} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (\frac {\sin \left (x\right )}{\cos \left (x\right )}\right )^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) - {\left (\cos \left (x\right ) + \sin \left (x\right )\right )} \log \left (\left (\frac {\sin \left (x\right )}{\cos \left (x\right )}\right )^{\frac {2}{3}} - \left (\frac {\sin \left (x\right )}{\cos \left (x\right )}\right )^{\frac {1}{3}} + 1\right ) + 2 \, {\left (\cos \left (x\right ) + \sin \left (x\right )\right )} \log \left (\left (\frac {\sin \left (x\right )}{\cos \left (x\right )}\right )^{\frac {1}{3}} + 1\right ) - 6 \, \left (\frac {\sin \left (x\right )}{\cos \left (x\right )}\right )^{\frac {1}{3}} \cos \left (x\right )}{6 \, {\left (\cos \left (x\right ) + \sin \left (x\right )\right )}} \] Input:

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

Output:

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) + sin(x))*log((sin(x)/cos(x))^(2/3) - (si 
n(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]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt [3]{\tan (x)}}{(\cos (x)+\sin (x))^2} \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \tan \left (x\right )^{\frac {1}{3}} - 1\right )}\right ) - \frac {\tan \left (x\right )^{\frac {1}{3}}}{\tan \left (x\right ) + 1} - \frac {1}{6} \, \log \left (\tan \left (x\right )^{\frac {2}{3}} - \tan \left (x\right )^{\frac {1}{3}} + 1\right ) + \frac {1}{3} \, \log \left (\tan \left (x\right )^{\frac {1}{3}} + 1\right ) \] Input:

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

Output:

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)

Time = 0.13 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt [3]{\tan (x)}}{(\cos (x)+\sin (x))^2} \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \tan \left (x\right )^{\frac {1}{3}} - 1\right )}\right ) - \frac {\tan \left (x\right )^{\frac {1}{3}}}{\tan \left (x\right ) + 1} - \frac {1}{6} \, \log \left (\tan \left (x\right )^{\frac {2}{3}} - \tan \left (x\right )^{\frac {1}{3}} + 1\right ) + \frac {1}{3} \, \log \left ({\left | \tan \left (x\right )^{\frac {1}{3}} + 1 \right |}\right ) \] Input:

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

Output:

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

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt [3]{\tan (x)}}{(\cos (x)+\sin (x))^2} \, dx=\int \frac {\tan \left (x \right )^{\frac {1}{3}}}{\cos \left (x \right )^{2}+2 \cos \left (x \right ) \sin \left (x \right )+\sin \left (x \right )^{2}}d x \] Input:

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

Output:

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