\(\int \frac {\sqrt [3]{x}}{1-x^6} \, dx\) [1379]

Optimal result

Integrand size = 15, antiderivative size = 244 \[ \int \frac {\sqrt [3]{x}}{1-x^6} \, dx=-\frac {\arctan \left (\frac {1+2 x^{2/3}}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {1}{3} \arctan \left (\left (x^{2/3}-\cos \left (\frac {2 \pi }{9}\right )\right ) \csc \left (\frac {2 \pi }{9}\right )\right ) \cos \left (\frac {\pi }{18}\right )-\frac {1}{6} \log \left (1-x^{2/3}\right )+\frac {1}{12} \log \left (1+x^{2/3}+x^{4/3}\right )-\frac {1}{6} \cos \left (\frac {2 \pi }{9}\right ) \log \left (1+x^{4/3}+2 x^{2/3} \cos \left (\frac {\pi }{9}\right )\right )+\frac {1}{6} \cos \left (\frac {\pi }{9}\right ) \log \left (1+x^{4/3}-2 x^{2/3} \sin \left (\frac {\pi }{18}\right )\right )-\frac {1}{6} \log \left (1+x^{4/3}-2 x^{2/3} \cos \left (\frac {2 \pi }{9}\right )\right ) \sin \left (\frac {\pi }{18}\right )+\frac {1}{3} \arctan \left (\sec \left (\frac {\pi }{18}\right ) \left (x^{2/3}-\sin \left (\frac {\pi }{18}\right )\right )\right ) \sin \left (\frac {\pi }{9}\right )-\frac {1}{3} \arctan \left (\left (x^{2/3}+\cos \left (\frac {\pi }{9}\right )\right ) \csc \left (\frac {\pi }{9}\right )\right ) \sin \left (\frac {2 \pi }{9}\right ) \]

[Out]

1/3*arctan((x^(2/3)-cos(2/9*Pi))*csc(2/9*Pi))*cos(1/18*Pi)-1/6*ln(1-x^(2/3))+1/12*ln(1+x^(2/3)+x^(4/3))-1/6*co
s(2/9*Pi)*ln(1+x^(4/3)+2*x^(2/3)*cos(1/9*Pi))+1/6*cos(1/9*Pi)*ln(1+x^(4/3)-2*x^(2/3)*sin(1/18*Pi))-1/6*ln(1+x^
(4/3)-2*x^(2/3)*cos(2/9*Pi))*sin(1/18*Pi)+1/3*arctan(sec(1/18*Pi)*(x^(2/3)-sin(1/18*Pi)))*sin(1/9*Pi)-1/3*arct
an((x^(2/3)+cos(1/9*Pi))*csc(1/9*Pi))*sin(2/9*Pi)-1/6*arctan(1/3*(1+2*x^(2/3))*3^(1/2))*3^(1/2)

Rubi [A] (warning: unable to verify)

Time = 0.23 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {335, 281, 300, 648, 632, 210, 642, 31} \[ \int \frac {\sqrt [3]{x}}{1-x^6} \, dx=-\frac {\arctan \left (\frac {2 x^{2/3}+1}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {1}{3} \cos \left (\frac {\pi }{18}\right ) \arctan \left (\csc \left (\frac {2 \pi }{9}\right ) \left (x^{2/3}-\cos \left (\frac {2 \pi }{9}\right )\right )\right )+\frac {1}{3} \sin \left (\frac {\pi }{9}\right ) \arctan \left (\sec \left (\frac {\pi }{18}\right ) \left (x^{2/3}-\sin \left (\frac {\pi }{18}\right )\right )\right )-\frac {1}{3} \sin \left (\frac {2 \pi }{9}\right ) \arctan \left (\csc \left (\frac {\pi }{9}\right ) \left (x^{2/3}+\cos \left (\frac {\pi }{9}\right )\right )\right )-\frac {1}{6} \log \left (1-x^{2/3}\right )+\frac {1}{12} \log \left (x^{4/3}+x^{2/3}+1\right )-\frac {1}{6} \cos \left (\frac {2 \pi }{9}\right ) \log \left (x^{4/3}+2 x^{2/3} \cos \left (\frac {\pi }{9}\right )+1\right )+\frac {1}{6} \cos \left (\frac {\pi }{9}\right ) \log \left (x^{4/3}-2 x^{2/3} \sin \left (\frac {\pi }{18}\right )+1\right )-\frac {1}{6} \sin \left (\frac {\pi }{18}\right ) \log \left (x^{4/3}-2 x^{2/3} \cos \left (\frac {2 \pi }{9}\right )+1\right ) \]

[In]

Int[x^(1/3)/(1 - x^6),x]

[Out]

-1/2*ArcTan[(1 + 2*x^(2/3))/Sqrt[3]]/Sqrt[3] + (ArcTan[(x^(2/3) - Cos[(2*Pi)/9])*Csc[(2*Pi)/9]]*Cos[Pi/18])/3
- Log[1 - x^(2/3)]/6 + Log[1 + x^(2/3) + x^(4/3)]/12 - (Cos[(2*Pi)/9]*Log[1 + x^(4/3) + 2*x^(2/3)*Cos[Pi/9]])/
6 + (Cos[Pi/9]*Log[1 + x^(4/3) - 2*x^(2/3)*Sin[Pi/18]])/6 - (Log[1 + x^(4/3) - 2*x^(2/3)*Cos[(2*Pi)/9]]*Sin[Pi
/18])/6 + (ArcTan[Sec[Pi/18]*(x^(2/3) - Sin[Pi/18])]*Sin[Pi/9])/3 - (ArcTan[(x^(2/3) + Cos[Pi/9])*Csc[Pi/9]]*S
in[(2*Pi)/9])/3

Rule 31

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

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 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 300

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator[Rt[-a/b, n]], s = Denominator[Rt[-
a/b, n]], k, u}, Simp[u = Int[(r*Cos[(2*k - 1)*m*(Pi/n)] + s*Cos[(2*k - 1)*(m + 1)*(Pi/n)]*x)/(r^2 + 2*r*s*Cos
[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x]; (r^(m + 1)/(a*n*s^m))*Int[1/(r - s*x), x] - Dist[2*((-r)^(m + 1)/(a*n*s^m
)), Sum[u, {k, 1, (n - 1)/2}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 1)/2, 0] && IGtQ[m, 0] && LtQ[m, n - 1
] && NegQ[a/b]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

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]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

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

Rubi steps \begin{align*} \text {integral}& = 3 \text {Subst}\left (\int \frac {x^3}{1-x^{18}} \, dx,x,\sqrt [3]{x}\right ) \\ & = \frac {3}{2} \text {Subst}\left (\int \frac {x}{1-x^9} \, dx,x,x^{2/3}\right ) \\ & = \frac {1}{6} \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,x^{2/3}\right )-\frac {1}{3} \text {Subst}\left (\int \frac {\frac {1}{2}-\frac {x}{2}}{1+x+x^2} \, dx,x,x^{2/3}\right )-\frac {1}{3} \text {Subst}\left (\int \frac {\cos \left (\frac {\pi }{9}\right )+x \cos \left (\frac {2 \pi }{9}\right )}{1+x^2+2 x \cos \left (\frac {\pi }{9}\right )} \, dx,x,x^{2/3}\right )-\frac {1}{3} \text {Subst}\left (\int \frac {-x \cos \left (\frac {\pi }{9}\right )-\sin \left (\frac {\pi }{18}\right )}{1+x^2-2 x \sin \left (\frac {\pi }{18}\right )} \, dx,x,x^{2/3}\right )-\frac {1}{3} \text {Subst}\left (\int \frac {-\cos \left (\frac {2 \pi }{9}\right )+x \sin \left (\frac {\pi }{18}\right )}{1+x^2-2 x \cos \left (\frac {2 \pi }{9}\right )} \, dx,x,x^{2/3}\right ) \\ & = -\frac {1}{6} \log \left (1-x^{2/3}\right )+\frac {1}{12} \text {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,x^{2/3}\right )-\frac {1}{4} \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,x^{2/3}\right )+\frac {1}{6} \cos \left (\frac {\pi }{9}\right ) \text {Subst}\left (\int \frac {2 x-2 \sin \left (\frac {\pi }{18}\right )}{1+x^2-2 x \sin \left (\frac {\pi }{18}\right )} \, dx,x,x^{2/3}\right )-\frac {1}{6} \cos \left (\frac {2 \pi }{9}\right ) \text {Subst}\left (\int \frac {2 x+2 \cos \left (\frac {\pi }{9}\right )}{1+x^2+2 x \cos \left (\frac {\pi }{9}\right )} \, dx,x,x^{2/3}\right )+\frac {1}{3} \left (\cos \left (\frac {2 \pi }{9}\right ) \left (1-\sin \left (\frac {\pi }{18}\right )\right )\right ) \text {Subst}\left (\int \frac {1}{1+x^2-2 x \cos \left (\frac {2 \pi }{9}\right )} \, dx,x,x^{2/3}\right )-\frac {1}{6} \sin \left (\frac {\pi }{18}\right ) \text {Subst}\left (\int \frac {2 x-2 \cos \left (\frac {2 \pi }{9}\right )}{1+x^2-2 x \cos \left (\frac {2 \pi }{9}\right )} \, dx,x,x^{2/3}\right )+\frac {1}{3} \left (\left (1+\cos \left (\frac {\pi }{9}\right )\right ) \sin \left (\frac {\pi }{18}\right )\right ) \text {Subst}\left (\int \frac {1}{1+x^2-2 x \sin \left (\frac {\pi }{18}\right )} \, dx,x,x^{2/3}\right )-\frac {1}{3} \left (\sin \left (\frac {\pi }{9}\right ) \sin \left (\frac {2 \pi }{9}\right )\right ) \text {Subst}\left (\int \frac {1}{1+x^2+2 x \cos \left (\frac {\pi }{9}\right )} \, dx,x,x^{2/3}\right ) \\ & = -\frac {1}{6} \log \left (1-x^{2/3}\right )+\frac {1}{12} \log \left (1+x^{2/3}+x^{4/3}\right )-\frac {1}{6} \cos \left (\frac {2 \pi }{9}\right ) \log \left (1+x^{4/3}+2 x^{2/3} \cos \left (\frac {\pi }{9}\right )\right )+\frac {1}{6} \cos \left (\frac {\pi }{9}\right ) \log \left (1+x^{4/3}-2 x^{2/3} \sin \left (\frac {\pi }{18}\right )\right )-\frac {1}{6} \log \left (1+x^{4/3}-2 x^{2/3} \cos \left (\frac {2 \pi }{9}\right )\right ) \sin \left (\frac {\pi }{18}\right )+\frac {1}{2} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x^{2/3}\right )-\frac {1}{3} \left (2 \cos \left (\frac {2 \pi }{9}\right ) \left (1-\sin \left (\frac {\pi }{18}\right )\right )\right ) \text {Subst}\left (\int \frac {1}{-x^2-4 \sin ^2\left (\frac {2 \pi }{9}\right )} \, dx,x,2 x^{2/3}-2 \cos \left (\frac {2 \pi }{9}\right )\right )-\frac {1}{3} \left (2 \left (1+\cos \left (\frac {\pi }{9}\right )\right ) \sin \left (\frac {\pi }{18}\right )\right ) \text {Subst}\left (\int \frac {1}{-x^2-4 \cos ^2\left (\frac {\pi }{18}\right )} \, dx,x,2 x^{2/3}-2 \sin \left (\frac {\pi }{18}\right )\right )+\frac {1}{3} \left (2 \sin \left (\frac {\pi }{9}\right ) \sin \left (\frac {2 \pi }{9}\right )\right ) \text {Subst}\left (\int \frac {1}{-x^2-4 \sin ^2\left (\frac {\pi }{9}\right )} \, dx,x,2 \left (x^{2/3}+\cos \left (\frac {\pi }{9}\right )\right )\right ) \\ & = -\frac {\tan ^{-1}\left (\frac {1+2 x^{2/3}}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{6} \log \left (1-x^{2/3}\right )+\frac {1}{12} \log \left (1+x^{2/3}+x^{4/3}\right )-\frac {1}{6} \cos \left (\frac {2 \pi }{9}\right ) \log \left (1+x^{4/3}+2 x^{2/3} \cos \left (\frac {\pi }{9}\right )\right )+\frac {1}{6} \cos \left (\frac {\pi }{9}\right ) \log \left (1+x^{4/3}-2 x^{2/3} \sin \left (\frac {\pi }{18}\right )\right )+\frac {1}{3} \tan ^{-1}\left (\left (x^{2/3}-\cos \left (\frac {2 \pi }{9}\right )\right ) \csc \left (\frac {2 \pi }{9}\right )\right ) \cot \left (\frac {2 \pi }{9}\right ) \left (1-\sin \left (\frac {\pi }{18}\right )\right )-\frac {1}{6} \log \left (1+x^{4/3}-2 x^{2/3} \cos \left (\frac {2 \pi }{9}\right )\right ) \sin \left (\frac {\pi }{18}\right )+\frac {1}{3} \tan ^{-1}\left (\sec \left (\frac {\pi }{18}\right ) \left (x^{2/3}-\sin \left (\frac {\pi }{18}\right )\right )\right ) \sin \left (\frac {\pi }{9}\right )-\frac {1}{3} \tan ^{-1}\left (\left (x^{2/3}+\cos \left (\frac {\pi }{9}\right )\right ) \csc \left (\frac {\pi }{9}\right )\right ) \sin \left (\frac {2 \pi }{9}\right ) \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.18 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt [3]{x}}{1-x^6} \, dx=\frac {1}{12} \left (2 \sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{x}}{\sqrt {3}}\right )+2 \sqrt {3} \arctan \left (\frac {1+2 \sqrt [3]{x}}{\sqrt {3}}\right )-2 \log \left (-1+\sqrt [3]{x}\right )-2 \log \left (1+\sqrt [3]{x}\right )+\log \left (1-\sqrt [3]{x}+x^{2/3}\right )+\log \left (1+\sqrt [3]{x}+x^{2/3}\right )+2 \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {\log \left (\sqrt [3]{x}-\text {$\#$1}\right )+\log \left (\sqrt [3]{x}-\text {$\#$1}\right ) \text {$\#$1}^3}{-\text {$\#$1}^2+2 \text {$\#$1}^5}\&\right ]+2 \text {RootSum}\left [1+\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log \left (\sqrt [3]{x}-\text {$\#$1}\right )+\log \left (\sqrt [3]{x}-\text {$\#$1}\right ) \text {$\#$1}^3}{\text {$\#$1}^2+2 \text {$\#$1}^5}\&\right ]\right ) \]

[In]

Integrate[x^(1/3)/(1 - x^6),x]

[Out]

(2*Sqrt[3]*ArcTan[(1 - 2*x^(1/3))/Sqrt[3]] + 2*Sqrt[3]*ArcTan[(1 + 2*x^(1/3))/Sqrt[3]] - 2*Log[-1 + x^(1/3)] -
2*Log[1 + x^(1/3)] + Log[1 - x^(1/3) + x^(2/3)] + Log[1 + x^(1/3) + x^(2/3)] + 2*RootSum[1 - #1^3 + #1^6 & ,
(Log[x^(1/3) - #1] + Log[x^(1/3) - #1]*#1^3)/(-#1^2 + 2*#1^5) & ] + 2*RootSum[1 + #1^3 + #1^6 & , (-Log[x^(1/3
) - #1] + Log[x^(1/3) - #1]*#1^3)/(#1^2 + 2*#1^5) & ])/12

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 5.90 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.66

method	result	size
derivativedivides	\(-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+\textit {\_Z}^{3}+1\right )}{\sum }\frac {\left (-\textit {\_R}^{3}+1\right ) \ln \left (x^{\frac {1}{3}}-\textit {\_R} \right )}{2 \textit {\_R}^{5}+\textit {\_R}^{2}}\right )}{6}+\frac {\ln \left (x^{\frac {2}{3}}+x^{\frac {1}{3}}+1\right )}{12}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x^{\frac {1}{3}}+1\right ) \sqrt {3}}{3}\right )}{6}-\frac {\ln \left (x^{\frac {1}{3}}-1\right )}{6}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )}{\sum }\frac {\left (\textit {\_R}^{3}+1\right ) \ln \left (x^{\frac {1}{3}}-\textit {\_R} \right )}{2 \textit {\_R}^{5}-\textit {\_R}^{2}}\right )}{6}-\frac {\ln \left (1+x^{\frac {1}{3}}\right )}{6}+\frac {\ln \left (x^{\frac {2}{3}}-x^{\frac {1}{3}}+1\right )}{12}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x^{\frac {1}{3}}-1\right ) \sqrt {3}}{3}\right )}{6}\)	\(162\)
default	\(-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+\textit {\_Z}^{3}+1\right )}{\sum }\frac {\left (-\textit {\_R}^{3}+1\right ) \ln \left (x^{\frac {1}{3}}-\textit {\_R} \right )}{2 \textit {\_R}^{5}+\textit {\_R}^{2}}\right )}{6}+\frac {\ln \left (x^{\frac {2}{3}}+x^{\frac {1}{3}}+1\right )}{12}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x^{\frac {1}{3}}+1\right ) \sqrt {3}}{3}\right )}{6}-\frac {\ln \left (x^{\frac {1}{3}}-1\right )}{6}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )}{\sum }\frac {\left (\textit {\_R}^{3}+1\right ) \ln \left (x^{\frac {1}{3}}-\textit {\_R} \right )}{2 \textit {\_R}^{5}-\textit {\_R}^{2}}\right )}{6}-\frac {\ln \left (1+x^{\frac {1}{3}}\right )}{6}+\frac {\ln \left (x^{\frac {2}{3}}-x^{\frac {1}{3}}+1\right )}{12}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x^{\frac {1}{3}}-1\right ) \sqrt {3}}{3}\right )}{6}\)	\(162\)
meijerg	\(-\frac {x^{\frac {4}{3}} \left (\ln \left (1-\left (x^{6}\right )^{\frac {1}{9}}\right )+\cos \left (\frac {4 \pi }{9}\right ) \ln \left (1-2 \cos \left (\frac {2 \pi }{9}\right ) \left (x^{6}\right )^{\frac {1}{9}}+\left (x^{6}\right )^{\frac {2}{9}}\right )-2 \sin \left (\frac {4 \pi }{9}\right ) \arctan \left (\frac {\sin \left (\frac {2 \pi }{9}\right ) \left (x^{6}\right )^{\frac {1}{9}}}{1-\cos \left (\frac {2 \pi }{9}\right ) \left (x^{6}\right )^{\frac {1}{9}}}\right )-\cos \left (\frac {\pi }{9}\right ) \ln \left (1-2 \cos \left (\frac {4 \pi }{9}\right ) \left (x^{6}\right )^{\frac {1}{9}}+\left (x^{6}\right )^{\frac {2}{9}}\right )-2 \sin \left (\frac {\pi }{9}\right ) \arctan \left (\frac {\sin \left (\frac {4 \pi }{9}\right ) \left (x^{6}\right )^{\frac {1}{9}}}{1-\cos \left (\frac {4 \pi }{9}\right ) \left (x^{6}\right )^{\frac {1}{9}}}\right )-\frac {\ln \left (1+\left (x^{6}\right )^{\frac {1}{9}}+\left (x^{6}\right )^{\frac {2}{9}}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{9}}}{2+\left (x^{6}\right )^{\frac {1}{9}}}\right )+\cos \left (\frac {2 \pi }{9}\right ) \ln \left (1+2 \cos \left (\frac {\pi }{9}\right ) \left (x^{6}\right )^{\frac {1}{9}}+\left (x^{6}\right )^{\frac {2}{9}}\right )+2 \sin \left (\frac {2 \pi }{9}\right ) \arctan \left (\frac {\sin \left (\frac {\pi }{9}\right ) \left (x^{6}\right )^{\frac {1}{9}}}{1+\cos \left (\frac {\pi }{9}\right ) \left (x^{6}\right )^{\frac {1}{9}}}\right )\right )}{6 \left (x^{6}\right )^{\frac {2}{9}}}\)	\(231\)
trager	\(\text {Expression too large to display}\)	\(1301\)

[In]

int(x^(1/3)/(-x^6+1),x,method=_RETURNVERBOSE)

[Out]

-1/6*sum((-_R^3+1)/(2*_R^5+_R^2)*ln(x^(1/3)-_R),_R=RootOf(_Z^6+_Z^3+1))+1/12*ln(x^(2/3)+x^(1/3)+1)+1/6*3^(1/2)
*arctan(1/3*(2*x^(1/3)+1)*3^(1/2))-1/6*ln(x^(1/3)-1)+1/6*sum((_R^3+1)/(2*_R^5-_R^2)*ln(x^(1/3)-_R),_R=RootOf(_
Z^6-_Z^3+1))-1/6*ln(1+x^(1/3))+1/12*ln(x^(2/3)-x^(1/3)+1)-1/6*3^(1/2)*arctan(1/3*(2*x^(1/3)-1)*3^(1/2))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.53 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.51 \[ \int \frac {\sqrt [3]{x}}{1-x^6} \, dx=-\frac {1}{24} \cdot 2^{\frac {2}{3}} {\left (i \, \sqrt {3} + 1\right )}^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} \log \left ({\left (\sqrt {3} 2^{\frac {1}{3}} {\left (i \, \sqrt {-3} - i\right )} + 2^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )}\right )} {\left (i \, \sqrt {3} + 1\right )}^{\frac {2}{3}} + 8 \, x^{\frac {2}{3}}\right ) + \frac {1}{24} \cdot 2^{\frac {2}{3}} {\left (i \, \sqrt {3} + 1\right )}^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} \log \left ({\left (\sqrt {3} 2^{\frac {1}{3}} {\left (-i \, \sqrt {-3} - i\right )} - 2^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )}\right )} {\left (i \, \sqrt {3} + 1\right )}^{\frac {2}{3}} + 8 \, x^{\frac {2}{3}}\right ) + \frac {1}{24} \cdot 2^{\frac {2}{3}} {\left (-i \, \sqrt {3} + 1\right )}^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} \log \left ({\left (\sqrt {3} 2^{\frac {1}{3}} {\left (i \, \sqrt {-3} + i\right )} - 2^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )}\right )} {\left (-i \, \sqrt {3} + 1\right )}^{\frac {2}{3}} + 8 \, x^{\frac {2}{3}}\right ) - \frac {1}{24} \cdot 2^{\frac {2}{3}} {\left (-i \, \sqrt {3} + 1\right )}^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} \log \left ({\left (\sqrt {3} 2^{\frac {1}{3}} {\left (-i \, \sqrt {-3} + i\right )} + 2^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )}\right )} {\left (-i \, \sqrt {3} + 1\right )}^{\frac {2}{3}} + 8 \, x^{\frac {2}{3}}\right ) + \frac {1}{12} \cdot 2^{\frac {2}{3}} {\left (i \, \sqrt {3} + 1\right )}^{\frac {1}{3}} \log \left ({\left (i \, \sqrt {3} 2^{\frac {1}{3}} + 2^{\frac {1}{3}}\right )} {\left (i \, \sqrt {3} + 1\right )}^{\frac {2}{3}} + 4 \, x^{\frac {2}{3}}\right ) + \frac {1}{12} \cdot 2^{\frac {2}{3}} {\left (-i \, \sqrt {3} + 1\right )}^{\frac {1}{3}} \log \left ({\left (-i \, \sqrt {3} 2^{\frac {1}{3}} + 2^{\frac {1}{3}}\right )} {\left (-i \, \sqrt {3} + 1\right )}^{\frac {2}{3}} + 4 \, x^{\frac {2}{3}}\right ) - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} x^{\frac {2}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + \frac {1}{12} \, \log \left (x^{\frac {4}{3}} + x^{\frac {2}{3}} + 1\right ) - \frac {1}{6} \, \log \left (x^{\frac {2}{3}} - 1\right ) \]

[In]

integrate(x^(1/3)/(-x^6+1),x, algorithm="fricas")

[Out]

-1/24*2^(2/3)*(I*sqrt(3) + 1)^(1/3)*(sqrt(-3) + 1)*log((sqrt(3)*2^(1/3)*(I*sqrt(-3) - I) + 2^(1/3)*(sqrt(-3) -
1))*(I*sqrt(3) + 1)^(2/3) + 8*x^(2/3)) + 1/24*2^(2/3)*(I*sqrt(3) + 1)^(1/3)*(sqrt(-3) - 1)*log((sqrt(3)*2^(1/
3)*(-I*sqrt(-3) - I) - 2^(1/3)*(sqrt(-3) + 1))*(I*sqrt(3) + 1)^(2/3) + 8*x^(2/3)) + 1/24*2^(2/3)*(-I*sqrt(3) +
1)^(1/3)*(sqrt(-3) - 1)*log((sqrt(3)*2^(1/3)*(I*sqrt(-3) + I) - 2^(1/3)*(sqrt(-3) + 1))*(-I*sqrt(3) + 1)^(2/3
) + 8*x^(2/3)) - 1/24*2^(2/3)*(-I*sqrt(3) + 1)^(1/3)*(sqrt(-3) + 1)*log((sqrt(3)*2^(1/3)*(-I*sqrt(-3) + I) + 2
^(1/3)*(sqrt(-3) - 1))*(-I*sqrt(3) + 1)^(2/3) + 8*x^(2/3)) + 1/12*2^(2/3)*(I*sqrt(3) + 1)^(1/3)*log((I*sqrt(3)
*2^(1/3) + 2^(1/3))*(I*sqrt(3) + 1)^(2/3) + 4*x^(2/3)) + 1/12*2^(2/3)*(-I*sqrt(3) + 1)^(1/3)*log((-I*sqrt(3)*2
^(1/3) + 2^(1/3))*(-I*sqrt(3) + 1)^(2/3) + 4*x^(2/3)) - 1/6*sqrt(3)*arctan(2/3*sqrt(3)*x^(2/3) + 1/3*sqrt(3))
+ 1/12*log(x^(4/3) + x^(2/3) + 1) - 1/6*log(x^(2/3) - 1)

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt [3]{x}}{1-x^6} \, dx=\text {Timed out} \]

[In]

integrate(x**(1/3)/(-x**6+1),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\sqrt [3]{x}}{1-x^6} \, dx=\int { -\frac {x^{\frac {1}{3}}}{x^{6} - 1} \,d x } \]

[In]

integrate(x^(1/3)/(-x^6+1),x, algorithm="maxima")

[Out]

1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^(1/3) + 1)) - 1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^(1/3) - 1)) + integrate(
1/6*(x^(4/3) + 2*x^(1/3))/(x^2 + x + 1), x) - integrate(1/6*(x^(4/3) - 2*x^(1/3))/(x^2 - x + 1), x) + 1/12*log
(x^(2/3) + x^(1/3) + 1) + 1/12*log(x^(2/3) - x^(1/3) + 1) - 1/6*log(x^(1/3) + 1) - 1/6*log(x^(1/3) - 1)

Giac [A] (verification not implemented)

none

Time = 0.90 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt [3]{x}}{1-x^6} \, dx=\frac {2}{3} \, \arctan \left (\frac {x^{\frac {2}{3}} - \cos \left (\frac {4}{9} \, \pi \right )}{\sin \left (\frac {4}{9} \, \pi \right )}\right ) \cos \left (\frac {4}{9} \, \pi \right ) \sin \left (\frac {4}{9} \, \pi \right ) + \frac {2}{3} \, \arctan \left (\frac {x^{\frac {2}{3}} - \cos \left (\frac {2}{9} \, \pi \right )}{\sin \left (\frac {2}{9} \, \pi \right )}\right ) \cos \left (\frac {2}{9} \, \pi \right ) \sin \left (\frac {2}{9} \, \pi \right ) - \frac {2}{3} \, \arctan \left (\frac {x^{\frac {2}{3}} + \cos \left (\frac {1}{9} \, \pi \right )}{\sin \left (\frac {1}{9} \, \pi \right )}\right ) \cos \left (\frac {1}{9} \, \pi \right ) \sin \left (\frac {1}{9} \, \pi \right ) - \frac {1}{6} \, {\left (\cos \left (\frac {4}{9} \, \pi \right )^{2} - \sin \left (\frac {4}{9} \, \pi \right )^{2}\right )} \log \left (-2 \, x^{\frac {2}{3}} \cos \left (\frac {4}{9} \, \pi \right ) + x^{\frac {4}{3}} + 1\right ) - \frac {1}{6} \, {\left (\cos \left (\frac {2}{9} \, \pi \right )^{2} - \sin \left (\frac {2}{9} \, \pi \right )^{2}\right )} \log \left (-2 \, x^{\frac {2}{3}} \cos \left (\frac {2}{9} \, \pi \right ) + x^{\frac {4}{3}} + 1\right ) - \frac {1}{6} \, {\left (\cos \left (\frac {1}{9} \, \pi \right )^{2} - \sin \left (\frac {1}{9} \, \pi \right )^{2}\right )} \log \left (2 \, x^{\frac {2}{3}} \cos \left (\frac {1}{9} \, \pi \right ) + x^{\frac {4}{3}} + 1\right ) - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{\frac {2}{3}} + 1\right )}\right ) + \frac {1}{12} \, \log \left (x^{\frac {4}{3}} + x^{\frac {2}{3}} + 1\right ) - \frac {1}{6} \, \log \left ({\left | x^{\frac {2}{3}} - 1 \right |}\right ) \]

[In]

integrate(x^(1/3)/(-x^6+1),x, algorithm="giac")

[Out]

2/3*arctan((x^(2/3) - cos(4/9*pi))/sin(4/9*pi))*cos(4/9*pi)*sin(4/9*pi) + 2/3*arctan((x^(2/3) - cos(2/9*pi))/s
in(2/9*pi))*cos(2/9*pi)*sin(2/9*pi) - 2/3*arctan((x^(2/3) + cos(1/9*pi))/sin(1/9*pi))*cos(1/9*pi)*sin(1/9*pi)
- 1/6*(cos(4/9*pi)^2 - sin(4/9*pi)^2)*log(-2*x^(2/3)*cos(4/9*pi) + x^(4/3) + 1) - 1/6*(cos(2/9*pi)^2 - sin(2/9
*pi)^2)*log(-2*x^(2/3)*cos(2/9*pi) + x^(4/3) + 1) - 1/6*(cos(1/9*pi)^2 - sin(1/9*pi)^2)*log(2*x^(2/3)*cos(1/9*
pi) + x^(4/3) + 1) - 1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^(2/3) + 1)) + 1/12*log(x^(4/3) + x^(2/3) + 1) - 1/6*l
og(abs(x^(2/3) - 1))

Mupad [B] (verification not implemented)

Time = 5.76 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt [3]{x}}{1-x^6} \, dx=-\frac {\ln \left (43046721\,x^{2/3}-43046721\right )}{6}+\frac {\ln \left (43046721\,x^{2/3}\,{\mathrm {e}}^{\frac {\pi \,4{}\mathrm {i}}{9}}-43046721\right )\,{\mathrm {e}}^{\frac {\pi \,1{}\mathrm {i}}{9}}}{6}+\frac {\ln \left (43046721\,x^{2/3}\,{\mathrm {e}}^{\frac {\pi \,2{}\mathrm {i}}{9}}-43046721\right )\,{\mathrm {e}}^{\frac {\pi \,5{}\mathrm {i}}{9}}}{6}+\frac {\ln \left (-43046721\,x^{2/3}\,{\mathrm {e}}^{\frac {\pi \,1{}\mathrm {i}}{9}}-43046721\right )\,{\mathrm {e}}^{\frac {\pi \,7{}\mathrm {i}}{9}}}{6}-\frac {\ln \left (43046721\,x^{2/3}\,{\mathrm {e}}^{\frac {\pi \,8{}\mathrm {i}}{9}}-43046721\right )\,{\mathrm {e}}^{\frac {\pi \,2{}\mathrm {i}}{9}}}{6}-\frac {\ln \left (-43046721\,x^{2/3}\,{\mathrm {e}}^{\frac {\pi \,7{}\mathrm {i}}{9}}-43046721\right )\,{\mathrm {e}}^{\frac {\pi \,4{}\mathrm {i}}{9}}}{6}-\frac {\ln \left (-43046721\,x^{2/3}\,{\mathrm {e}}^{\frac {\pi \,5{}\mathrm {i}}{9}}-43046721\right )\,{\mathrm {e}}^{\frac {\pi \,8{}\mathrm {i}}{9}}}{6}-\ln \left (55788550416\,x^{2/3}\,{\left (-\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )}^4-43046721\right )\,\left (-\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )+\ln \left (55788550416\,x^{2/3}\,{\left (\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )}^4-43046721\right )\,\left (\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right ) \]

[In]

int(-x^(1/3)/(x^6 - 1),x)

[Out]

(log(43046721*x^(2/3)*exp((pi*4i)/9) - 43046721)*exp((pi*1i)/9))/6 - log(43046721*x^(2/3) - 43046721)/6 + (log
(43046721*x^(2/3)*exp((pi*2i)/9) - 43046721)*exp((pi*5i)/9))/6 + (log(- 43046721*x^(2/3)*exp((pi*1i)/9) - 4304
6721)*exp((pi*7i)/9))/6 - (log(43046721*x^(2/3)*exp((pi*8i)/9) - 43046721)*exp((pi*2i)/9))/6 - (log(- 43046721
*x^(2/3)*exp((pi*7i)/9) - 43046721)*exp((pi*4i)/9))/6 - (log(- 43046721*x^(2/3)*exp((pi*5i)/9) - 43046721)*exp
((pi*8i)/9))/6 - log(55788550416*x^(2/3)*((3^(1/2)*1i)/12 - 1/12)^4 - 43046721)*((3^(1/2)*1i)/12 - 1/12) + log
(55788550416*x^(2/3)*((3^(1/2)*1i)/12 + 1/12)^4 - 43046721)*((3^(1/2)*1i)/12 + 1/12)