\(\int \frac {\sqrt [3]{1-x^3}}{1-x+x^2} \, dx\) [59]

Optimal result

Integrand size = 22, antiderivative size = 280 \[ \int \frac {\sqrt [3]{1-x^3}}{1-x+x^2} \, dx=\frac {\sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{2} (-1+x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2^{2/3}}+\frac {\arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}-\frac {\arctan \left (\frac {1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}-\frac {\log \left (-3 (-1+x) \left (1-x+x^2\right )\right )}{2\ 2^{2/3}}+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}+\frac {3 \log \left (-\sqrt [3]{2} (-1+x)+\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}+\frac {1}{2} \log \left (x+\sqrt [3]{1-x^3}\right )-\frac {\log \left (\sqrt [3]{2} x+\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}} \]

[Out]

-1/4*ln(-3*(-1+x)*(x^2-x+1))*2^(1/3)+1/4*ln(2^(1/3)-(-x^3+1)^(1/3))*2^(1/3)+3/4*ln(-2^(1/3)*(-1+x)+(-x^3+1)^(1
/3))*2^(1/3)+1/2*ln(x+(-x^3+1)^(1/3))-1/4*ln(2^(1/3)*x+(-x^3+1)^(1/3))*2^(1/3)+1/3*arctan(1/3*(1-2*x/(-x^3+1)^
(1/3))*3^(1/2))*3^(1/2)-1/6*2^(1/3)*arctan(1/3*(1-2*2^(1/3)*x/(-x^3+1)^(1/3))*3^(1/2))*3^(1/2)-1/6*2^(1/3)*arc
tan(1/3*(1+2^(2/3)*(-x^3+1)^(1/3))*3^(1/2))*3^(1/2)+1/2*arctan(1/3*(1+2*2^(1/3)*(-1+x)/(-x^3+1)^(1/3))*3^(1/2)
)*3^(1/2)*2^(1/3)

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.46, number of steps used = 19, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {2183, 420, 493, 298, 31, 648, 631, 210, 642, 495, 337, 503} \[ \int \frac {\sqrt [3]{1-x^3}}{1-x+x^2} \, dx=\frac {\sqrt [3]{2} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\arctan \left (\frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}+\frac {\arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\sqrt [3]{2} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\log \left (x^3+1\right )}{3\ 2^{2/3}}+\frac {\log \left (2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}-\frac {\log \left (\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3\ 2^{2/3}}+\frac {1}{3} \sqrt [3]{2} \log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )-\frac {\log \left (\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}\right )}{6\ 2^{2/3}}+\frac {1}{2} \log \left (-\sqrt [3]{1-x^3}-x\right )-\frac {\log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )}{2^{2/3}} \]

[In]

Int[(1 - x^3)^(1/3)/(1 - x + x^2),x]

[Out]

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

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 298

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Dist[-(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3
]^2*x^2), x], x] /; FreeQ[{a, b}, x]

Rule 337

Int[(x_)/((a_) + (b_.)*(x_)^3)^(2/3), x_Symbol] :> With[{q = Rt[b, 3]}, Simp[-ArcTan[(1 + 2*q*(x/(a + b*x^3)^(
1/3)))/Sqrt[3]]/(Sqrt[3]*q^2), x] - Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*q^2), x]] /; FreeQ[{a, b}, x]

Rule 420

Int[((a_) + (b_.)*(x_)^3)^(1/3)/((c_) + (d_.)*(x_)^3), x_Symbol] :> With[{q = Rt[b/a, 3]}, Dist[9*(a/(c*q)), S
ubst[Int[x/((4 - a*x^3)*(1 + 2*a*x^3)), x], x, (1 + q*x)/(a + b*x^3)^(1/3)], x]] /; FreeQ[{a, b, c, d}, x] &&
NeQ[b*c - a*d, 0] && EqQ[b*c + a*d, 0]

Rule 493

Int[((e_.)*(x_))^(m_.)/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[b/(b*c - a*d), I
nt[(e*x)^m/(a + b*x^n), x], x] - Dist[d/(b*c - a*d), Int[(e*x)^m/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e,
m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0]

Rule 495

Int[((x_)*((a_) + (b_.)*(x_)^(n_))^(p_))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Dist[b/d, Int[x*(a + b*x^n)^(p
- 1), x], x] - Dist[(b*c - a*d)/d, Int[x*((a + b*x^n)^(p - 1)/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d}, x]
&& NeQ[b*c - a*d, 0] && IGtQ[n, 0] && GtQ[p, 0] && IntBinomialQ[a, b, c, d, 1, 1, n, p, -1, x]

Rule 503

Int[(x_)/(((a_) + (b_.)*(x_)^3)^(2/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/c, 3]}, Si
mp[-ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sqrt[3]*c*q^2), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/
(2*c*q^2), x] + Simp[Log[c + d*x^3]/(6*c*q^2), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
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]

Rule 2183

Int[(Px_.)*((c_) + (d_.)*(x_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^3)^(p_.), x_Symbol] :> Dist[1/c^q, Int[E
xpandIntegrand[(c^3 - d^3*x^3)^q*(a + b*x^3)^p, Px/(c - d*x)^q, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] &&
PolyQ[Px, x] && EqQ[d^2 - c*e, 0] && ILtQ[q, 0] && RationalQ[p] && EqQ[Denominator[p], 3]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\sqrt [3]{1-x^3}}{1+x^3}+\frac {x \sqrt [3]{1-x^3}}{1+x^3}\right ) \, dx \\ & = \int \frac {\sqrt [3]{1-x^3}}{1+x^3} \, dx+\int \frac {x \sqrt [3]{1-x^3}}{1+x^3} \, dx \\ & = 2 \int \frac {x}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx-9 \text {Subst}\left (\int \frac {x}{\left (4-x^3\right ) \left (1+2 x^3\right )} \, dx,x,\frac {1-x}{\sqrt [3]{1-x^3}}\right )-\int \frac {x}{\left (1-x^3\right )^{2/3}} \, dx \\ & = \frac {\arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\sqrt [3]{2} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\log \left (1+x^3\right )}{3\ 2^{2/3}}+\frac {1}{2} \log \left (-x-\sqrt [3]{1-x^3}\right )-\frac {\log \left (-\sqrt [3]{2} x-\sqrt [3]{1-x^3}\right )}{2^{2/3}}-2 \text {Subst}\left (\int \frac {x}{1+2 x^3} \, dx,x,\frac {1-x}{\sqrt [3]{1-x^3}}\right )-\text {Subst}\left (\int \frac {x}{4-x^3} \, dx,x,\frac {1-x}{\sqrt [3]{1-x^3}}\right ) \\ & = \frac {\arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\sqrt [3]{2} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\log \left (1+x^3\right )}{3\ 2^{2/3}}+\frac {1}{2} \log \left (-x-\sqrt [3]{1-x^3}\right )-\frac {\log \left (-\sqrt [3]{2} x-\sqrt [3]{1-x^3}\right )}{2^{2/3}}-\frac {\text {Subst}\left (\int \frac {1}{2^{2/3}-x} \, dx,x,\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}+\frac {\text {Subst}\left (\int \frac {2^{2/3}-x}{2 \sqrt [3]{2}+2^{2/3} x+x^2} \, dx,x,\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}+\frac {1}{3} 2^{2/3} \text {Subst}\left (\int \frac {1}{1+\sqrt [3]{2} x} \, dx,x,\frac {1-x}{\sqrt [3]{1-x^3}}\right )-\frac {1}{3} 2^{2/3} \text {Subst}\left (\int \frac {1+\sqrt [3]{2} x}{1-\sqrt [3]{2} x+2^{2/3} x^2} \, dx,x,\frac {1-x}{\sqrt [3]{1-x^3}}\right ) \\ & = \frac {\arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\sqrt [3]{2} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\log \left (1+x^3\right )}{3\ 2^{2/3}}+\frac {\log \left (2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}+\frac {1}{3} \sqrt [3]{2} \log \left (1+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )+\frac {1}{2} \log \left (-x-\sqrt [3]{1-x^3}\right )-\frac {\log \left (-\sqrt [3]{2} x-\sqrt [3]{1-x^3}\right )}{2^{2/3}}+\frac {1}{2} \text {Subst}\left (\int \frac {1}{2 \sqrt [3]{2}+2^{2/3} x+x^2} \, dx,x,\frac {1-x}{\sqrt [3]{1-x^3}}\right )-\frac {\text {Subst}\left (\int \frac {2^{2/3}+2 x}{2 \sqrt [3]{2}+2^{2/3} x+x^2} \, dx,x,\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{6\ 2^{2/3}}-\frac {\text {Subst}\left (\int \frac {-\sqrt [3]{2}+2\ 2^{2/3} x}{1-\sqrt [3]{2} x+2^{2/3} x^2} \, dx,x,\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}-\frac {\text {Subst}\left (\int \frac {1}{1-\sqrt [3]{2} x+2^{2/3} x^2} \, dx,x,\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{\sqrt [3]{2}} \\ & = \frac {\arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\sqrt [3]{2} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\log \left (1+x^3\right )}{3\ 2^{2/3}}+\frac {\log \left (2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}-\frac {\log \left (1+\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}+\frac {1}{3} \sqrt [3]{2} \log \left (1+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )-\frac {\log \left (2 \sqrt [3]{2}+\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}\right )}{6\ 2^{2/3}}+\frac {1}{2} \log \left (-x-\sqrt [3]{1-x^3}\right )-\frac {\log \left (-\sqrt [3]{2} x-\sqrt [3]{1-x^3}\right )}{2^{2/3}}-\frac {\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )}{2^{2/3}}-\sqrt [3]{2} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right ) \\ & = \frac {\sqrt [3]{2} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\arctan \left (\frac {1+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}+\frac {\arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\sqrt [3]{2} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\log \left (1+x^3\right )}{3\ 2^{2/3}}+\frac {\log \left (2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}-\frac {\log \left (1+\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}+\frac {1}{3} \sqrt [3]{2} \log \left (1+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )-\frac {\log \left (2 \sqrt [3]{2}+\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}\right )}{6\ 2^{2/3}}+\frac {1}{2} \log \left (-x-\sqrt [3]{1-x^3}\right )-\frac {\log \left (-\sqrt [3]{2} x-\sqrt [3]{1-x^3}\right )}{2^{2/3}} \\ \end{align*}

Mathematica [F]

\[ \int \frac {\sqrt [3]{1-x^3}}{1-x+x^2} \, dx=\int \frac {\sqrt [3]{1-x^3}}{1-x+x^2} \, dx \]

[In]

Integrate[(1 - x^3)^(1/3)/(1 - x + x^2),x]

[Out]

Integrate[(1 - x^3)^(1/3)/(1 - x + x^2), x]

Maple [C] (warning: unable to verify)

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

Time = 21.32 (sec) , antiderivative size = 1410, normalized size of antiderivative = 5.04

method	result	size
trager	\(\text {Expression too large to display}\)	\(1410\)

[In]

int((-x^3+1)^(1/3)/(x^2-x+1),x,method=_RETURNVERBOSE)

[Out]

1/3*ln(RootOf(_Z^2+_Z+1)^2*x^3-3*RootOf(_Z^2+_Z+1)*(-x^3+1)^(2/3)*x+3*RootOf(_Z^2+_Z+1)*(-x^3+1)^(1/3)*x^2-2*R
ootOf(_Z^2+_Z+1)*x^3+x^3+RootOf(_Z^2+_Z+1)-1)*RootOf(_Z^2+_Z+1)-1/3*RootOf(_Z^2+_Z+1)*ln(RootOf(_Z^2+_Z+1)^2*x
^3+3*RootOf(_Z^2+_Z+1)*(-x^3+1)^(2/3)*x-3*RootOf(_Z^2+_Z+1)*(-x^3+1)^(1/3)*x^2+4*RootOf(_Z^2+_Z+1)*x^3+3*x*(-x
^3+1)^(2/3)-3*x^2*(-x^3+1)^(1/3)+4*x^3-RootOf(_Z^2+_Z+1)-2)-1/3*ln(RootOf(_Z^2+_Z+1)^2*x^3+3*RootOf(_Z^2+_Z+1)
*(-x^3+1)^(2/3)*x-3*RootOf(_Z^2+_Z+1)*(-x^3+1)^(1/3)*x^2+4*RootOf(_Z^2+_Z+1)*x^3+3*x*(-x^3+1)^(2/3)-3*x^2*(-x^
3+1)^(1/3)+4*x^3-RootOf(_Z^2+_Z+1)-2)-1/18*ln(-(12*(-x^3+1)^(1/3)*RootOf(_Z^3-324*RootOf(_Z^2+_Z+1)-162)*RootO
f(_Z^2+_Z+1)*x^3+RootOf(_Z^3-324*RootOf(_Z^2+_Z+1)-162)^2*x^4-36*(-x^3+1)^(1/3)*RootOf(_Z^3-324*RootOf(_Z^2+_Z
+1)-162)*RootOf(_Z^2+_Z+1)*x^2+6*(-x^3+1)^(1/3)*RootOf(_Z^3-324*RootOf(_Z^2+_Z+1)-162)*x^3+2*x^3*RootOf(_Z^3-3
24*RootOf(_Z^2+_Z+1)-162)^2+108*(-x^3+1)^(2/3)*x^2+12*(-x^3+1)^(1/3)*RootOf(_Z^3-324*RootOf(_Z^2+_Z+1)-162)*Ro
otOf(_Z^2+_Z+1)*x-18*(-x^3+1)^(1/3)*RootOf(_Z^3-324*RootOf(_Z^2+_Z+1)-162)*x^2-RootOf(_Z^3-324*RootOf(_Z^2+_Z+
1)-162)^2*x^2-108*x*(-x^3+1)^(2/3)+6*(-x^3+1)^(1/3)*RootOf(_Z^3-324*RootOf(_Z^2+_Z+1)-162)*x-2*RootOf(_Z^3-324
*RootOf(_Z^2+_Z+1)-162)^2*x+RootOf(_Z^3-324*RootOf(_Z^2+_Z+1)-162)^2)/(x^2-x+1)^2)*RootOf(_Z^3-324*RootOf(_Z^2
+_Z+1)-162)*RootOf(_Z^2+_Z+1)-1/18*ln(-(12*(-x^3+1)^(1/3)*RootOf(_Z^3-324*RootOf(_Z^2+_Z+1)-162)*RootOf(_Z^2+_
Z+1)*x^3+RootOf(_Z^3-324*RootOf(_Z^2+_Z+1)-162)^2*x^4-36*(-x^3+1)^(1/3)*RootOf(_Z^3-324*RootOf(_Z^2+_Z+1)-162)
*RootOf(_Z^2+_Z+1)*x^2+6*(-x^3+1)^(1/3)*RootOf(_Z^3-324*RootOf(_Z^2+_Z+1)-162)*x^3+2*x^3*RootOf(_Z^3-324*RootO
f(_Z^2+_Z+1)-162)^2+108*(-x^3+1)^(2/3)*x^2+12*(-x^3+1)^(1/3)*RootOf(_Z^3-324*RootOf(_Z^2+_Z+1)-162)*RootOf(_Z^
2+_Z+1)*x-18*(-x^3+1)^(1/3)*RootOf(_Z^3-324*RootOf(_Z^2+_Z+1)-162)*x^2-RootOf(_Z^3-324*RootOf(_Z^2+_Z+1)-162)^
2*x^2-108*x*(-x^3+1)^(2/3)+6*(-x^3+1)^(1/3)*RootOf(_Z^3-324*RootOf(_Z^2+_Z+1)-162)*x-2*RootOf(_Z^3-324*RootOf(
_Z^2+_Z+1)-162)^2*x+RootOf(_Z^3-324*RootOf(_Z^2+_Z+1)-162)^2)/(x^2-x+1)^2)*RootOf(_Z^3-324*RootOf(_Z^2+_Z+1)-1
62)+1/18*RootOf(_Z^3-324*RootOf(_Z^2+_Z+1)-162)*ln(-(-5*RootOf(_Z^2+_Z+1)*RootOf(_Z^3-324*RootOf(_Z^2+_Z+1)-16
2)^2*x^4+18*(-x^3+1)^(1/3)*RootOf(_Z^3-324*RootOf(_Z^2+_Z+1)-162)*RootOf(_Z^2+_Z+1)*x^3+2*RootOf(_Z^2+_Z+1)*Ro
otOf(_Z^3-324*RootOf(_Z^2+_Z+1)-162)^2*x^3-6*(-x^3+1)^(1/3)*RootOf(_Z^3-324*RootOf(_Z^2+_Z+1)-162)*RootOf(_Z^2
+_Z+1)*x^2-18*(-x^3+1)^(1/3)*RootOf(_Z^3-324*RootOf(_Z^2+_Z+1)-162)*x^3+RootOf(_Z^2+_Z+1)*RootOf(_Z^3-324*Root
Of(_Z^2+_Z+1)-162)^2*x^2+216*(-x^3+1)^(2/3)*x^2-6*(-x^3+1)^(1/3)*RootOf(_Z^3-324*RootOf(_Z^2+_Z+1)-162)*RootOf
(_Z^2+_Z+1)*x+6*(-x^3+1)^(1/3)*RootOf(_Z^3-324*RootOf(_Z^2+_Z+1)-162)*x^2+2*RootOf(_Z^2+_Z+1)*RootOf(_Z^3-324*
RootOf(_Z^2+_Z+1)-162)^2*x-108*x*(-x^3+1)^(2/3)+6*(-x^3+1)^(1/3)*RootOf(_Z^3-324*RootOf(_Z^2+_Z+1)-162)*x-Root
Of(_Z^3-324*RootOf(_Z^2+_Z+1)-162)^2*RootOf(_Z^2+_Z+1))/(x^2-x+1)^2)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 5.35 (sec) , antiderivative size = 3880, normalized size of antiderivative = 13.86 \[ \int \frac {\sqrt [3]{1-x^3}}{1-x+x^2} \, dx=\text {Too large to display} \]

[In]

integrate((-x^3+1)^(1/3)/(x^2-x+1),x, algorithm="fricas")

[Out]

1/72*sqrt(3)*(sqrt(-3)*(-4)^(1/6) + (-4)^(1/6))*log(3*(6*sqrt(-3)*(-4)^(1/3)*(26655*x^12 - 185476*x^11 + 15587
2*x^10 + 361508*x^9 - 363117*x^8 - 87612*x^7 - 197936*x^6 + 492924*x^5 - 182367*x^4 - 39436*x^3 + 18254*x^2 +
3652*x - 1278) + 48*(11866*x^10 - 16425*x^9 - 125794*x^8 + 251931*x^7 - 71187*x^6 - 79049*x^5 - 2745*x^4 + 520
32*x^3 - 20629*x^2 - sqrt(3)*(3104*I*x^10 - 43815*I*x^9 + 84520*I*x^8 + 11329*I*x^7 - 92013*I*x^6 + 5291*I*x^5
+ 53855*I*x^4 - 20262*I*x^3 - 2009*I*x^2 + 1278*I*x) + 2008*x)*(-x^3 + 1)^(2/3) + sqrt(3)*(sqrt(-3)*(-4)^(5/6
)*(31397*x^12 + 113940*x^11 - 831396*x^10 + 973364*x^9 - 140709*x^8 + 407484*x^7 - 1009896*x^6 + 313212*x^5 +
248121*x^4 - 75940*x^3 - 48198*x^2 + 19716*x - 2008) - (-4)^(5/6)*(31397*x^12 + 113940*x^11 - 831396*x^10 + 97
3364*x^9 - 140709*x^8 + 407484*x^7 - 1009896*x^6 + 313212*x^5 + 248121*x^4 - 75940*x^3 - 48198*x^2 + 19716*x -
2008)) - 6*(-4)^(1/3)*(26655*x^12 - 185476*x^11 + 155872*x^10 + 361508*x^9 - 363117*x^8 - 87612*x^7 - 197936*
x^6 + 492924*x^5 - 182367*x^4 - 39436*x^3 + 18254*x^2 + 3652*x - 1278) - 6*(-x^3 + 1)^(1/3)*(sqrt(-3)*(-4)^(2/
3)*(1459*x^11 + 94937*x^10 - 314364*x^9 + 204807*x^8 + 73586*x^7 + 103515*x^6 - 263973*x^5 + 67714*x^4 + 54774
*x^3 - 25376*x^2 + 2008*x) + (-4)^(2/3)*(1459*x^11 + 94937*x^10 - 314364*x^9 + 204807*x^8 + 73586*x^7 + 103515
*x^6 - 263973*x^5 + 67714*x^4 + 54774*x^3 - 25376*x^2 + 2008*x) + 2*sqrt(3)*(sqrt(-3)*(-4)^(1/6)*(12049*x^11 -
48557*x^10 - 31048*x^9 + 203745*x^8 - 117748*x^7 - 29753*x^6 - 67923*x^5 + 127612*x^4 - 49654*x^3 + 1642*x^2
+ 1278*x) + (-4)^(1/6)*(12049*x^11 - 48557*x^10 - 31048*x^9 + 203745*x^8 - 117748*x^7 - 29753*x^6 - 67923*x^5
+ 127612*x^4 - 49654*x^3 + 1642*x^2 + 1278*x))))/(x^12 - 6*x^11 + 21*x^10 - 50*x^9 + 90*x^8 - 126*x^7 + 141*x^
6 - 126*x^5 + 90*x^4 - 50*x^3 + 21*x^2 - 6*x + 1)) - 1/72*sqrt(3)*(sqrt(-3)*(-4)^(1/6) - (-4)^(1/6))*log(-3*(6
*sqrt(-3)*(-4)^(1/3)*(26655*x^12 - 185476*x^11 + 155872*x^10 + 361508*x^9 - 363117*x^8 - 87612*x^7 - 197936*x^
6 + 492924*x^5 - 182367*x^4 - 39436*x^3 + 18254*x^2 + 3652*x - 1278) - 48*(11866*x^10 - 16425*x^9 - 125794*x^8
+ 251931*x^7 - 71187*x^6 - 79049*x^5 - 2745*x^4 + 52032*x^3 - 20629*x^2 - sqrt(3)*(3104*I*x^10 - 43815*I*x^9
+ 84520*I*x^8 + 11329*I*x^7 - 92013*I*x^6 + 5291*I*x^5 + 53855*I*x^4 - 20262*I*x^3 - 2009*I*x^2 + 1278*I*x) +
2008*x)*(-x^3 + 1)^(2/3) + sqrt(3)*(sqrt(-3)*(-4)^(5/6)*(31397*x^12 + 113940*x^11 - 831396*x^10 + 973364*x^9 -
140709*x^8 + 407484*x^7 - 1009896*x^6 + 313212*x^5 + 248121*x^4 - 75940*x^3 - 48198*x^2 + 19716*x - 2008) + (
-4)^(5/6)*(31397*x^12 + 113940*x^11 - 831396*x^10 + 973364*x^9 - 140709*x^8 + 407484*x^7 - 1009896*x^6 + 31321
2*x^5 + 248121*x^4 - 75940*x^3 - 48198*x^2 + 19716*x - 2008)) + 6*(-4)^(1/3)*(26655*x^12 - 185476*x^11 + 15587
2*x^10 + 361508*x^9 - 363117*x^8 - 87612*x^7 - 197936*x^6 + 492924*x^5 - 182367*x^4 - 39436*x^3 + 18254*x^2 +
3652*x - 1278) - 6*(-x^3 + 1)^(1/3)*(sqrt(-3)*(-4)^(2/3)*(1459*x^11 + 94937*x^10 - 314364*x^9 + 204807*x^8 + 7
3586*x^7 + 103515*x^6 - 263973*x^5 + 67714*x^4 + 54774*x^3 - 25376*x^2 + 2008*x) - (-4)^(2/3)*(1459*x^11 + 949
37*x^10 - 314364*x^9 + 204807*x^8 + 73586*x^7 + 103515*x^6 - 263973*x^5 + 67714*x^4 + 54774*x^3 - 25376*x^2 +
2008*x) + 2*sqrt(3)*(sqrt(-3)*(-4)^(1/6)*(12049*x^11 - 48557*x^10 - 31048*x^9 + 203745*x^8 - 117748*x^7 - 2975
3*x^6 - 67923*x^5 + 127612*x^4 - 49654*x^3 + 1642*x^2 + 1278*x) - (-4)^(1/6)*(12049*x^11 - 48557*x^10 - 31048*
x^9 + 203745*x^8 - 117748*x^7 - 29753*x^6 - 67923*x^5 + 127612*x^4 - 49654*x^3 + 1642*x^2 + 1278*x))))/(x^12 -
6*x^11 + 21*x^10 - 50*x^9 + 90*x^8 - 126*x^7 + 141*x^6 - 126*x^5 + 90*x^4 - 50*x^3 + 21*x^2 - 6*x + 1)) - 1/7
2*sqrt(3)*(sqrt(-3)*(-4)^(1/6) + (-4)^(1/6))*log(3*(6*sqrt(-3)*(-4)^(1/3)*(26655*x^12 - 185476*x^11 + 155872*x
^10 + 361508*x^9 - 363117*x^8 - 87612*x^7 - 197936*x^6 + 492924*x^5 - 182367*x^4 - 39436*x^3 + 18254*x^2 + 365
2*x - 1278) + 48*(11866*x^10 - 16425*x^9 - 125794*x^8 + 251931*x^7 - 71187*x^6 - 79049*x^5 - 2745*x^4 + 52032*
x^3 - 20629*x^2 - sqrt(3)*(-3104*I*x^10 + 43815*I*x^9 - 84520*I*x^8 - 11329*I*x^7 + 92013*I*x^6 - 5291*I*x^5 -
53855*I*x^4 + 20262*I*x^3 + 2009*I*x^2 - 1278*I*x) + 2008*x)*(-x^3 + 1)^(2/3) - sqrt(3)*(sqrt(-3)*(-4)^(5/6)*
(31397*x^12 + 113940*x^11 - 831396*x^10 + 973364*x^9 - 140709*x^8 + 407484*x^7 - 1009896*x^6 + 313212*x^5 + 24
8121*x^4 - 75940*x^3 - 48198*x^2 + 19716*x - 2008) - (-4)^(5/6)*(31397*x^12 + 113940*x^11 - 831396*x^10 + 9733
64*x^9 - 140709*x^8 + 407484*x^7 - 1009896*x^6 + 313212*x^5 + 248121*x^4 - 75940*x^3 - 48198*x^2 + 19716*x - 2
008)) - 6*(-4)^(1/3)*(26655*x^12 - 185476*x^11 + 155872*x^10 + 361508*x^9 - 363117*x^8 - 87612*x^7 - 197936*x^
6 + 492924*x^5 - 182367*x^4 - 39436*x^3 + 18254*x^2 + 3652*x - 1278) - 6*(-x^3 + 1)^(1/3)*(sqrt(-3)*(-4)^(2/3)
*(1459*x^11 + 94937*x^10 - 314364*x^9 + 204807*x^8 + 73586*x^7 + 103515*x^6 - 263973*x^5 + 67714*x^4 + 54774*x
^3 - 25376*x^2 + 2008*x) + (-4)^(2/3)*(1459*x^11 + 94937*x^10 - 314364*x^9 + 204807*x^8 + 73586*x^7 + 103515*x
^6 - 263973*x^5 + 67714*x^4 + 54774*x^3 - 25376*x^2 + 2008*x) - 2*sqrt(3)*(sqrt(-3)*(-4)^(1/6)*(12049*x^11 - 4
8557*x^10 - 31048*x^9 + 203745*x^8 - 117748*x^7 - 29753*x^6 - 67923*x^5 + 127612*x^4 - 49654*x^3 + 1642*x^2 +
1278*x) + (-4)^(1/6)*(12049*x^11 - 48557*x^10 - 31048*x^9 + 203745*x^8 - 117748*x^7 - 29753*x^6 - 67923*x^5 +
127612*x^4 - 49654*x^3 + 1642*x^2 + 1278*x))))/(x^12 - 6*x^11 + 21*x^10 - 50*x^9 + 90*x^8 - 126*x^7 + 141*x^6
- 126*x^5 + 90*x^4 - 50*x^3 + 21*x^2 - 6*x + 1)) + 1/72*sqrt(3)*(sqrt(-3)*(-4)^(1/6) - (-4)^(1/6))*log(-3*(6*s
qrt(-3)*(-4)^(1/3)*(26655*x^12 - 185476*x^11 + 155872*x^10 + 361508*x^9 - 363117*x^8 - 87612*x^7 - 197936*x^6
+ 492924*x^5 - 182367*x^4 - 39436*x^3 + 18254*x^2 + 3652*x - 1278) - 48*(11866*x^10 - 16425*x^9 - 125794*x^8 +
251931*x^7 - 71187*x^6 - 79049*x^5 - 2745*x^4 + 52032*x^3 - 20629*x^2 - sqrt(3)*(-3104*I*x^10 + 43815*I*x^9 -
84520*I*x^8 - 11329*I*x^7 + 92013*I*x^6 - 5291*I*x^5 - 53855*I*x^4 + 20262*I*x^3 + 2009*I*x^2 - 1278*I*x) + 2
008*x)*(-x^3 + 1)^(2/3) - sqrt(3)*(sqrt(-3)*(-4)^(5/6)*(31397*x^12 + 113940*x^11 - 831396*x^10 + 973364*x^9 -
140709*x^8 + 407484*x^7 - 1009896*x^6 + 313212*x^5 + 248121*x^4 - 75940*x^3 - 48198*x^2 + 19716*x - 2008) + (-
4)^(5/6)*(31397*x^12 + 113940*x^11 - 831396*x^10 + 973364*x^9 - 140709*x^8 + 407484*x^7 - 1009896*x^6 + 313212
*x^5 + 248121*x^4 - 75940*x^3 - 48198*x^2 + 19716*x - 2008)) + 6*(-4)^(1/3)*(26655*x^12 - 185476*x^11 + 155872
*x^10 + 361508*x^9 - 363117*x^8 - 87612*x^7 - 197936*x^6 + 492924*x^5 - 182367*x^4 - 39436*x^3 + 18254*x^2 + 3
652*x - 1278) - 6*(-x^3 + 1)^(1/3)*(sqrt(-3)*(-4)^(2/3)*(1459*x^11 + 94937*x^10 - 314364*x^9 + 204807*x^8 + 73
586*x^7 + 103515*x^6 - 263973*x^5 + 67714*x^4 + 54774*x^3 - 25376*x^2 + 2008*x) - (-4)^(2/3)*(1459*x^11 + 9493
7*x^10 - 314364*x^9 + 204807*x^8 + 73586*x^7 + 103515*x^6 - 263973*x^5 + 67714*x^4 + 54774*x^3 - 25376*x^2 + 2
008*x) - 2*sqrt(3)*(sqrt(-3)*(-4)^(1/6)*(12049*x^11 - 48557*x^10 - 31048*x^9 + 203745*x^8 - 117748*x^7 - 29753
*x^6 - 67923*x^5 + 127612*x^4 - 49654*x^3 + 1642*x^2 + 1278*x) - (-4)^(1/6)*(12049*x^11 - 48557*x^10 - 31048*x
^9 + 203745*x^8 - 117748*x^7 - 29753*x^6 - 67923*x^5 + 127612*x^4 - 49654*x^3 + 1642*x^2 + 1278*x))))/(x^12 -
6*x^11 + 21*x^10 - 50*x^9 + 90*x^8 - 126*x^7 + 141*x^6 - 126*x^5 + 90*x^4 - 50*x^3 + 21*x^2 - 6*x + 1)) - 1/36
*sqrt(3)*(-4)^(1/6)*log(3*(sqrt(3)*(-4)^(5/6)*(31397*x^12 + 113940*x^11 - 831396*x^10 + 973364*x^9 - 140709*x^
8 + 407484*x^7 - 1009896*x^6 + 313212*x^5 + 248121*x^4 - 75940*x^3 - 48198*x^2 + 19716*x - 2008) + 24*(11866*x
^10 - 16425*x^9 - 125794*x^8 + 251931*x^7 - 71187*x^6 - 79049*x^5 - 2745*x^4 + 52032*x^3 - 20629*x^2 - sqrt(3)
*(3104*I*x^10 - 43815*I*x^9 + 84520*I*x^8 + 11329*I*x^7 - 92013*I*x^6 + 5291*I*x^5 + 53855*I*x^4 - 20262*I*x^3
- 2009*I*x^2 + 1278*I*x) + 2008*x)*(-x^3 + 1)^(2/3) + 6*(-4)^(1/3)*(26655*x^12 - 185476*x^11 + 155872*x^10 +
361508*x^9 - 363117*x^8 - 87612*x^7 - 197936*x^6 + 492924*x^5 - 182367*x^4 - 39436*x^3 + 18254*x^2 + 3652*x -
1278) + 6*(-x^3 + 1)^(1/3)*(2*sqrt(3)*(-4)^(1/6)*(12049*x^11 - 48557*x^10 - 31048*x^9 + 203745*x^8 - 117748*x^
7 - 29753*x^6 - 67923*x^5 + 127612*x^4 - 49654*x^3 + 1642*x^2 + 1278*x) + (-4)^(2/3)*(1459*x^11 + 94937*x^10 -
314364*x^9 + 204807*x^8 + 73586*x^7 + 103515*x^6 - 263973*x^5 + 67714*x^4 + 54774*x^3 - 25376*x^2 + 2008*x)))
/(x^12 - 6*x^11 + 21*x^10 - 50*x^9 + 90*x^8 - 126*x^7 + 141*x^6 - 126*x^5 + 90*x^4 - 50*x^3 + 21*x^2 - 6*x + 1
)) + 1/36*sqrt(3)*(-4)^(1/6)*log(-3*(sqrt(3)*(-4)^(5/6)*(31397*x^12 + 113940*x^11 - 831396*x^10 + 973364*x^9 -
140709*x^8 + 407484*x^7 - 1009896*x^6 + 313212*x^5 + 248121*x^4 - 75940*x^3 - 48198*x^2 + 19716*x - 2008) - 2
4*(11866*x^10 - 16425*x^9 - 125794*x^8 + 251931*x^7 - 71187*x^6 - 79049*x^5 - 2745*x^4 + 52032*x^3 - 20629*x^2
- sqrt(3)*(-3104*I*x^10 + 43815*I*x^9 - 84520*I*x^8 - 11329*I*x^7 + 92013*I*x^6 - 5291*I*x^5 - 53855*I*x^4 +
20262*I*x^3 + 2009*I*x^2 - 1278*I*x) + 2008*x)*(-x^3 + 1)^(2/3) - 6*(-4)^(1/3)*(26655*x^12 - 185476*x^11 + 155
872*x^10 + 361508*x^9 - 363117*x^8 - 87612*x^7 - 197936*x^6 + 492924*x^5 - 182367*x^4 - 39436*x^3 + 18254*x^2
+ 3652*x - 1278) + 6*(-x^3 + 1)^(1/3)*(2*sqrt(3)*(-4)^(1/6)*(12049*x^11 - 48557*x^10 - 31048*x^9 + 203745*x^8
- 117748*x^7 - 29753*x^6 - 67923*x^5 + 127612*x^4 - 49654*x^3 + 1642*x^2 + 1278*x) - (-4)^(2/3)*(1459*x^11 + 9
4937*x^10 - 314364*x^9 + 204807*x^8 + 73586*x^7 + 103515*x^6 - 263973*x^5 + 67714*x^4 + 54774*x^3 - 25376*x^2
+ 2008*x)))/(x^12 - 6*x^11 + 21*x^10 - 50*x^9 + 90*x^8 - 126*x^7 + 141*x^6 - 126*x^5 + 90*x^4 - 50*x^3 + 21*x^
2 - 6*x + 1)) + 1/3*sqrt(3)*arctan((4*sqrt(3)*(-x^3 + 1)^(1/3)*x^2 + 2*sqrt(3)*(-x^3 + 1)^(2/3)*x - sqrt(3)*(x
^3 - 1))/(9*x^3 - 1)) + 1/6*log(3*(-x^3 + 1)^(1/3)*x^2 + 3*(-x^3 + 1)^(2/3)*x + 1)

Sympy [F]

\[ \int \frac {\sqrt [3]{1-x^3}}{1-x+x^2} \, dx=\int \frac {\sqrt [3]{- \left (x - 1\right ) \left (x^{2} + x + 1\right )}}{x^{2} - x + 1}\, dx \]

[In]

integrate((-x**3+1)**(1/3)/(x**2-x+1),x)

[Out]

Integral((-(x - 1)*(x**2 + x + 1))**(1/3)/(x**2 - x + 1), x)

Maxima [F]

\[ \int \frac {\sqrt [3]{1-x^3}}{1-x+x^2} \, dx=\int { \frac {{\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x^{2} - x + 1} \,d x } \]

[In]

integrate((-x^3+1)^(1/3)/(x^2-x+1),x, algorithm="maxima")

[Out]

integrate((-x^3 + 1)^(1/3)/(x^2 - x + 1), x)

Giac [F]

\[ \int \frac {\sqrt [3]{1-x^3}}{1-x+x^2} \, dx=\int { \frac {{\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x^{2} - x + 1} \,d x } \]

[In]

integrate((-x^3+1)^(1/3)/(x^2-x+1),x, algorithm="giac")

[Out]

integrate((-x^3 + 1)^(1/3)/(x^2 - x + 1), x)

Mupad [F(-1)]

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

[In]

int((1 - x^3)^(1/3)/(x^2 - x + 1),x)

[Out]

int((1 - x^3)^(1/3)/(x^2 - x + 1), x)