\(\int \frac {(1-2 x) (1-x^3)^{2/3}}{(1-x+x^2)^2} \, dx\) [110]

Optimal result

Integrand size = 27, antiderivative size = 199 \[ \int \frac {(1-2 x) \left (1-x^3\right )^{2/3}}{\left (1-x+x^2\right )^2} \, dx=\frac {\left (1-x^3\right )^{2/3}}{1-x+x^2}-\frac {2 \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {2^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {2^{2/3} \arctan \left (\frac {1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{\sqrt [3]{2}}-\frac {\log \left (-\sqrt [3]{2} x-\sqrt [3]{1-x^3}\right )}{\sqrt [3]{2}}+\log \left (x+\sqrt [3]{1-x^3}\right ) \]

[Out]

(-x^3+1)^(2/3)/(x^2-x+1)+1/2*ln(2^(1/3)-(-x^3+1)^(1/3))*2^(2/3)-1/2*ln(-2^(1/3)*x-(-x^3+1)^(1/3))*2^(2/3)+ln(x
+(-x^3+1)^(1/3))-2/3*arctan(1/3*(1-2*x/(-x^3+1)^(1/3))*3^(1/2))*3^(1/2)+1/3*arctan(1/3*(1-2*2^(1/3)*x/(-x^3+1)
^(1/3))*3^(1/2))*2^(2/3)*3^(1/2)+1/3*arctan(1/3*(1+2^(2/3)*(-x^3+1)^(1/3))*3^(1/2))*2^(2/3)*3^(1/2)

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.26, number of steps used = 14, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {2183, 386, 384, 455, 43, 57, 631, 210, 31, 478, 544, 245} \[ \int \frac {(1-2 x) \left (1-x^3\right )^{2/3}}{\left (1-x+x^2\right )^2} \, dx=-\frac {2 \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {2^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {2^{2/3} \arctan \left (\frac {2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\left (1-x^3\right )^{2/3} x}{x^3+1}+\frac {\left (1-x^3\right )^{2/3}}{x^3+1}+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{\sqrt [3]{2}}-\frac {2}{3} 2^{2/3} \log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )+\frac {\log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )}{3 \sqrt [3]{2}}+\log \left (\sqrt [3]{1-x^3}+x\right ) \]

[In]

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

[Out]

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

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 57

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, Simp[-L
og[RemoveContent[a + b*x, x]]/(2*b*q), x] + (Dist[3/(2*b), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/
3)], x] - Dist[3/(2*b*q), Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && PosQ
[(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 245

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

Rule 384

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

Rule 386

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

Rule 455

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

Rule 478

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

Rule 544

Int[(((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Dist[f/d,
Int[(a + b*x^n)^p, x], x] + Dist[(d*e - c*f)/d, Int[(a + b*x^n)^p/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e,
f, p, n}, x]

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 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 {\left (1-x^3\right )^{2/3}}{\left (1+x^3\right )^2}-\frac {3 x^2 \left (1-x^3\right )^{2/3}}{\left (1+x^3\right )^2}-\frac {2 x^3 \left (1-x^3\right )^{2/3}}{\left (1+x^3\right )^2}\right ) \, dx \\ & = -\left (2 \int \frac {x^3 \left (1-x^3\right )^{2/3}}{\left (1+x^3\right )^2} \, dx\right )-3 \int \frac {x^2 \left (1-x^3\right )^{2/3}}{\left (1+x^3\right )^2} \, dx+\int \frac {\left (1-x^3\right )^{2/3}}{\left (1+x^3\right )^2} \, dx \\ & = \frac {x \left (1-x^3\right )^{2/3}}{1+x^3}+\frac {2}{3} \int \frac {1}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx-\frac {2}{3} \int \frac {1-3 x^3}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx-\text {Subst}\left (\int \frac {(1-x)^{2/3}}{(1+x)^2} \, dx,x,x^3\right ) \\ & = \frac {\left (1-x^3\right )^{2/3}}{1+x^3}+\frac {x \left (1-x^3\right )^{2/3}}{1+x^3}-\frac {2^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\log \left (1+x^3\right )}{9 \sqrt [3]{2}}+\frac {\log \left (-\sqrt [3]{2} x-\sqrt [3]{1-x^3}\right )}{3 \sqrt [3]{2}}+\frac {2}{3} \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} (1+x)} \, dx,x,x^3\right )+2 \int \frac {1}{\sqrt [3]{1-x^3}} \, dx-\frac {8}{3} \int \frac {1}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx \\ & = \frac {\left (1-x^3\right )^{2/3}}{1+x^3}+\frac {x \left (1-x^3\right )^{2/3}}{1+x^3}-\frac {2 \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {2^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\log \left (-\sqrt [3]{2} x-\sqrt [3]{1-x^3}\right )}{3 \sqrt [3]{2}}-\frac {2}{3} 2^{2/3} \log \left (-\sqrt [3]{2} x-\sqrt [3]{1-x^3}\right )+\log \left (x+\sqrt [3]{1-x^3}\right )-\frac {\text {Subst}\left (\int \frac {1}{\sqrt [3]{2}-x} \, dx,x,\sqrt [3]{1-x^3}\right )}{\sqrt [3]{2}}+\text {Subst}\left (\int \frac {1}{2^{2/3}+\sqrt [3]{2} x+x^2} \, dx,x,\sqrt [3]{1-x^3}\right ) \\ & = \frac {\left (1-x^3\right )^{2/3}}{1+x^3}+\frac {x \left (1-x^3\right )^{2/3}}{1+x^3}-\frac {2 \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {2^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{\sqrt [3]{2}}+\frac {\log \left (-\sqrt [3]{2} x-\sqrt [3]{1-x^3}\right )}{3 \sqrt [3]{2}}-\frac {2}{3} 2^{2/3} \log \left (-\sqrt [3]{2} x-\sqrt [3]{1-x^3}\right )+\log \left (x+\sqrt [3]{1-x^3}\right )-2^{2/3} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2^{2/3} \sqrt [3]{1-x^3}\right ) \\ & = \frac {\left (1-x^3\right )^{2/3}}{1+x^3}+\frac {x \left (1-x^3\right )^{2/3}}{1+x^3}-\frac {2 \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {2^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {2^{2/3} \arctan \left (\frac {1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{\sqrt [3]{2}}+\frac {\log \left (-\sqrt [3]{2} x-\sqrt [3]{1-x^3}\right )}{3 \sqrt [3]{2}}-\frac {2}{3} 2^{2/3} \log \left (-\sqrt [3]{2} x-\sqrt [3]{1-x^3}\right )+\log \left (x+\sqrt [3]{1-x^3}\right ) \\ \end{align*}

Mathematica [F]

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

[In]

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

[Out]

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

Maple [C] (verified)

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

Time = 26.80 (sec) , antiderivative size = 618, normalized size of antiderivative = 3.11

method	result	size
trager	\(\frac {\left (-x^{3}+1\right )^{\frac {2}{3}}}{x^{2}-x +1}+\frac {2 \ln \left (-\operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{6} x^{3}-36 \operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{3} \left (-x^{3}+1\right )^{\frac {2}{3}} x +36 \operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{3} x^{3}-24 \operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{3}+432 x \left (-x^{3}+1\right )^{\frac {2}{3}}+864 x^{2} \left (-x^{3}+1\right )^{\frac {1}{3}}\right )}{3}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{5} \left (-x^{3}+1\right )^{\frac {1}{3}}+\operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{4} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{4} x -\operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{4}+12 \operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{2} \left (-x^{3}+1\right )^{\frac {1}{3}}+36 \operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right ) x^{2}-36 \operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right ) x +144 \left (-x^{3}+1\right )^{\frac {2}{3}}-36 \operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )}{x^{2}-x +1}\right )}{3}+\frac {\ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{4} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{4} x -\operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{4}+12 \operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{2} \left (-x^{3}+1\right )^{\frac {1}{3}} x +72 \left (-x^{3}+1\right )^{\frac {2}{3}}}{x^{2}-x +1}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{4}}{72}-\frac {\ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{4} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{4} x -\operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{4}+12 \operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{2} \left (-x^{3}+1\right )^{\frac {1}{3}} x +72 \left (-x^{3}+1\right )^{\frac {2}{3}}}{x^{2}-x +1}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )}{6}-\frac {\ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{6} x^{3}+72 \operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{3} \left (-x^{3}+1\right )^{\frac {2}{3}} x +72 \operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{3} \left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}-24 \operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{3}+864 x \left (-x^{3}+1\right )^{\frac {2}{3}}-864 x^{2} \left (-x^{3}+1\right )^{\frac {1}{3}}-1296 x^{3}+864\right ) \operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{3}}{36}-\frac {\ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{6} x^{3}+72 \operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{3} \left (-x^{3}+1\right )^{\frac {2}{3}} x +72 \operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{3} \left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}-24 \operatorname {RootOf}\left (\textit {\_Z}^{6}+432\right )^{3}+864 x \left (-x^{3}+1\right )^{\frac {2}{3}}-864 x^{2} \left (-x^{3}+1\right )^{\frac {1}{3}}-1296 x^{3}+864\right )}{3}\)	\(618\)
risch	\(\text {Expression too large to display}\)	\(1119\)

[In]

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

[Out]

(-x^3+1)^(2/3)/(x^2-x+1)+2/3*ln(-RootOf(_Z^6+432)^6*x^3-36*RootOf(_Z^6+432)^3*(-x^3+1)^(2/3)*x+36*RootOf(_Z^6+
432)^3*x^3-24*RootOf(_Z^6+432)^3+432*x*(-x^3+1)^(2/3)+864*x^2*(-x^3+1)^(1/3))+1/3*RootOf(_Z^6+432)*ln((RootOf(
_Z^6+432)^5*(-x^3+1)^(1/3)+RootOf(_Z^6+432)^4*x^2-RootOf(_Z^6+432)^4*x-RootOf(_Z^6+432)^4+12*RootOf(_Z^6+432)^
2*(-x^3+1)^(1/3)+36*RootOf(_Z^6+432)*x^2-36*RootOf(_Z^6+432)*x+144*(-x^3+1)^(2/3)-36*RootOf(_Z^6+432))/(x^2-x+
1))+1/72*ln(-(RootOf(_Z^6+432)^4*x^2+RootOf(_Z^6+432)^4*x-RootOf(_Z^6+432)^4+12*RootOf(_Z^6+432)^2*(-x^3+1)^(1
/3)*x+72*(-x^3+1)^(2/3))/(x^2-x+1))*RootOf(_Z^6+432)^4-1/6*ln(-(RootOf(_Z^6+432)^4*x^2+RootOf(_Z^6+432)^4*x-Ro
otOf(_Z^6+432)^4+12*RootOf(_Z^6+432)^2*(-x^3+1)^(1/3)*x+72*(-x^3+1)^(2/3))/(x^2-x+1))*RootOf(_Z^6+432)-1/36*ln
(RootOf(_Z^6+432)^6*x^3+72*RootOf(_Z^6+432)^3*(-x^3+1)^(2/3)*x+72*RootOf(_Z^6+432)^3*(-x^3+1)^(1/3)*x^2-24*Roo
tOf(_Z^6+432)^3+864*x*(-x^3+1)^(2/3)-864*x^2*(-x^3+1)^(1/3)-1296*x^3+864)*RootOf(_Z^6+432)^3-1/3*ln(RootOf(_Z^
6+432)^6*x^3+72*RootOf(_Z^6+432)^3*(-x^3+1)^(2/3)*x+72*RootOf(_Z^6+432)^3*(-x^3+1)^(1/3)*x^2-24*RootOf(_Z^6+43
2)^3+864*x*(-x^3+1)^(2/3)-864*x^2*(-x^3+1)^(1/3)-1296*x^3+864)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.18 (sec) , antiderivative size = 2298, normalized size of antiderivative = 11.55 \[ \int \frac {(1-2 x) \left (1-x^3\right )^{2/3}}{\left (1-x+x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/36*(2*sqrt(3)*(-16)^(1/6)*(x^2 - x + 1)*log(9*(4*sqrt(3)*(-16)^(1/6)*(131*x^6 - 48*x^5 - 381*x^4 - 152*x^3 +
267*x^2 + 276*x - 112) + 12*2^(2/3)*(15*x^6 - 200*x^5 + 5*x^4 + 216*x^3 + 157*x^2 - 124*x - 42) + 24*(31*x^4
+ 107*x^3 - 243*x^2 - sqrt(3)*(-23*I*x^4 + 85*I*x^3 + 57*I*x^2 - 104*I*x + 4*I) - 26*x + 50)*(-x^3 + 1)^(2/3)
+ 3*(-x^3 + 1)^(1/3)*(sqrt(3)*(-16)^(5/6)*(4*x^5 + 69*x^4 - 58*x^3 - 77*x^2 + 12*x + 23) - 8*(-2)^(1/3)*(50*x^
5 - 93*x^4 - 88*x^3 - 7*x^2 + 150*x - 31)))/(x^6 - 3*x^5 + 6*x^4 - 7*x^3 + 6*x^2 - 3*x + 1)) - 2*sqrt(3)*(-16)
^(1/6)*(x^2 - x + 1)*log(-9*(4*sqrt(3)*(-16)^(1/6)*(131*x^6 - 48*x^5 - 381*x^4 - 152*x^3 + 267*x^2 + 276*x - 1
12) - 12*2^(2/3)*(15*x^6 - 200*x^5 + 5*x^4 + 216*x^3 + 157*x^2 - 124*x - 42) - 24*(31*x^4 + 107*x^3 - 243*x^2
- sqrt(3)*(23*I*x^4 - 85*I*x^3 - 57*I*x^2 + 104*I*x - 4*I) - 26*x + 50)*(-x^3 + 1)^(2/3) + 3*(-x^3 + 1)^(1/3)*
(sqrt(3)*(-16)^(5/6)*(4*x^5 + 69*x^4 - 58*x^3 - 77*x^2 + 12*x + 23) + 8*(-2)^(1/3)*(50*x^5 - 93*x^4 - 88*x^3 -
7*x^2 + 150*x - 31)))/(x^6 - 3*x^5 + 6*x^4 - 7*x^3 + 6*x^2 - 3*x + 1)) - 24*sqrt(3)*(x^2 - x + 1)*arctan((4*s
qrt(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)) + sqrt(3)*(sqrt(-
3)*(-16)^(1/6)*(x^2 - x + 1) + (-16)^(1/6)*(x^2 - x + 1))*log(-9*(12*2^(2/3)*sqrt(-3)*(15*x^6 - 200*x^5 + 5*x^
4 + 216*x^3 + 157*x^2 - 124*x - 42) + 12*2^(2/3)*(15*x^6 - 200*x^5 + 5*x^4 + 216*x^3 + 157*x^2 - 124*x - 42) -
48*(31*x^4 + 107*x^3 - 243*x^2 - sqrt(3)*(23*I*x^4 - 85*I*x^3 - 57*I*x^2 + 104*I*x - 4*I) - 26*x + 50)*(-x^3
+ 1)^(2/3) - 4*sqrt(3)*(sqrt(-3)*(-16)^(1/6)*(131*x^6 - 48*x^5 - 381*x^4 - 152*x^3 + 267*x^2 + 276*x - 112) +
(-16)^(1/6)*(131*x^6 - 48*x^5 - 381*x^4 - 152*x^3 + 267*x^2 + 276*x - 112)) + 3*(-x^3 + 1)^(1/3)*(8*(-2)^(1/3)
*sqrt(-3)*(50*x^5 - 93*x^4 - 88*x^3 - 7*x^2 + 150*x - 31) + sqrt(3)*(sqrt(-3)*(-16)^(5/6)*(4*x^5 + 69*x^4 - 58
*x^3 - 77*x^2 + 12*x + 23) - (-16)^(5/6)*(4*x^5 + 69*x^4 - 58*x^3 - 77*x^2 + 12*x + 23)) - 8*(-2)^(1/3)*(50*x^
5 - 93*x^4 - 88*x^3 - 7*x^2 + 150*x - 31)))/(x^6 - 3*x^5 + 6*x^4 - 7*x^3 + 6*x^2 - 3*x + 1)) - sqrt(3)*(sqrt(-
3)*(-16)^(1/6)*(x^2 - x + 1) + (-16)^(1/6)*(x^2 - x + 1))*log(-9*(12*2^(2/3)*sqrt(-3)*(15*x^6 - 200*x^5 + 5*x^
4 + 216*x^3 + 157*x^2 - 124*x - 42) + 12*2^(2/3)*(15*x^6 - 200*x^5 + 5*x^4 + 216*x^3 + 157*x^2 - 124*x - 42) -
48*(31*x^4 + 107*x^3 - 243*x^2 - sqrt(3)*(-23*I*x^4 + 85*I*x^3 + 57*I*x^2 - 104*I*x + 4*I) - 26*x + 50)*(-x^3
+ 1)^(2/3) + 4*sqrt(3)*(sqrt(-3)*(-16)^(1/6)*(131*x^6 - 48*x^5 - 381*x^4 - 152*x^3 + 267*x^2 + 276*x - 112) +
(-16)^(1/6)*(131*x^6 - 48*x^5 - 381*x^4 - 152*x^3 + 267*x^2 + 276*x - 112)) + 3*(-x^3 + 1)^(1/3)*(8*(-2)^(1/3
)*sqrt(-3)*(50*x^5 - 93*x^4 - 88*x^3 - 7*x^2 + 150*x - 31) - sqrt(3)*(sqrt(-3)*(-16)^(5/6)*(4*x^5 + 69*x^4 - 5
8*x^3 - 77*x^2 + 12*x + 23) - (-16)^(5/6)*(4*x^5 + 69*x^4 - 58*x^3 - 77*x^2 + 12*x + 23)) - 8*(-2)^(1/3)*(50*x
^5 - 93*x^4 - 88*x^3 - 7*x^2 + 150*x - 31)))/(x^6 - 3*x^5 + 6*x^4 - 7*x^3 + 6*x^2 - 3*x + 1)) - sqrt(3)*(sqrt(
-3)*(-16)^(1/6)*(x^2 - x + 1) - (-16)^(1/6)*(x^2 - x + 1))*log(9*(12*2^(2/3)*sqrt(-3)*(15*x^6 - 200*x^5 + 5*x^
4 + 216*x^3 + 157*x^2 - 124*x - 42) - 12*2^(2/3)*(15*x^6 - 200*x^5 + 5*x^4 + 216*x^3 + 157*x^2 - 124*x - 42) +
48*(31*x^4 + 107*x^3 - 243*x^2 - sqrt(3)*(23*I*x^4 - 85*I*x^3 - 57*I*x^2 + 104*I*x - 4*I) - 26*x + 50)*(-x^3
+ 1)^(2/3) - 4*sqrt(3)*(sqrt(-3)*(-16)^(1/6)*(131*x^6 - 48*x^5 - 381*x^4 - 152*x^3 + 267*x^2 + 276*x - 112) -
(-16)^(1/6)*(131*x^6 - 48*x^5 - 381*x^4 - 152*x^3 + 267*x^2 + 276*x - 112)) + 3*(-x^3 + 1)^(1/3)*(8*(-2)^(1/3)
*sqrt(-3)*(50*x^5 - 93*x^4 - 88*x^3 - 7*x^2 + 150*x - 31) + sqrt(3)*(sqrt(-3)*(-16)^(5/6)*(4*x^5 + 69*x^4 - 58
*x^3 - 77*x^2 + 12*x + 23) + (-16)^(5/6)*(4*x^5 + 69*x^4 - 58*x^3 - 77*x^2 + 12*x + 23)) + 8*(-2)^(1/3)*(50*x^
5 - 93*x^4 - 88*x^3 - 7*x^2 + 150*x - 31)))/(x^6 - 3*x^5 + 6*x^4 - 7*x^3 + 6*x^2 - 3*x + 1)) + sqrt(3)*(sqrt(-
3)*(-16)^(1/6)*(x^2 - x + 1) - (-16)^(1/6)*(x^2 - x + 1))*log(9*(12*2^(2/3)*sqrt(-3)*(15*x^6 - 200*x^5 + 5*x^4
+ 216*x^3 + 157*x^2 - 124*x - 42) - 12*2^(2/3)*(15*x^6 - 200*x^5 + 5*x^4 + 216*x^3 + 157*x^2 - 124*x - 42) +
48*(31*x^4 + 107*x^3 - 243*x^2 - sqrt(3)*(-23*I*x^4 + 85*I*x^3 + 57*I*x^2 - 104*I*x + 4*I) - 26*x + 50)*(-x^3
+ 1)^(2/3) + 4*sqrt(3)*(sqrt(-3)*(-16)^(1/6)*(131*x^6 - 48*x^5 - 381*x^4 - 152*x^3 + 267*x^2 + 276*x - 112) -
(-16)^(1/6)*(131*x^6 - 48*x^5 - 381*x^4 - 152*x^3 + 267*x^2 + 276*x - 112)) + 3*(-x^3 + 1)^(1/3)*(8*(-2)^(1/3)
*sqrt(-3)*(50*x^5 - 93*x^4 - 88*x^3 - 7*x^2 + 150*x - 31) - sqrt(3)*(sqrt(-3)*(-16)^(5/6)*(4*x^5 + 69*x^4 - 58
*x^3 - 77*x^2 + 12*x + 23) + (-16)^(5/6)*(4*x^5 + 69*x^4 - 58*x^3 - 77*x^2 + 12*x + 23)) + 8*(-2)^(1/3)*(50*x^
5 - 93*x^4 - 88*x^3 - 7*x^2 + 150*x - 31)))/(x^6 - 3*x^5 + 6*x^4 - 7*x^3 + 6*x^2 - 3*x + 1)) + 12*(x^2 - x + 1
)*log(3*(-x^3 + 1)^(1/3)*x^2 + 3*(-x^3 + 1)^(2/3)*x + 1) + 36*(-x^3 + 1)^(2/3))/(x^2 - x + 1)

Sympy [F]

\[ \int \frac {(1-2 x) \left (1-x^3\right )^{2/3}}{\left (1-x+x^2\right )^2} \, dx=- \int \left (- \frac {\left (1 - x^{3}\right )^{\frac {2}{3}}}{x^{4} - 2 x^{3} + 3 x^{2} - 2 x + 1}\right )\, dx - \int \frac {2 x \left (1 - x^{3}\right )^{\frac {2}{3}}}{x^{4} - 2 x^{3} + 3 x^{2} - 2 x + 1}\, dx \]

[In]

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

[Out]

-Integral(-(1 - x**3)**(2/3)/(x**4 - 2*x**3 + 3*x**2 - 2*x + 1), x) - Integral(2*x*(1 - x**3)**(2/3)/(x**4 - 2
*x**3 + 3*x**2 - 2*x + 1), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(-((2*x - 1)*(1 - x^3)^(2/3))/(x^2 - x + 1)^2,x)

[Out]

-int(((2*x - 1)*(1 - x^3)^(2/3))/(x^2 - x + 1)^2, x)