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

Optimal. Leaf size=199 \[ \frac {\left (1-x^3\right )^{2/3}}{1-x+x^2}-\frac {2 \tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {2^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {2^{2/3} \tan ^{-1}\left (\frac {1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{\sqrt [3]{2}}-\frac {\log \left (-\sqrt [3]{2} x-\sqrt [3]{1-x^3}\right )}{\sqrt [3]{2}}+\log \left (x+\sqrt [3]{1-x^3}\right ) \]

[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)

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Rubi [A]
time = 0.17, antiderivative size = 250, normalized size of antiderivative = 1.26, number of steps used = 14, number of rules used = 12, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {2183, 386, 384, 455, 43, 57, 631, 210, 31, 478, 544, 245} \begin {gather*} \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 )-\frac {2 \tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {2^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {2^{2/3} \tan ^{-1}\left (\frac {2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )}{\sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[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*} \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 )^{2/3}}{\left (1-x+x^2\right )^2}-\frac {2 x \left (1-x^3\right )^{2/3}}{\left (1-x+x^2\right )^2}\right ) \, dx\\ &=-\left (2 \int \frac {x \left (1-x^3\right )^{2/3}}{\left (1-x+x^2\right )^2} \, dx\right )+\int \frac {\left (1-x^3\right )^{2/3}}{\left (1-x+x^2\right )^2} \, dx\\ &=-\left (2 \int \left (-\frac {2 \left (1+i \sqrt {3}\right ) \left (1-x^3\right )^{2/3}}{3 \left (1+i \sqrt {3}-2 x\right )^2}+\frac {2 i \left (1-x^3\right )^{2/3}}{3 \sqrt {3} \left (1+i \sqrt {3}-2 x\right )}-\frac {2 \left (1-i \sqrt {3}\right ) \left (1-x^3\right )^{2/3}}{3 \left (-1+i \sqrt {3}+2 x\right )^2}+\frac {2 i \left (1-x^3\right )^{2/3}}{3 \sqrt {3} \left (-1+i \sqrt {3}+2 x\right )}\right ) \, dx\right )+\int \left (-\frac {4 \left (1-x^3\right )^{2/3}}{3 \left (1+i \sqrt {3}-2 x\right )^2}+\frac {4 i \left (1-x^3\right )^{2/3}}{3 \sqrt {3} \left (1+i \sqrt {3}-2 x\right )}-\frac {4 \left (1-x^3\right )^{2/3}}{3 \left (-1+i \sqrt {3}+2 x\right )^2}+\frac {4 i \left (1-x^3\right )^{2/3}}{3 \sqrt {3} \left (-1+i \sqrt {3}+2 x\right )}\right ) \, dx\\ &=-\left (\frac {4}{3} \int \frac {\left (1-x^3\right )^{2/3}}{\left (1+i \sqrt {3}-2 x\right )^2} \, dx\right )-\frac {4}{3} \int \frac {\left (1-x^3\right )^{2/3}}{\left (-1+i \sqrt {3}+2 x\right )^2} \, dx+\frac {1}{3} \left (4 \left (1-i \sqrt {3}\right )\right ) \int \frac {\left (1-x^3\right )^{2/3}}{\left (-1+i \sqrt {3}+2 x\right )^2} \, dx+\frac {1}{3} \left (4 \left (1+i \sqrt {3}\right )\right ) \int \frac {\left (1-x^3\right )^{2/3}}{\left (1+i \sqrt {3}-2 x\right )^2} \, dx\\ \end {align*}

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Mathematica [F]
time = 12.06, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(1-2 x) \left (1-x^3\right )^{2/3}}{\left (1-x+x^2\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[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]

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Mathics [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Timed out

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 24.62, size = 457, normalized size = 2.30

method	result	size
trager	\(\frac {\left (-x^{3}+1\right )^{\frac {2}{3}}}{x^{2}-x +1}+\frac {\ln \left (-\frac {\RootOf \left (\textit {\_Z}^{6}+432\right )^{4} x^{2}+\RootOf \left (\textit {\_Z}^{6}+432\right )^{4} x -\RootOf \left (\textit {\_Z}^{6}+432\right )^{4}+12 \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 ) \RootOf \left (\textit {\_Z}^{6}+432\right )^{4}}{72}+\frac {\ln \left (-\frac {\RootOf \left (\textit {\_Z}^{6}+432\right )^{4} x^{2}+\RootOf \left (\textit {\_Z}^{6}+432\right )^{4} x -\RootOf \left (\textit {\_Z}^{6}+432\right )^{4}+12 \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 ) \RootOf \left (\textit {\_Z}^{6}+432\right )}{6}+\frac {\RootOf \left (\textit {\_Z}^{6}+432\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{6}+432\right )^{5} \left (-x^{3}+1\right )^{\frac {1}{3}} x -\RootOf \left (\textit {\_Z}^{6}+432\right )^{4} x^{2}-\RootOf \left (\textit {\_Z}^{6}+432\right )^{4} x +\RootOf \left (\textit {\_Z}^{6}+432\right )^{4}-12 \RootOf \left (\textit {\_Z}^{6}+432\right )^{2} \left (-x^{3}+1\right )^{\frac {1}{3}} x +36 \RootOf \left (\textit {\_Z}^{6}+432\right ) x^{2}+36 \RootOf \left (\textit {\_Z}^{6}+432\right ) x +144 \left (-x^{3}+1\right )^{\frac {2}{3}}-36 \RootOf \left (\textit {\_Z}^{6}+432\right )}{x^{2}-x +1}\right )}{3}-\frac {\ln \left (\left (-x^{3}+1\right )^{\frac {2}{3}}-x \left (-x^{3}+1\right )^{\frac {1}{3}}+x^{2}\right ) \RootOf \left (\textit {\_Z}^{6}+432\right )^{3}}{12}-\ln \left (\left (-x^{3}+1\right )^{\frac {2}{3}}-x \left (-x^{3}+1\right )^{\frac {1}{3}}+x^{2}\right )+\frac {\RootOf \left (\textit {\_Z}^{6}+432\right )^{3} \ln \left (-\RootOf \left (\textit {\_Z}^{6}+432\right )^{6} x^{3}+24 \RootOf \left (\textit {\_Z}^{6}+432\right )^{3}+1728 x \left (-x^{3}+1\right )^{\frac {2}{3}}-1728 x^{2} \left (-x^{3}+1\right )^{\frac {1}{3}}+1296 x^{3}-864\right )}{18}\)	\(457\)
risch	\(\text {Expression too large to display}\)	\(1033\)

Verification of antiderivative is not currently implemented for this CAS.

[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)+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+Ro
otOf(_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))*Roo
tOf(_Z^6+432)+1/3*RootOf(_Z^6+432)*ln(-(RootOf(_Z^6+432)^5*(-x^3+1)^(1/3)*x-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+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))-1/12*ln((-x^3+1)^(2/3)-x*(-x^3+1)^(1/3)+x^2)*RootOf(_Z^
6+432)^3-ln((-x^3+1)^(2/3)-x*(-x^3+1)^(1/3)+x^2)+1/18*RootOf(_Z^6+432)^3*ln(-RootOf(_Z^6+432)^6*x^3+24*RootOf(
_Z^6+432)^3+1728*x*(-x^3+1)^(2/3)-1728*x^2*(-x^3+1)^(1/3)+1296*x^3-864)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1827 vs. \(2 (163) = 326\).
time = 1.34, size = 1827, normalized size = 9.18

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/72*(8*4^(1/3)*sqrt(3)*(x^2 - x + 1)*arctan(-1/6*(3822*4^(2/3)*sqrt(3)*(50*x^4 - 74*x^3 - 207*x^2 + 143*x +
19)*(-x^3 + 1)^(2/3) + 7644*4^(1/3)*sqrt(3)*(19*x^5 - 150*x^4 + 43*x^3 + 112*x^2 + 57*x - 50)*(-x^3 + 1)^(1/3)
- 7*sqrt(39)*(6*4^(1/3)*sqrt(3)*(1150*x^4 - 3974*x^3 - 1911*x^2 + 1522*x + 3898)*(-x^3 + 1)^(2/3) - 4^(2/3)*s
qrt(3)*(1778*x^6 - 6366*x^5 - 8412*x^4 + 17254*x^3 + 15117*x^2 - 4227*x - 16105) + 12*sqrt(3)*(437*x^5 - 1539*
x^4 - 333*x^3 - 2074*x^2 + 372*x + 3261)*(-x^3 + 1)^(1/3))*sqrt((6*4^(1/3)*(5*x^4 + 4*x^3 - 3*x^2 - 4*x + 1)*(
-x^3 + 1)^(2/3) + 4^(2/3)*(19*x^6 + 15*x^5 - 12*x^4 - 25*x^3 - 12*x^2 + 15*x + 1) - 12*(4*x^5 + 3*x^4 - 2*x^3
- 5*x^2 + 1)*(-x^3 + 1)^(1/3))/(x^6 - 3*x^5 + 6*x^4 - 7*x^3 + 6*x^2 - 3*x + 1)) + 6*sqrt(3)*(29494*x^6 - 17582
*x^5 + 153824*x^4 - 266248*x^3 - 129950*x^2 + 238106*x - 29747))/(138718*x^6 - 463746*x^5 - 296508*x^4 - 11507
2*x^3 + 1093704*x^2 - 70446*x - 256859)) + 8*4^(1/3)*sqrt(3)*(x^2 - x + 1)*arctan(1/6*(3822*4^(2/3)*sqrt(3)*(1
9*x^4 - 181*x^3 + 36*x^2 + 169*x - 31)*(-x^3 + 1)^(2/3) - 7644*4^(1/3)*sqrt(3)*(31*x^5 + 57*x^4 - 131*x^3 - 11
9*x^2 + 93*x + 19)*(-x^3 + 1)^(1/3) + 7*sqrt(39)*(6*4^(1/3)*sqrt(3)*(3385*x^4 + 3574*x^3 - 1911*x^2 - 2948*x +
124)*(-x^3 + 1)^(2/3) + 4^(2/3)*sqrt(3)*(13027*x^6 + 16539*x^5 - 8961*x^4 - 32644*x^3 - 2361*x^2 + 17139*x -
239) - 12*sqrt(3)*(2748*x^5 + 3450*x^4 - 4126*x^3 - 2385*x^2 + 1539*x - 76)*(-x^3 + 1)^(1/3))*sqrt((6*4^(1/3)*
(x^4 - 4*x^3 - 3*x^2 + 4*x + 5)*(-x^3 + 1)^(2/3) + 4^(2/3)*(x^6 + 15*x^5 - 12*x^4 - 25*x^3 - 12*x^2 + 15*x + 1
9) + 12*(x^5 - 5*x^3 - 2*x^2 + 3*x + 4)*(-x^3 + 1)^(1/3))/(x^6 - 3*x^5 + 6*x^4 - 7*x^3 + 6*x^2 - 3*x + 1)) + 6
*sqrt(3)*(53953*x^6 - 12994*x^5 - 396521*x^4 + 169424*x^3 + 300029*x^2 - 62294*x - 41597))/(52723*x^6 + 682854
*x^5 - 325173*x^4 - 1353400*x^3 + 193623*x^2 + 640446*x - 16073)) + 16*4^(1/3)*sqrt(3)*(x^2 - x + 1)*arctan(1/
6*(7644*4^(2/3)*sqrt(3)*(5*x^4 - 107*x^3 - 243*x^2 + 26*x + 157)*(-x^3 + 1)^(2/3) - 7644*4^(1/3)*sqrt(3)*(307*
x^5 + 300*x^4 - 140*x^3 - 221*x^2 - 186*x - 98)*(-x^3 + 1)^(1/3) + 7*sqrt(39)*4^(1/3)*(6*4^(1/3)*sqrt(3)*(3109
*x^4 + 400*x^3 - 3822*x^2 + 1426*x + 3622)*(-x^3 + 1)^(2/3) + 4^(2/3)*sqrt(3)*(15505*x^6 + 11493*x^5 - 22383*x
^4 - 22720*x^3 - 5454*x^2 + 13032*x + 10888) - 12*sqrt(3)*(2111*x^5 + 3450*x^4 - 941*x^3 - 1111*x^2 - 372*x -
2624)*(-x^3 + 1)^(1/3)) + 6*sqrt(3)*(307479*x^6 + 239258*x^5 - 543668*x^4 - 607716*x^3 + 19112*x^2 + 232000*x
+ 343788))/(933353*x^6 + 1472754*x^5 + 285042*x^4 - 1008596*x^3 - 1598208*x^2 - 560184*x + 468980)) + 48*sqrt(
3)*(x^2 - x + 1)*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)) - 3*4^(1/3)*(x^2 - x + 1)*log(39626496*(6*4^(1/3)*(5*x^4 + 4*x^3 - 3*x^2 - 4*x + 1)*(-x^3 + 1)^(2/3
) + 4^(2/3)*(19*x^6 + 15*x^5 - 12*x^4 - 25*x^3 - 12*x^2 + 15*x + 1) - 12*(4*x^5 + 3*x^4 - 2*x^3 - 5*x^2 + 1)*(
-x^3 + 1)^(1/3))/(x^6 - 3*x^5 + 6*x^4 - 7*x^3 + 6*x^2 - 3*x + 1)) - 3*4^(1/3)*(x^2 - x + 1)*log(9906624*(6*4^(
1/3)*(5*x^4 + 4*x^3 - 3*x^2 - 4*x + 1)*(-x^3 + 1)^(2/3) + 4^(2/3)*(19*x^6 + 15*x^5 - 12*x^4 - 25*x^3 - 12*x^2
+ 15*x + 1) - 12*(4*x^5 + 3*x^4 - 2*x^3 - 5*x^2 + 1)*(-x^3 + 1)^(1/3))/(x^6 - 3*x^5 + 6*x^4 - 7*x^3 + 6*x^2 -
3*x + 1)) + 3*4^(1/3)*(x^2 - x + 1)*log(39626496*(6*4^(1/3)*(x^4 - 4*x^3 - 3*x^2 + 4*x + 5)*(-x^3 + 1)^(2/3) +
4^(2/3)*(x^6 + 15*x^5 - 12*x^4 - 25*x^3 - 12*x^2 + 15*x + 19) + 12*(x^5 - 5*x^3 - 2*x^2 + 3*x + 4)*(-x^3 + 1)
^(1/3))/(x^6 - 3*x^5 + 6*x^4 - 7*x^3 + 6*x^2 - 3*x + 1)) + 3*4^(1/3)*(x^2 - x + 1)*log(9906624*(6*4^(1/3)*(x^4
- 4*x^3 - 3*x^2 + 4*x + 5)*(-x^3 + 1)^(2/3) + 4^(2/3)*(x^6 + 15*x^5 - 12*x^4 - 25*x^3 - 12*x^2 + 15*x + 19) +
12*(x^5 - 5*x^3 - 2*x^2 + 3*x + 4)*(-x^3 + 1)^(1/3))/(x^6 - 3*x^5 + 6*x^4 - 7*x^3 + 6*x^2 - 3*x + 1)) - 24*(x
^2 - x + 1)*log(3*(-x^3 + 1)^(1/3)*x^2 + 3*(-x^3 + 1)^(2/3)*x + 1) - 72*(-x^3 + 1)^(2/3))/(x^2 - x + 1)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F] N/A
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Could not integrate

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {\left (2\,x-1\right )\,{\left (1-x^3\right )}^{2/3}}{{\left (x^2-x+1\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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