3.1.59 \(\int \frac {\sqrt [3]{1-x^3}}{1-x+x^2} \, dx\) [59]
Optimal. Leaf size=280 \[ \frac {\sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{2} (-1+x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2^{2/3}}+\frac {\tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}-\frac {\tan ^{-1}\left (\frac {1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}-\frac {\log \left (-3 (-1+x) \left (1-x+x^2\right )\right )}{2\ 2^{2/3}}+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}+\frac {3 \log \left (-\sqrt [3]{2} (-1+x)+\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}+\frac {1}{2} \log \left (x+\sqrt [3]{1-x^3}\right )-\frac {\log \left (\sqrt [3]{2} x+\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}} \]
[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)
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Rubi [A]
time = 0.15, antiderivative size = 408, normalized size of antiderivative = 1.46, number of steps
used = 19, number of rules used = 12, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.546, Rules used = {2183, 420,
493, 298, 31, 648, 631, 210, 642, 495, 337, 503} \begin {gather*} \frac {\sqrt [3]{2} \text {ArcTan}\left (\frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\text {ArcTan}\left (\frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}+\frac {\text {ArcTan}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\sqrt [3]{2} \text {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}} \end {gather*}
Antiderivative was successfully verified.
[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*} \int \frac {\sqrt [3]{1-x^3}}{1-x+x^2} \, dx &=\int \left (\frac {2 i \sqrt [3]{1-x^3}}{\sqrt {3} \left (1+i \sqrt {3}-2 x\right )}+\frac {2 i \sqrt [3]{1-x^3}}{\sqrt {3} \left (-1+i \sqrt {3}+2 x\right )}\right ) \, dx\\ &=\frac {(2 i) \int \frac {\sqrt [3]{1-x^3}}{1+i \sqrt {3}-2 x} \, dx}{\sqrt {3}}+\frac {(2 i) \int \frac {\sqrt [3]{1-x^3}}{-1+i \sqrt {3}+2 x} \, dx}{\sqrt {3}}\\ \end {align*}
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Mathematica [F]
time = 9.23, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{1-x^3}}{1-x+x^2} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Integrate[(1 - x^3)^(1/3)/(1 - x + x^2),x]
[Out]
Integrate[(1 - x^3)^(1/3)/(1 - x + x^2), x]
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 7.06, size = 925, normalized size = 3.30
| | |
| method |
result |
size |
| | |
| trager |
\(\text {Expression too large to display}\) |
\(925\) |
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|
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int((-x^3+1)^(1/3)/(x^2-x+1),x,method=_RETURNVERBOSE)
[Out]
1/3*ln(-RootOf(_Z^6+108)^6*x^3+18*RootOf(_Z^6+108)^3*(-x^3+1)^(2/3)*x-18*RootOf(_Z^6+108)^3*x^3+108*x*(-x^3+1)
^(2/3)+216*x^2*(-x^3+1)^(1/3)+12*RootOf(_Z^6+108)^3)-1/6*RootOf(_Z^6+108)*ln(-(RootOf(_Z^6+108)^5*x^4-2*RootOf
(_Z^6+108)^4*(-x^3+1)^(1/3)*x^3+2*RootOf(_Z^6+108)^5*x^3+6*RootOf(_Z^6+108)^4*(-x^3+1)^(1/3)*x^2-x^2*RootOf(_Z
^6+108)^5-2*RootOf(_Z^6+108)^4*(-x^3+1)^(1/3)*x-2*RootOf(_Z^6+108)^5*x-6*RootOf(_Z^6+108)^2*x^4+36*RootOf(_Z^6
+108)*(-x^3+1)^(1/3)*x^3+RootOf(_Z^6+108)^5-12*RootOf(_Z^6+108)^2*x^3+144*(-x^3+1)^(2/3)*x^2-108*RootOf(_Z^6+1
08)*(-x^3+1)^(1/3)*x^2+6*x^2*RootOf(_Z^6+108)^2-144*x*(-x^3+1)^(2/3)+36*RootOf(_Z^6+108)*(-x^3+1)^(1/3)*x+12*R
ootOf(_Z^6+108)^2*x-6*RootOf(_Z^6+108)^2)/(x^2-x+1)^2)+1/72*ln(-(-3*RootOf(_Z^6+108)^4*(-x^3+1)^(1/3)*x^3+Root
Of(_Z^6+108)^4*(-x^3+1)^(1/3)*x^2+RootOf(_Z^6+108)^4*(-x^3+1)^(1/3)*x-15*RootOf(_Z^6+108)^2*x^4+6*RootOf(_Z^6+
108)^2*x^3+72*(-x^3+1)^(2/3)*x^2+3*x^2*RootOf(_Z^6+108)^2-36*x*(-x^3+1)^(2/3)+6*RootOf(_Z^6+108)^2*x-3*RootOf(
_Z^6+108)^2)/(x^2-x+1)^2)*RootOf(_Z^6+108)^4+1/12*ln(-(-3*RootOf(_Z^6+108)^4*(-x^3+1)^(1/3)*x^3+RootOf(_Z^6+10
8)^4*(-x^3+1)^(1/3)*x^2+RootOf(_Z^6+108)^4*(-x^3+1)^(1/3)*x-15*RootOf(_Z^6+108)^2*x^4+6*RootOf(_Z^6+108)^2*x^3
+72*(-x^3+1)^(2/3)*x^2+3*x^2*RootOf(_Z^6+108)^2-36*x*(-x^3+1)^(2/3)+6*RootOf(_Z^6+108)^2*x-3*RootOf(_Z^6+108)^
2)/(x^2-x+1)^2)*RootOf(_Z^6+108)-1/36*ln(RootOf(_Z^6+108)^6*x^3-36*RootOf(_Z^6+108)^3*(-x^3+1)^(2/3)*x-36*(-x^
3+1)^(1/3)*RootOf(_Z^6+108)^3*x^2+216*x*(-x^3+1)^(2/3)-216*x^2*(-x^3+1)^(1/3)+12*RootOf(_Z^6+108)^3-324*x^3+21
6)*RootOf(_Z^6+108)^3-1/6*ln(RootOf(_Z^6+108)^6*x^3-36*RootOf(_Z^6+108)^3*(-x^3+1)^(2/3)*x-36*(-x^3+1)^(1/3)*R
ootOf(_Z^6+108)^3*x^2+216*x*(-x^3+1)^(2/3)-216*x^2*(-x^3+1)^(1/3)+12*RootOf(_Z^6+108)^3-324*x^3+216)
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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((-x^3+1)^(1/3)/(x^2-x+1),x, algorithm="maxima")
[Out]
integrate((-x^3 + 1)^(1/3)/(x^2 - x + 1), x)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3085 vs.
\(2 (218) = 436\).
time = 5.42, size = 3085, normalized size = 11.02 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-x^3+1)^(1/3)/(x^2-x+1),x, algorithm="fricas")
[Out]
-1/9*sqrt(3)*2^(1/3)*arctan(1/3*(26795748*sqrt(3)*2^(2/3)*(586745*x^11 - 706109*x^10 - 191742*x^9 - 43779*x^8
+ 396304*x^7 + 323715*x^6 - 462255*x^5 + 73568*x^4 + 24102*x^3 + 2372*x^2 - 2008*x)*(-x^3 + 1)^(1/3) + 2679574
8*sqrt(3)*2^(1/3)*(340975*x^10 + 46080*x^9 - 970873*x^8 + 685704*x^7 - 289743*x^6 + 397966*x^5 - 203166*x^4 -
21912*x^3 + 29756*x^2 - 4016*x)*(-x^3 + 1)^(2/3) + 7*sqrt(273426)*2^(1/6)*(6*sqrt(3)*2^(2/3)*(338078915*x^10 -
459916473*x^9 - 111133574*x^8 + 235674676*x^7 + 297312537*x^6 - 494815414*x^5 + 244815194*x^4 - 34383000*x^3
- 8933924*x^2 + 2566224*x)*(-x^3 + 1)^(2/3) + sqrt(3)*2^(1/3)*(2332175065*x^12 - 3283524318*x^11 + 1882024851*
x^10 - 3919300970*x^9 + 2796090405*x^8 + 610770276*x^7 + 98233512*x^6 + 140867400*x^5 - 1145424564*x^4 + 43098
7096*x^3 + 108889824*x^2 - 54987072*x + 4032064) - 6*sqrt(3)*(493920245*x^11 - 452201839*x^10 - 276972599*x^9
- 661557480*x^8 + 1375964914*x^7 - 191435014*x^6 - 333786162*x^5 - 47180632*x^4 + 107411572*x^3 - 13096840*x^2
- 2566224*x)*(-x^3 + 1)^(1/3)) - 3*sqrt(3)*(2247079524645*x^12 - 5276442179264*x^11 + 3816306322874*x^10 - 32
80399521884*x^9 + 6278089258290*x^8 - 6181108351032*x^7 + 2698150339136*x^6 + 1210170331680*x^5 - 255854124396
0*x^4 + 1136906331664*x^3 - 42652634816*x^2 - 54080708992*x + 5152977792))/(18230538112975*x^12 - 141157161884
40*x^11 - 20854883745366*x^10 + 1856205891292*x^9 + 11854156958820*x^8 + 23868971173080*x^7 - 27900743059560*x
^6 + 8785124358048*x^5 - 2880050871456*x^4 + 1047429829408*x^3 + 242964112512*x^2 - 141331907328*x + 809638451
2)) + 1/18*sqrt(3)*2^(1/3)*arctan(-1/3*(13397874*sqrt(3)*2^(2/3)*(18803*x^11 - 25367*x^10 - 203754*x^9 + 40802
1*x^8 - 139829*x^7 + 7128*x^6 - 233871*x^5 + 225275*x^4 - 47094*x^3 - 10225*x^2 + 2921*x)*(-x^3 + 1)^(1/3) + 2
6795748*sqrt(3)*2^(1/3)*(10589*x^10 - 73935*x^9 + 63883*x^8 + 142959*x^7 - 173613*x^6 - 31588*x^5 + 79410*x^4
- 4377*x^3 - 13328*x^2 + 2921*x)*(-x^3 + 1)^(2/3) - 7*sqrt(273426)*(6*sqrt(3)*2^(2/3)*(309683372*x^10 - 328552
599*x^9 - 24698630*x^8 - 422031122*x^7 + 702164163*x^6 - 95703451*x^5 - 206316094*x^4 + 60985482*x^3 + 1116781
6*x^2 - 3733038*x)*(-x^3 + 1)^(2/3) + sqrt(3)*2^(1/3)*(2345654785*x^12 - 2502234618*x^11 - 252041853*x^10 - 44
16416426*x^9 + 6899968311*x^8 - 1680852528*x^7 + 1576960038*x^6 - 2990585436*x^5 + 642930363*x^4 + 528479914*x
^3 - 117963261*x^2 - 38399466*x + 8532241) - 6*sqrt(3)*(491687266*x^11 - 516958230*x^10 - 69305552*x^9 - 80893
4094*x^8 + 1418391515*x^7 - 385704187*x^6 - 112721241*x^5 - 69510422*x^4 + 47121139*x^3 + 11465929*x^2 - 47992
03*x)*(-x^3 + 1)^(1/3))*sqrt((6*2^(2/3)*(4*x^10 - 27*x^9 + 32*x^8 + 6*x^7 + 12*x^6 - 65*x^5 + 48*x^4 - 6*x^3 -
4*x^2 + x)*(-x^3 + 1)^(2/3) - 2^(1/3)*(35*x^12 - 66*x^11 - 201*x^10 + 338*x^9 + 90*x^8 - 90*x^7 - 249*x^6 - 1
8*x^5 + 306*x^4 - 166*x^3 + 15*x^2 + 6*x - 1) - 6*(x^11 + 29*x^10 - 93*x^9 + 66*x^8 - 19*x^7 + 87*x^6 - 99*x^5
+ 10*x^4 + 27*x^3 - 11*x^2 + x)*(-x^3 + 1)^(1/3))/(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)) - 3*sqrt(3)*(2995162579*x^12 + 315959718008*x^11 - 849682
072424*x^10 + 177300060912*x^9 - 508006765899*x^8 + 3583876884636*x^7 - 3031033916540*x^6 - 1410763301208*x^5
+ 2375077456341*x^4 - 546587071308*x^3 - 175036021936*x^2 + 63861157012*x - 3114267965))/(367648430113*x^12 -
1408582980384*x^11 - 1269375810828*x^10 + 5714713216048*x^9 - 1087485936795*x^8 - 126379999188*x^7 - 103196508
60540*x^6 + 10854292018608*x^5 - 1383220291365*x^4 - 1828745373668*x^3 + 426327416076*x^2 + 93479232396*x - 24
922675961)) - 1/18*sqrt(3)*2^(1/3)*arctan(1/3*(13397874*sqrt(3)*2^(2/3)*(17344*x^11 - 120304*x^10 + 110610*x^9
+ 203214*x^8 - 213415*x^7 - 96387*x^6 + 30102*x^5 + 157561*x^4 - 101868*x^3 + 15151*x^2 + 913*x)*(-x^3 + 1)^(
1/3) - 26795748*sqrt(3)*2^(1/3)*(1277*x^10 + 57510*x^9 - 189677*x^8 + 108972*x^7 + 102426*x^6 - 47461*x^5 - 82
155*x^4 + 56409*x^3 - 7301*x^2 - 913*x)*(-x^3 + 1)^(2/3) + 7*sqrt(273426)*(6*sqrt(3)*2^(2/3)*(8733539*x^10 - 1
22586360*x^9 + 269810944*x^8 - 28009538*x^7 - 316185126*x^6 + 161786897*x^5 + 95479640*x^4 - 80193978*x^3 + 11
163982*x^2 + 1166814*x)*(-x^3 + 1)^(2/3) - sqrt(3)*2^(1/3)*(1971824*x^12 - 78264612*x^11 + 705529692*x^10 - 15
56393152*x^9 + 933849120*x^8 + 135726408*x^7 - 213906684*x^6 + 446158968*x^5 - 582881445*x^4 + 182390318*x^3 +
31120185*x^2 - 12999294*x - 833569) + 6*sqrt(3)*(12965988*x^11 - 175265260*x^10 + 270273662*x^9 + 299814882*x
^8 - 663644613*x^7 + 77553085*x^6 + 286893603*x^5 - 82332150*x^4 - 33723265*x^3 + 10863861*x^2 + 333245*x)*(-x
^3 + 1)^(1/3))*sqrt((6*2^(2/3)*(143*x^10 - 177*x^9 - 2*x^8 - 54*x^7 + 141*x^6 - 31*x^5 - 18*x^4 - 6*x^3 + 7*x^
2 - x)*(-x^3 + 1)^(2/3) + 2^(1/3)*(1081*x^12 - 1338*x^11 - 15*x^10 - 1130*x^9 + 1962*x^8 - 234*x^7 + 33*x^6 -
630*x^5 + 234*x^4 + 58*x^3 - 15*x^2 - 6*x + 1) - 6*(227*x^11 - 281*x^10 - 3*x^9 - 162*x^8 + 319*x^7 - 51*x^6 -
21*x^5 - 58*x^4 + 33*x^3 - x^2 - x)*(-x^3 + 1)^(1/3))/(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 - ...
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{- \left (x - 1\right ) \left (x^{2} + x + 1\right )}}{x^{2} - x + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
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Giac [F]
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((-x^3+1)^(1/3)/(x^2-x+1),x, algorithm="giac")
[Out]
integrate((-x^3 + 1)^(1/3)/(x^2 - x + 1), x)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (1-x^3\right )}^{1/3}}{x^2-x+1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((1 - x^3)^(1/3)/(x^2 - x + 1),x)
[Out]
int((1 - x^3)^(1/3)/(x^2 - x + 1), x)
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