∫ 3 √ ___ 1−x3

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

Optimal. Leaf size=280 \[ \frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}+\frac {3 \log \left (\sqrt [3]{1-x^3}-\sqrt [3]{2} (x-1)\right )}{2\ 2^{2/3}}+\frac {1}{2} \log \left (\sqrt [3]{1-x^3}+x\right )-\frac {\log \left (\sqrt [3]{1-x^3}+\sqrt [3]{2} x\right )}{2\ 2^{2/3}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{2} (x-1)}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )}{2^{2/3}}+\frac {\tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}-\frac {\tan ^{-1}\left (\frac {2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}-\frac {\log \left (-3 (x-1) \left (x^2-x+1\right )\right )}{2\ 2^{2/3}} \]

[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 [F] time = 0.24, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\sqrt [3]{1-x^3}}{1-x+x^2} \, dx \]

Veriﬁcation is Not applicable to the result.

[In]

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

[Out]

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

I*Sqrt[3] + 2*x), x])/Sqrt[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 = 0.11, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt [3]{1-x^3}}{1-x+x^2} \, dx \]

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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fricas [B] time = 16.51, size = 3085, normalized size = 11.02 \[ \text {result too large to display} \]

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 - 6*x + 1)) - 3*sqrt(3)*(67113679084*x^12 - 61534090748*x^11 - 10

06807736260*x^10 + 1996201310444*x^9 + 193806523788*x^8 - 2673973669800*x^7 + 775957356356*x^6 + 2110159119756

*x^5 - 1821028473882*x^4 + 377014646048*x^3 + 67410900094*x^2 - 19835743048*x - 1369553867))/(168032067092*x^1

2 - 2318893136652*x^11 + 4401905935020*x^10 + 1550444734940*x^9 - 6210007783092*x^8 - 1634341806144*x^7 + 6341

768478444*x^6 - 948091553244*x^5 - 2281774840272*x^4 + 1036207535072*x^3 - 59480228082*x^2 - 20085678624*x - 7

61048497)) + 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)) + 1/48*2^(1/3)*log(7717175424*(6*2^(2/3)*(143*x^10 - 177*x^9 - 2*x^8 - 54*x^7 + 141*x^6 - 3

1*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 + 1

962*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 - 6*x + 1)) + 1/48*2^(1/3)*log(19292

93856*(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 - 6*x + 1)) - 1/48*2^(1/3)*log(7717175424*(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 - 18*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)) - 1/48*2^(1/3

)*log(1929293856*(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 - 18*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)) + 1/6*log(3*(-x^3 + 1)^(1/3)*x^2 + 3*(-x^3 + 1)^(2/3)*x + 1)

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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x^{2} - x + 1}\,{d x} \]

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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maple [C] time = 33.31, size = 1404, normalized size = 5.01 \[ \text {Expression too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

/3)*RootOf(_Z^2+_Z+1)*x^2+4*x^3*RootOf(_Z^2+_Z+1)+3*x*(-x^3+1)^(2/3)-3*x^2*(-x^3+1)^(1/3)+4*x^3-RootOf(_Z^2+_Z

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

RootOf(_Z^2+_Z+1)*x^2+4*x^3*RootOf(_Z^2+_Z+1)+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(-(36*RootOf(_Z^3+324*RootOf(_Z^2+_Z+1)+162)*RootOf(_Z^2+_Z+1)*(-x^3+1)^(1/3)*x^3-5*RootOf(_Z^3+324*

RootOf(_Z^2+_Z+1)+162)^2*x^4-12*RootOf(_Z^3+324*RootOf(_Z^2+_Z+1)+162)*RootOf(_Z^2+_Z+1)*(-x^3+1)^(1/3)*x^2+2*

RootOf(_Z^3+324*RootOf(_Z^2+_Z+1)+162)^2*x^3+18*RootOf(_Z^3+324*RootOf(_Z^2+_Z+1)+162)*(-x^3+1)^(1/3)*x^3-12*R

ootOf(_Z^3+324*RootOf(_Z^2+_Z+1)+162)*RootOf(_Z^2+_Z+1)*(-x^3+1)^(1/3)*x+RootOf(_Z^3+324*RootOf(_Z^2+_Z+1)+162

)^2*x^2-6*RootOf(_Z^3+324*RootOf(_Z^2+_Z+1)+162)*(-x^3+1)^(1/3)*x^2+216*(-x^3+1)^(2/3)*x^2+2*RootOf(_Z^3+324*R

ootOf(_Z^2+_Z+1)+162)^2*x-6*RootOf(_Z^3+324*RootOf(_Z^2+_Z+1)+162)*(-x^3+1)^(1/3)*x-108*x*(-x^3+1)^(2/3)-RootO

f(_Z^3+324*RootOf(_Z^2+_Z+1)+162)^2)/(x^2-x+1)^2)*RootOf(_Z^2+_Z+1)*RootOf(_Z^3+324*RootOf(_Z^2+_Z+1)+162)-1/1

8*ln(-(36*RootOf(_Z^3+324*RootOf(_Z^2+_Z+1)+162)*RootOf(_Z^2+_Z+1)*(-x^3+1)^(1/3)*x^3-5*RootOf(_Z^3+324*RootOf

(_Z^2+_Z+1)+162)^2*x^4-12*RootOf(_Z^3+324*RootOf(_Z^2+_Z+1)+162)*RootOf(_Z^2+_Z+1)*(-x^3+1)^(1/3)*x^2+2*RootOf

(_Z^3+324*RootOf(_Z^2+_Z+1)+162)^2*x^3+18*RootOf(_Z^3+324*RootOf(_Z^2+_Z+1)+162)*(-x^3+1)^(1/3)*x^3-12*RootOf(

_Z^3+324*RootOf(_Z^2+_Z+1)+162)*RootOf(_Z^2+_Z+1)*(-x^3+1)^(1/3)*x+RootOf(_Z^3+324*RootOf(_Z^2+_Z+1)+162)^2*x^

2-6*RootOf(_Z^3+324*RootOf(_Z^2+_Z+1)+162)*(-x^3+1)^(1/3)*x^2+216*(-x^3+1)^(2/3)*x^2+2*RootOf(_Z^3+324*RootOf(

_Z^2+_Z+1)+162)^2*x-6*RootOf(_Z^3+324*RootOf(_Z^2+_Z+1)+162)*(-x^3+1)^(1/3)*x-108*x*(-x^3+1)^(2/3)-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)+1/18*RootOf(_Z^3+324*RootOf

(_Z^2+_Z+1)+162)*ln(-(RootOf(_Z^2+_Z+1)*RootOf(_Z^3+324*RootOf(_Z^2+_Z+1)+162)^2*x^4+2*RootOf(_Z^2+_Z+1)*RootO

f(_Z^3+324*RootOf(_Z^2+_Z+1)+162)^2*x^3+6*RootOf(_Z^3+324*RootOf(_Z^2+_Z+1)+162)*RootOf(_Z^2+_Z+1)*(-x^3+1)^(1

/3)*x^3-RootOf(_Z^2+_Z+1)*RootOf(_Z^3+324*RootOf(_Z^2+_Z+1)+162)^2*x^2-18*RootOf(_Z^3+324*RootOf(_Z^2+_Z+1)+16

2)*RootOf(_Z^2+_Z+1)*(-x^3+1)^(1/3)*x^2-6*RootOf(_Z^3+324*RootOf(_Z^2+_Z+1)+162)*(-x^3+1)^(1/3)*x^3-2*RootOf(_

Z^2+_Z+1)*RootOf(_Z^3+324*RootOf(_Z^2+_Z+1)+162)^2*x+6*RootOf(_Z^3+324*RootOf(_Z^2+_Z+1)+162)*RootOf(_Z^2+_Z+1

)*(-x^3+1)^(1/3)*x+18*RootOf(_Z^3+324*RootOf(_Z^2+_Z+1)+162)*(-x^3+1)^(1/3)*x^2+108*(-x^3+1)^(2/3)*x^2+RootOf(

_Z^2+_Z+1)*RootOf(_Z^3+324*RootOf(_Z^2+_Z+1)+162)^2-6*RootOf(_Z^3+324*RootOf(_Z^2+_Z+1)+162)*(-x^3+1)^(1/3)*x-

108*x*(-x^3+1)^(2/3))/(x^2-x+1)^2)

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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x^{2} - x + 1}\,{d x} \]

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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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (1-x^3\right )}^{1/3}}{x^2-x+1} \,d x \]

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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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt [3]{- \left (x - 1\right ) \left (x^{2} + x + 1\right )}}{x^{2} - x + 1}\, dx \]

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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