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

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

[Out]

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

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Rubi [F]  time = 0.235482, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., 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*}

Mathematica [F]  time = 0.106666, size = 0, normalized size = 0. \[ \int \frac{\sqrt [3]{1-x^3}}{1-x+x^2} \, dx \]

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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Maple [F]  time = 0.086, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}-x+1}\sqrt [3]{-{x}^{3}+1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{x^{2} - x + 1}\,{d x} \end{align*}

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]  time = 51.3796, size = 10386, normalized size = 37.09 \begin{align*} \text{result too large to display} \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt [3]{- \left (x - 1\right ) \left (x^{2} + x + 1\right )}}{x^{2} - x + 1}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{x^{2} - x + 1}\,{d x} \end{align*}

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)