3.9.9 \(\int \frac {(1+x^6) (-1+x^3+x^6)^{2/3}}{1-x^6+x^{12}} \, dx\)

Optimal. Leaf size=61 \[ \frac {1}{6} \text {RootSum}\left [\text {$\#$1}^6-2 \text {$\#$1}^3+2\& ,\frac {\text {$\#$1}^2 \log \left (\sqrt [3]{x^6+x^3-1}-\text {$\#$1} x\right )-\text {$\#$1}^2 \log (x)}{\text {$\#$1}^3-1}\& \right ] \]

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Rubi [F]  time = 0.41, 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 = {} \begin {gather*} \int \frac {\left (1+x^6\right ) \left (-1+x^3+x^6\right )^{2/3}}{1-x^6+x^{12}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((1 + x^6)*(-1 + x^3 + x^6)^(2/3))/(1 - x^6 + x^12),x]

[Out]

-1/2*((1 + I*Sqrt[3])*Defer[Int][(-1 + x^3 + x^6)^(2/3)/(Sqrt[1 - I*Sqrt[3]] - Sqrt[2]*x^3), x])/Sqrt[1 - I*Sq
rt[3]] - ((1 - I*Sqrt[3])*Defer[Int][(-1 + x^3 + x^6)^(2/3)/(Sqrt[1 + I*Sqrt[3]] - Sqrt[2]*x^3), x])/(2*Sqrt[1
+ I*Sqrt[3]]) - ((1 + I*Sqrt[3])*Defer[Int][(-1 + x^3 + x^6)^(2/3)/(Sqrt[1 - I*Sqrt[3]] + Sqrt[2]*x^3), x])/(
2*Sqrt[1 - I*Sqrt[3]]) - ((1 - I*Sqrt[3])*Defer[Int][(-1 + x^3 + x^6)^(2/3)/(Sqrt[1 + I*Sqrt[3]] + Sqrt[2]*x^3
), x])/(2*Sqrt[1 + I*Sqrt[3]])

Rubi steps

\begin {align*} \int \frac {\left (1+x^6\right ) \left (-1+x^3+x^6\right )^{2/3}}{1-x^6+x^{12}} \, dx &=\int \left (\frac {\left (1-i \sqrt {3}\right ) \left (-1+x^3+x^6\right )^{2/3}}{-1-i \sqrt {3}+2 x^6}+\frac {\left (1+i \sqrt {3}\right ) \left (-1+x^3+x^6\right )^{2/3}}{-1+i \sqrt {3}+2 x^6}\right ) \, dx\\ &=\left (1-i \sqrt {3}\right ) \int \frac {\left (-1+x^3+x^6\right )^{2/3}}{-1-i \sqrt {3}+2 x^6} \, dx+\left (1+i \sqrt {3}\right ) \int \frac {\left (-1+x^3+x^6\right )^{2/3}}{-1+i \sqrt {3}+2 x^6} \, dx\\ &=\left (1-i \sqrt {3}\right ) \int \left (\frac {\sqrt {1+i \sqrt {3}} \left (-1+x^3+x^6\right )^{2/3}}{2 \left (-1-i \sqrt {3}\right ) \left (\sqrt {1+i \sqrt {3}}-\sqrt {2} x^3\right )}+\frac {\sqrt {1+i \sqrt {3}} \left (-1+x^3+x^6\right )^{2/3}}{2 \left (-1-i \sqrt {3}\right ) \left (\sqrt {1+i \sqrt {3}}+\sqrt {2} x^3\right )}\right ) \, dx+\left (1+i \sqrt {3}\right ) \int \left (\frac {\sqrt {1-i \sqrt {3}} \left (-1+x^3+x^6\right )^{2/3}}{2 \left (-1+i \sqrt {3}\right ) \left (\sqrt {1-i \sqrt {3}}-\sqrt {2} x^3\right )}+\frac {\sqrt {1-i \sqrt {3}} \left (-1+x^3+x^6\right )^{2/3}}{2 \left (-1+i \sqrt {3}\right ) \left (\sqrt {1-i \sqrt {3}}+\sqrt {2} x^3\right )}\right ) \, dx\\ &=-\frac {\left (1-i \sqrt {3}\right ) \int \frac {\left (-1+x^3+x^6\right )^{2/3}}{\sqrt {1+i \sqrt {3}}-\sqrt {2} x^3} \, dx}{2 \sqrt {1+i \sqrt {3}}}-\frac {\left (1-i \sqrt {3}\right ) \int \frac {\left (-1+x^3+x^6\right )^{2/3}}{\sqrt {1+i \sqrt {3}}+\sqrt {2} x^3} \, dx}{2 \sqrt {1+i \sqrt {3}}}-\frac {\left (1+i \sqrt {3}\right ) \int \frac {\left (-1+x^3+x^6\right )^{2/3}}{\sqrt {1-i \sqrt {3}}-\sqrt {2} x^3} \, dx}{2 \sqrt {1-i \sqrt {3}}}-\frac {\left (1+i \sqrt {3}\right ) \int \frac {\left (-1+x^3+x^6\right )^{2/3}}{\sqrt {1-i \sqrt {3}}+\sqrt {2} x^3} \, dx}{2 \sqrt {1-i \sqrt {3}}}\\ \end {align*}

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Mathematica [F]  time = 0.49, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1+x^6\right ) \left (-1+x^3+x^6\right )^{2/3}}{1-x^6+x^{12}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((1 + x^6)*(-1 + x^3 + x^6)^(2/3))/(1 - x^6 + x^12),x]

[Out]

Integrate[((1 + x^6)*(-1 + x^3 + x^6)^(2/3))/(1 - x^6 + x^12), x]

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IntegrateAlgebraic [A]  time = 0.00, size = 61, normalized size = 1.00 \begin {gather*} \frac {1}{6} \text {RootSum}\left [2-2 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x) \text {$\#$1}^2+\log \left (\sqrt [3]{-1+x^3+x^6}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-1+\text {$\#$1}^3}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((1 + x^6)*(-1 + x^3 + x^6)^(2/3))/(1 - x^6 + x^12),x]

[Out]

RootSum[2 - 2*#1^3 + #1^6 & , (-(Log[x]*#1^2) + Log[(-1 + x^3 + x^6)^(1/3) - x*#1]*#1^2)/(-1 + #1^3) & ]/6

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fricas [B]  time = 33.07, size = 583, normalized size = 9.56 \begin {gather*} \frac {1}{18} \cdot 4^{\frac {1}{6}} \sqrt {3} \left (-1\right )^{\frac {1}{3}} \arctan \left (\frac {4^{\frac {1}{6}} {\left (6 \cdot 4^{\frac {2}{3}} \sqrt {3} \left (-1\right )^{\frac {2}{3}} {\left (x^{31} - 14 \, x^{28} - 75 \, x^{25} + 82 \, x^{22} + 293 \, x^{19} - 132 \, x^{16} - 293 \, x^{13} + 82 \, x^{10} + 75 \, x^{7} - 14 \, x^{4} - x\right )} {\left (x^{6} + x^{3} - 1\right )}^{\frac {2}{3}} - 12 \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} {\left (3 \, x^{32} + 49 \, x^{29} - 51 \, x^{26} - 344 \, x^{23} + 99 \, x^{20} + 609 \, x^{17} - 99 \, x^{14} - 344 \, x^{11} + 51 \, x^{8} + 49 \, x^{5} - 3 \, x^{2}\right )} {\left (x^{6} + x^{3} - 1\right )}^{\frac {1}{3}} + 4^{\frac {1}{3}} \sqrt {3} {\left (x^{36} + 54 \, x^{33} - 129 \, x^{30} - 846 \, x^{27} + 258 \, x^{24} + 2502 \, x^{21} - 169 \, x^{18} - 2502 \, x^{15} + 258 \, x^{12} + 846 \, x^{9} - 129 \, x^{6} - 54 \, x^{3} + 1\right )}\right )}}{6 \, {\left (x^{36} - 54 \, x^{33} - 489 \, x^{30} + 270 \, x^{27} + 2922 \, x^{24} - 54 \, x^{21} - 4921 \, x^{18} + 54 \, x^{15} + 2922 \, x^{12} - 270 \, x^{9} - 489 \, x^{6} + 54 \, x^{3} + 1\right )}}\right ) + \frac {1}{36} \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (\frac {3 \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{7} + 4 \, x^{4} - x\right )} {\left (x^{6} + x^{3} - 1\right )}^{\frac {2}{3}} - 4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{12} + 18 \, x^{9} + 17 \, x^{6} - 18 \, x^{3} + 1\right )} - 6 \, {\left (3 \, x^{8} + 5 \, x^{5} - 3 \, x^{2}\right )} {\left (x^{6} + x^{3} - 1\right )}^{\frac {1}{3}}}{x^{12} - x^{6} + 1}\right ) - \frac {1}{72} \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (\frac {6 \cdot 4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (3 \, x^{20} - 5 \, x^{17} - 12 \, x^{14} + 11 \, x^{11} + 12 \, x^{8} - 5 \, x^{5} - 3 \, x^{2}\right )} {\left (x^{6} + x^{3} - 1\right )}^{\frac {1}{3}} - 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{24} - 18 \, x^{21} - 2 \, x^{18} + 72 \, x^{15} + 3 \, x^{12} - 72 \, x^{9} - 2 \, x^{6} + 18 \, x^{3} + 1\right )} - 12 \, {\left (x^{19} - 5 \, x^{16} - 2 \, x^{13} + 11 \, x^{10} + 2 \, x^{7} - 5 \, x^{4} - x\right )} {\left (x^{6} + x^{3} - 1\right )}^{\frac {2}{3}}}{x^{24} - 2 \, x^{18} + 3 \, x^{12} - 2 \, x^{6} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6+1)*(x^6+x^3-1)^(2/3)/(x^12-x^6+1),x, algorithm="fricas")

[Out]

1/18*4^(1/6)*sqrt(3)*(-1)^(1/3)*arctan(1/6*4^(1/6)*(6*4^(2/3)*sqrt(3)*(-1)^(2/3)*(x^31 - 14*x^28 - 75*x^25 + 8
2*x^22 + 293*x^19 - 132*x^16 - 293*x^13 + 82*x^10 + 75*x^7 - 14*x^4 - x)*(x^6 + x^3 - 1)^(2/3) - 12*sqrt(3)*(-
1)^(1/3)*(3*x^32 + 49*x^29 - 51*x^26 - 344*x^23 + 99*x^20 + 609*x^17 - 99*x^14 - 344*x^11 + 51*x^8 + 49*x^5 -
3*x^2)*(x^6 + x^3 - 1)^(1/3) + 4^(1/3)*sqrt(3)*(x^36 + 54*x^33 - 129*x^30 - 846*x^27 + 258*x^24 + 2502*x^21 -
169*x^18 - 2502*x^15 + 258*x^12 + 846*x^9 - 129*x^6 - 54*x^3 + 1))/(x^36 - 54*x^33 - 489*x^30 + 270*x^27 + 292
2*x^24 - 54*x^21 - 4921*x^18 + 54*x^15 + 2922*x^12 - 270*x^9 - 489*x^6 + 54*x^3 + 1)) + 1/36*4^(2/3)*(-1)^(1/3
)*log((3*4^(2/3)*(-1)^(1/3)*(x^7 + 4*x^4 - x)*(x^6 + x^3 - 1)^(2/3) - 4^(1/3)*(-1)^(2/3)*(x^12 + 18*x^9 + 17*x
^6 - 18*x^3 + 1) - 6*(3*x^8 + 5*x^5 - 3*x^2)*(x^6 + x^3 - 1)^(1/3))/(x^12 - x^6 + 1)) - 1/72*4^(2/3)*(-1)^(1/3
)*log((6*4^(1/3)*(-1)^(2/3)*(3*x^20 - 5*x^17 - 12*x^14 + 11*x^11 + 12*x^8 - 5*x^5 - 3*x^2)*(x^6 + x^3 - 1)^(1/
3) - 4^(2/3)*(-1)^(1/3)*(x^24 - 18*x^21 - 2*x^18 + 72*x^15 + 3*x^12 - 72*x^9 - 2*x^6 + 18*x^3 + 1) - 12*(x^19
- 5*x^16 - 2*x^13 + 11*x^10 + 2*x^7 - 5*x^4 - x)*(x^6 + x^3 - 1)^(2/3))/(x^24 - 2*x^18 + 3*x^12 - 2*x^6 + 1))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{6} + x^{3} - 1\right )}^{\frac {2}{3}} {\left (x^{6} + 1\right )}}{x^{12} - x^{6} + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6+1)*(x^6+x^3-1)^(2/3)/(x^12-x^6+1),x, algorithm="giac")

[Out]

integrate((x^6 + x^3 - 1)^(2/3)*(x^6 + 1)/(x^12 - x^6 + 1), x)

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maple [B]  time = 21.16, size = 1958, normalized size = 32.10

method	result	size
trager	\(\text {Expression too large to display}\)	\(1958\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6+1)*(x^6+x^3-1)^(2/3)/(x^12-x^6+1),x,method=_RETURNVERBOSE)

[Out]

1/6*RootOf(_Z^3+2)*ln(-(565*RootOf(_Z^3+2)*x^12+3594*RootOf(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)*x^12
-1130*RootOf(_Z^3+2)*x^9-7188*RootOf(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)*x^9+7188*RootOf(RootOf(_Z^3
+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)*x^3-2825*RootOf(_Z^3+2)*x^6-17970*RootOf(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+
2)+36*_Z^2)*x^6+565*RootOf(_Z^3+2)+3594*RootOf(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)+1130*RootOf(_Z^3+
2)*x^3+11394*RootOf(_Z^3+2)^4*(x^6+x^3-1)^(2/3)*RootOf(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)^2*x^4+379
8*RootOf(_Z^3+2)^2*(x^6+x^3-1)^(2/3)*RootOf(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)*x^7-1224*RootOf(_Z^3
+2)^3*(x^6+x^3-1)^(1/3)*RootOf(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)^2*x^5-3594*RootOf(_Z^3+2)^4*(x^6+
x^3-1)^(1/3)*RootOf(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)*x^5-612*(x^6+x^3-1)^(1/3)*RootOf(_Z^3+2)*Roo
tOf(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)*x^8-10782*RootOf(_Z^3+2)^2*(x^6+x^3-1)^(2/3)*RootOf(RootOf(_
Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)*x^4+204*(x^6+x^3-1)^(1/3)*RootOf(_Z^3+2)*RootOf(RootOf(_Z^3+2)^2+6*_Z*Ro
otOf(_Z^3+2)+36*_Z^2)*x^5-3798*RootOf(_Z^3+2)^2*(x^6+x^3-1)^(2/3)*RootOf(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+
36*_Z^2)*x+612*(x^6+x^3-1)^(1/3)*RootOf(_Z^3+2)*RootOf(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)*x^2-1797*
(x^6+x^3-1)^(1/3)*RootOf(_Z^3+2)^2*x^8+599*(x^6+x^3-1)^(1/3)*RootOf(_Z^3+2)^2*x^5+1797*(x^6+x^3-1)^(1/3)*RootO
f(_Z^3+2)^2*x^2+43128*RootOf(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)^2*RootOf(_Z^3+2)^2*x^9+6780*RootOf(
RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)*RootOf(_Z^3+2)^3*x^9+21564*RootOf(RootOf(_Z^3+2)^2+6*_Z*RootOf(_
Z^3+2)+36*_Z^2)^2*RootOf(_Z^3+2)^2*x^6+3390*RootOf(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)*RootOf(_Z^3+2
)^3*x^6-43128*RootOf(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)^2*RootOf(_Z^3+2)^2*x^3-6780*RootOf(RootOf(_
Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)*RootOf(_Z^3+2)^3*x^3-2328*(x^6+x^3-1)^(2/3)*x^7+2328*(x^6+x^3-1)^(2/3)*x
+2328*(x^6+x^3-1)^(2/3)*x^4)/(x^12-x^6+1))+RootOf(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)*ln(-(599*RootO
f(_Z^3+2)*x^12+3390*RootOf(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)*x^12+1198*RootOf(_Z^3+2)*x^9+6780*Roo
tOf(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)*x^9-6780*RootOf(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2
)*x^3-1797*RootOf(_Z^3+2)*x^6-10170*RootOf(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)*x^6+599*RootOf(_Z^3+2
)+3390*RootOf(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)-1198*RootOf(_Z^3+2)*x^3+11394*RootOf(_Z^3+2)^4*(x^
6+x^3-1)^(2/3)*RootOf(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)^2*x^4-3798*RootOf(_Z^3+2)^2*(x^6+x^3-1)^(2
/3)*RootOf(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)*x^7-20340*RootOf(_Z^3+2)^3*(x^6+x^3-1)^(1/3)*RootOf(R
ootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)^2*x^5-3594*RootOf(_Z^3+2)^4*(x^6+x^3-1)^(1/3)*RootOf(RootOf(_Z^3+
2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)*x^5+10170*(x^6+x^3-1)^(1/3)*RootOf(_Z^3+2)*RootOf(RootOf(_Z^3+2)^2+6*_Z*Root
Of(_Z^3+2)+36*_Z^2)*x^8+3186*RootOf(_Z^3+2)^2*(x^6+x^3-1)^(2/3)*RootOf(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36
*_Z^2)*x^4+3390*(x^6+x^3-1)^(1/3)*RootOf(_Z^3+2)*RootOf(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)*x^5+3798
*RootOf(_Z^3+2)^2*(x^6+x^3-1)^(2/3)*RootOf(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)*x-10170*(x^6+x^3-1)^(
1/3)*RootOf(_Z^3+2)*RootOf(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)*x^2+1797*(x^6+x^3-1)^(1/3)*RootOf(_Z^
3+2)^2*x^8+599*(x^6+x^3-1)^(1/3)*RootOf(_Z^3+2)^2*x^5-1797*(x^6+x^3-1)^(1/3)*RootOf(_Z^3+2)^2*x^2-40680*RootOf
(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)^2*RootOf(_Z^3+2)^2*x^9-7188*RootOf(RootOf(_Z^3+2)^2+6*_Z*RootOf
(_Z^3+2)+36*_Z^2)*RootOf(_Z^3+2)^3*x^9-20340*RootOf(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)^2*RootOf(_Z^
3+2)^2*x^6-3594*RootOf(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)*RootOf(_Z^3+2)^3*x^6+40680*RootOf(RootOf(
_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)^2*RootOf(_Z^3+2)^2*x^3+7188*RootOf(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)
+36*_Z^2)*RootOf(_Z^3+2)^3*x^3-1062*(x^6+x^3-1)^(2/3)*x^7+1062*(x^6+x^3-1)^(2/3)*x)/(x^12-x^6+1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{6} + x^{3} - 1\right )}^{\frac {2}{3}} {\left (x^{6} + 1\right )}}{x^{12} - x^{6} + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6+1)*(x^6+x^3-1)^(2/3)/(x^12-x^6+1),x, algorithm="maxima")

[Out]

integrate((x^6 + x^3 - 1)^(2/3)*(x^6 + 1)/(x^12 - x^6 + 1), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\left (x^6+1\right )\,{\left (x^6+x^3-1\right )}^{2/3}}{x^{12}-x^6+1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^6 + 1)*(x^3 + x^6 - 1)^(2/3))/(x^12 - x^6 + 1),x)

[Out]

int(((x^6 + 1)*(x^3 + x^6 - 1)^(2/3))/(x^12 - x^6 + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**6+1)*(x**6+x**3-1)**(2/3)/(x**12-x**6+1),x)

[Out]

Timed out

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