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

Optimal. Leaf size=128 \[ \frac {\log \left (2^{2/3} \sqrt [3]{x^6+x^3+1}-2 x\right )}{3 \sqrt [3]{2}}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^6+x^3+1}+x}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {\log \left (2 x^2+2^{2/3} \sqrt [3]{x^6+x^3+1} x+\sqrt [3]{2} \left (x^6+x^3+1\right )^{2/3}\right )}{6 \sqrt [3]{2}} \]

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Rubi [F]  time = 0.43, 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]

(x*(1 + (2*x^3)/(1 - I*Sqrt[3]))^(1/3)*(1 + (2*x^3)/(1 + I*Sqrt[3]))^(1/3)*AppellF1[1/3, 1/3, 1/3, 4/3, (-2*x^
3)/(1 - I*Sqrt[3]), (-2*x^3)/(1 + I*Sqrt[3])])/(1 + x^3 + x^6)^(1/3) + (1 + I*Sqrt[3])*Defer[Int][1/((-1 - I*S
qrt[3] + 2*x^3)*(1 + x^3 + x^6)^(1/3)), x] + (1 - I*Sqrt[3])*Defer[Int][1/((-1 + I*Sqrt[3] + 2*x^3)*(1 + x^3 +
x^6)^(1/3)), x]

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 \frac {-1+x^6}{\left (1-x^3+x^6\right ) \sqrt [3]{1+x^3+x^6}} \, dx\\ &=\int \left (\frac {1}{\sqrt [3]{1+x^3+x^6}}-\frac {2-x^3}{\left (1-x^3+x^6\right ) \sqrt [3]{1+x^3+x^6}}\right ) \, dx\\ &=\int \frac {1}{\sqrt [3]{1+x^3+x^6}} \, dx-\int \frac {2-x^3}{\left (1-x^3+x^6\right ) \sqrt [3]{1+x^3+x^6}} \, dx\\ &=\frac {\left (\sqrt [3]{1+\frac {2 x^3}{1-i \sqrt {3}}} \sqrt [3]{1+\frac {2 x^3}{1+i \sqrt {3}}}\right ) \int \frac {1}{\sqrt [3]{1+\frac {2 x^3}{1-i \sqrt {3}}} \sqrt [3]{1+\frac {2 x^3}{1+i \sqrt {3}}}} \, dx}{\sqrt [3]{1+x^3+x^6}}-\int \left (\frac {-1-i \sqrt {3}}{\left (-1-i \sqrt {3}+2 x^3\right ) \sqrt [3]{1+x^3+x^6}}+\frac {-1+i \sqrt {3}}{\left (-1+i \sqrt {3}+2 x^3\right ) \sqrt [3]{1+x^3+x^6}}\right ) \, dx\\ &=\frac {x \sqrt [3]{1+\frac {2 x^3}{1-i \sqrt {3}}} \sqrt [3]{1+\frac {2 x^3}{1+i \sqrt {3}}} F_1\left (\frac {1}{3};\frac {1}{3},\frac {1}{3};\frac {4}{3};-\frac {2 x^3}{1-i \sqrt {3}},-\frac {2 x^3}{1+i \sqrt {3}}\right )}{\sqrt [3]{1+x^3+x^6}}-\left (-1-i \sqrt {3}\right ) \int \frac {1}{\left (-1-i \sqrt {3}+2 x^3\right ) \sqrt [3]{1+x^3+x^6}} \, dx-\left (-1+i \sqrt {3}\right ) \int \frac {1}{\left (-1+i \sqrt {3}+2 x^3\right ) \sqrt [3]{1+x^3+x^6}} \, dx\\ \end {align*}

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Mathematica [F]  time = 1.23, 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 = 0.50, size = 128, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{1+x^3+x^6}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{1+x^3+x^6}\right )}{3 \sqrt [3]{2}}-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{1+x^3+x^6}+\sqrt [3]{2} \left (1+x^3+x^6\right )^{2/3}\right )}{6 \sqrt [3]{2}} \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]

-(ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(1 + x^3 + x^6)^(1/3))]/(2^(1/3)*Sqrt[3])) + Log[-2*x + 2^(2/3)*(1 + x^3 + x
^6)^(1/3)]/(3*2^(1/3)) - Log[2*x^2 + 2^(2/3)*x*(1 + x^3 + x^6)^(1/3) + 2^(1/3)*(1 + x^3 + x^6)^(2/3)]/(6*2^(1/
3))

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fricas [B]  time = 11.14, size = 322, normalized size = 2.52 \begin {gather*} -\frac {1}{18} \, \sqrt {6} 2^{\frac {1}{6}} \arctan \left (\frac {2^{\frac {1}{6}} {\left (6 \, \sqrt {6} 2^{\frac {2}{3}} {\left (x^{13} + 16 \, x^{10} + 21 \, x^{7} + 16 \, x^{4} + x\right )} {\left (x^{6} + x^{3} + 1\right )}^{\frac {2}{3}} - \sqrt {6} 2^{\frac {1}{3}} {\left (x^{18} - 21 \, x^{15} - 102 \, x^{12} - 133 \, x^{9} - 102 \, x^{6} - 21 \, x^{3} + 1\right )} - 24 \, \sqrt {6} {\left (x^{14} + x^{11} + x^{5} + x^{2}\right )} {\left (x^{6} + x^{3} + 1\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (x^{18} + 51 \, x^{15} + 114 \, x^{12} + 155 \, x^{9} + 114 \, x^{6} + 51 \, x^{3} + 1\right )}}\right ) + \frac {1}{18} \cdot 2^{\frac {2}{3}} \log \left (-\frac {3 \cdot 2^{\frac {2}{3}} {\left (x^{6} + x^{3} + 1\right )}^{\frac {2}{3}} x - 6 \, {\left (x^{6} + x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 2^{\frac {1}{3}} {\left (x^{6} - x^{3} + 1\right )}}{x^{6} - x^{3} + 1}\right ) - \frac {1}{36} \cdot 2^{\frac {2}{3}} \log \left (\frac {2^{\frac {2}{3}} {\left (x^{12} + 16 \, x^{9} + 21 \, x^{6} + 16 \, x^{3} + 1\right )} + 12 \cdot 2^{\frac {1}{3}} {\left (x^{8} + 2 \, x^{5} + x^{2}\right )} {\left (x^{6} + x^{3} + 1\right )}^{\frac {1}{3}} + 6 \, {\left (x^{7} + 5 \, x^{4} + x\right )} {\left (x^{6} + x^{3} + 1\right )}^{\frac {2}{3}}}{x^{12} - 2 \, x^{9} + 3 \, x^{6} - 2 \, x^{3} + 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*sqrt(6)*2^(1/6)*arctan(1/6*2^(1/6)*(6*sqrt(6)*2^(2/3)*(x^13 + 16*x^10 + 21*x^7 + 16*x^4 + x)*(x^6 + x^3
+ 1)^(2/3) - sqrt(6)*2^(1/3)*(x^18 - 21*x^15 - 102*x^12 - 133*x^9 - 102*x^6 - 21*x^3 + 1) - 24*sqrt(6)*(x^14 +
x^11 + x^5 + x^2)*(x^6 + x^3 + 1)^(1/3))/(x^18 + 51*x^15 + 114*x^12 + 155*x^9 + 114*x^6 + 51*x^3 + 1)) + 1/18
*2^(2/3)*log(-(3*2^(2/3)*(x^6 + x^3 + 1)^(2/3)*x - 6*(x^6 + x^3 + 1)^(1/3)*x^2 - 2^(1/3)*(x^6 - x^3 + 1))/(x^6
- x^3 + 1)) - 1/36*2^(2/3)*log((2^(2/3)*(x^12 + 16*x^9 + 21*x^6 + 16*x^3 + 1) + 12*2^(1/3)*(x^8 + 2*x^5 + x^2
)*(x^6 + x^3 + 1)^(1/3) + 6*(x^7 + 5*x^4 + x)*(x^6 + x^3 + 1)^(2/3))/(x^12 - 2*x^9 + 3*x^6 - 2*x^3 + 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 [C]  time = 8.09, size = 1054, normalized size = 8.23

method	result	size
trager	\(\RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) \ln \left (-\frac {3 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{3} x^{3}+90 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x^{3}+\RootOf \left (\textit {\_Z}^{3}-4\right ) x^{6}+30 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) x^{6}+18 \left (x^{6}+x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) x -3 \left (x^{6}+x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x^{2}+36 \left (x^{6}+x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{2}+3 \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{3}+90 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) x^{3}-6 \left (x^{6}+x^{3}+1\right )^{\frac {2}{3}} x +\RootOf \left (\textit {\_Z}^{3}-4\right )+30 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right )}{x^{6}-x^{3}+1}\right )-\frac {\ln \left (-\frac {12 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{3} x^{3}+90 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x^{3}-4 \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{6}-30 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) x^{6}-18 \left (x^{6}+x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) x -9 \left (x^{6}+x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x^{2}-36 \left (x^{6}+x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{2}-4 \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{3}-30 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) x^{3}-18 \left (x^{6}+x^{3}+1\right )^{\frac {2}{3}} x -4 \RootOf \left (\textit {\_Z}^{3}-4\right )-30 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right )}{x^{6}-x^{3}+1}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )}{6}-\ln \left (-\frac {12 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{3} x^{3}+90 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x^{3}-4 \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{6}-30 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) x^{6}-18 \left (x^{6}+x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) x -9 \left (x^{6}+x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x^{2}-36 \left (x^{6}+x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{2}-4 \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{3}-30 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) x^{3}-18 \left (x^{6}+x^{3}+1\right )^{\frac {2}{3}} x -4 \RootOf \left (\textit {\_Z}^{3}-4\right )-30 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right )}{x^{6}-x^{3}+1}\right ) \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right )\)	\(1054\)

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]

RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*ln(-(3*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^
2)*RootOf(_Z^3-4)^3*x^3+90*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)^2*RootOf(_Z^3-4)^2*x^3+RootOf(
_Z^3-4)*x^6+30*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*x^6+18*(x^6+x^3+1)^(2/3)*RootOf(_Z^3-4)^2*
RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*x-3*(x^6+x^3+1)^(1/3)*RootOf(_Z^3-4)^2*x^2+36*(x^6+x^3+1)
^(1/3)*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)*x^2+3*RootOf(_Z^3-4)*x^3+90*RootOf(
RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*x^3-6*(x^6+x^3+1)^(2/3)*x+RootOf(_Z^3-4)+30*RootOf(RootOf(_Z^3-4
)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2))/(x^6-x^3+1))-1/6*ln(-(12*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2
)*RootOf(_Z^3-4)^3*x^3+90*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)^2*RootOf(_Z^3-4)^2*x^3-4*RootOf
(_Z^3-4)*x^6-30*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*x^6-18*(x^6+x^3+1)^(2/3)*RootOf(_Z^3-4)^2
*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*x-9*(x^6+x^3+1)^(1/3)*RootOf(_Z^3-4)^2*x^2-36*(x^6+x^3+1
)^(1/3)*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)*x^2-4*RootOf(_Z^3-4)*x^3-30*RootOf
(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*x^3-18*(x^6+x^3+1)^(2/3)*x-4*RootOf(_Z^3-4)-30*RootOf(RootOf(_Z
^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2))/(x^6-x^3+1))*RootOf(_Z^3-4)-ln(-(12*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(
_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)^3*x^3+90*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)^2*RootOf(_Z^3-4)
^2*x^3-4*RootOf(_Z^3-4)*x^6-30*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*x^6-18*(x^6+x^3+1)^(2/3)*R
ootOf(_Z^3-4)^2*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*x-9*(x^6+x^3+1)^(1/3)*RootOf(_Z^3-4)^2*x^
2-36*(x^6+x^3+1)^(1/3)*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)*x^2-4*RootOf(_Z^3-4
)*x^3-30*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*x^3-18*(x^6+x^3+1)^(2/3)*x-4*RootOf(_Z^3-4)-30*R
ootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2))/(x^6-x^3+1))*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+
36*_Z^2)

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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 = -0.01 \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^6 + x^12 + 1),x)

[Out]

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