3.21.36 \(\int \frac {-3-4 x+3 x^6}{(1+2 x+x^6) \sqrt [3]{1+2 x+2 x^3+x^6}} \, dx\)

Optimal. Leaf size=145 \[ \frac {\log \left (2^{2/3} \sqrt [3]{x^6+2 x^3+2 x+1}-2 x\right )}{\sqrt [3]{2}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^6+2 x^3+2 x+1}+x}\right )}{\sqrt [3]{2}}-\frac {\log \left (2 x^2+2^{2/3} \sqrt [3]{x^6+2 x^3+2 x+1} x+\sqrt [3]{2} \left (x^6+2 x^3+2 x+1\right )^{2/3}\right )}{2 \sqrt [3]{2}} \]

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Rubi [F]  time = 1.94, 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 {-3-4 x+3 x^6}{\left (1+2 x+x^6\right ) \sqrt [3]{1+2 x+2 x^3+x^6}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-3 - 4*x + 3*x^6)/((1 + 2*x + x^6)*(1 + 2*x + 2*x^3 + x^6)^(1/3)),x]

[Out]

3*Defer[Int][(1 + 2*x + 2*x^3 + x^6)^(-1/3), x] - Defer[Int][1/((1 + x)*(1 + 2*x + 2*x^3 + x^6)^(1/3)), x] - 5
*Defer[Int][1/((1 + x - x^2 + x^3 - x^4 + x^5)*(1 + 2*x + 2*x^3 + x^6)^(1/3)), x] - 4*Defer[Int][x/((1 + x - x
^2 + x^3 - x^4 + x^5)*(1 + 2*x + 2*x^3 + x^6)^(1/3)), x] + 3*Defer[Int][x^2/((1 + x - x^2 + x^3 - x^4 + x^5)*(
1 + 2*x + 2*x^3 + x^6)^(1/3)), x] - 2*Defer[Int][x^3/((1 + x - x^2 + x^3 - x^4 + x^5)*(1 + 2*x + 2*x^3 + x^6)^
(1/3)), x] + Defer[Int][x^4/((1 + x - x^2 + x^3 - x^4 + x^5)*(1 + 2*x + 2*x^3 + x^6)^(1/3)), x]

Rubi steps

\begin {align*} \int \frac {-3-4 x+3 x^6}{\left (1+2 x+x^6\right ) \sqrt [3]{1+2 x+2 x^3+x^6}} \, dx &=\int \left (\frac {3}{\sqrt [3]{1+2 x+2 x^3+x^6}}-\frac {2 (3+5 x)}{\left (1+2 x+x^6\right ) \sqrt [3]{1+2 x+2 x^3+x^6}}\right ) \, dx\\ &=-\left (2 \int \frac {3+5 x}{\left (1+2 x+x^6\right ) \sqrt [3]{1+2 x+2 x^3+x^6}} \, dx\right )+3 \int \frac {1}{\sqrt [3]{1+2 x+2 x^3+x^6}} \, dx\\ &=-\left (2 \int \left (\frac {1}{2 (1+x) \sqrt [3]{1+2 x+2 x^3+x^6}}+\frac {5+4 x-3 x^2+2 x^3-x^4}{2 \left (1+x-x^2+x^3-x^4+x^5\right ) \sqrt [3]{1+2 x+2 x^3+x^6}}\right ) \, dx\right )+3 \int \frac {1}{\sqrt [3]{1+2 x+2 x^3+x^6}} \, dx\\ &=3 \int \frac {1}{\sqrt [3]{1+2 x+2 x^3+x^6}} \, dx-\int \frac {1}{(1+x) \sqrt [3]{1+2 x+2 x^3+x^6}} \, dx-\int \frac {5+4 x-3 x^2+2 x^3-x^4}{\left (1+x-x^2+x^3-x^4+x^5\right ) \sqrt [3]{1+2 x+2 x^3+x^6}} \, dx\\ &=3 \int \frac {1}{\sqrt [3]{1+2 x+2 x^3+x^6}} \, dx-\int \frac {1}{(1+x) \sqrt [3]{1+2 x+2 x^3+x^6}} \, dx-\int \left (\frac {5}{\left (1+x-x^2+x^3-x^4+x^5\right ) \sqrt [3]{1+2 x+2 x^3+x^6}}+\frac {4 x}{\left (1+x-x^2+x^3-x^4+x^5\right ) \sqrt [3]{1+2 x+2 x^3+x^6}}-\frac {3 x^2}{\left (1+x-x^2+x^3-x^4+x^5\right ) \sqrt [3]{1+2 x+2 x^3+x^6}}+\frac {2 x^3}{\left (1+x-x^2+x^3-x^4+x^5\right ) \sqrt [3]{1+2 x+2 x^3+x^6}}-\frac {x^4}{\left (1+x-x^2+x^3-x^4+x^5\right ) \sqrt [3]{1+2 x+2 x^3+x^6}}\right ) \, dx\\ &=-\left (2 \int \frac {x^3}{\left (1+x-x^2+x^3-x^4+x^5\right ) \sqrt [3]{1+2 x+2 x^3+x^6}} \, dx\right )+3 \int \frac {1}{\sqrt [3]{1+2 x+2 x^3+x^6}} \, dx+3 \int \frac {x^2}{\left (1+x-x^2+x^3-x^4+x^5\right ) \sqrt [3]{1+2 x+2 x^3+x^6}} \, dx-4 \int \frac {x}{\left (1+x-x^2+x^3-x^4+x^5\right ) \sqrt [3]{1+2 x+2 x^3+x^6}} \, dx-5 \int \frac {1}{\left (1+x-x^2+x^3-x^4+x^5\right ) \sqrt [3]{1+2 x+2 x^3+x^6}} \, dx-\int \frac {1}{(1+x) \sqrt [3]{1+2 x+2 x^3+x^6}} \, dx+\int \frac {x^4}{\left (1+x-x^2+x^3-x^4+x^5\right ) \sqrt [3]{1+2 x+2 x^3+x^6}} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.43, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-3-4 x+3 x^6}{\left (1+2 x+x^6\right ) \sqrt [3]{1+2 x+2 x^3+x^6}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(-3 - 4*x + 3*x^6)/((1 + 2*x + x^6)*(1 + 2*x + 2*x^3 + x^6)^(1/3)),x]

[Out]

Integrate[(-3 - 4*x + 3*x^6)/((1 + 2*x + x^6)*(1 + 2*x + 2*x^3 + x^6)^(1/3)), x]

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IntegrateAlgebraic [A]  time = 0.53, size = 145, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{1+2 x+2 x^3+x^6}}\right )}{\sqrt [3]{2}}+\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{1+2 x+2 x^3+x^6}\right )}{\sqrt [3]{2}}-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{1+2 x+2 x^3+x^6}+\sqrt [3]{2} \left (1+2 x+2 x^3+x^6\right )^{2/3}\right )}{2 \sqrt [3]{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-3 - 4*x + 3*x^6)/((1 + 2*x + x^6)*(1 + 2*x + 2*x^3 + x^6)^(1/3)),x]

[Out]

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

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fricas [B]  time = 45.98, size = 478, normalized size = 3.30 \begin {gather*} -\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {1}{6}} {\left (2^{\frac {5}{6}} {\left (x^{18} + 36 \, x^{15} + 6 \, x^{13} + 183 \, x^{12} + 144 \, x^{10} + 288 \, x^{9} + 12 \, x^{8} + 372 \, x^{7} + 183 \, x^{6} + 144 \, x^{5} + 144 \, x^{4} + 44 \, x^{3} + 12 \, x^{2} + 6 \, x + 1\right )} + 12 \, \sqrt {2} {\left (x^{14} + 18 \, x^{11} + 4 \, x^{9} + 38 \, x^{8} + 36 \, x^{6} + 18 \, x^{5} + 4 \, x^{4} + 4 \, x^{3} + x^{2}\right )} {\left (x^{6} + 2 \, x^{3} + 2 \, x + 1\right )}^{\frac {1}{3}} + 12 \cdot 2^{\frac {1}{6}} {\left (x^{13} + 6 \, x^{10} + 4 \, x^{8} + 2 \, x^{7} + 12 \, x^{5} + 6 \, x^{4} + 4 \, x^{3} + 4 \, x^{2} + x\right )} {\left (x^{6} + 2 \, x^{3} + 2 \, x + 1\right )}^{\frac {2}{3}}\right )}}{6 \, {\left (x^{18} + 6 \, x^{13} - 105 \, x^{12} - 216 \, x^{9} + 12 \, x^{8} - 204 \, x^{7} - 105 \, x^{6} + 8 \, x^{3} + 12 \, x^{2} + 6 \, x + 1\right )}}\right ) + \frac {1}{6} \cdot 2^{\frac {2}{3}} \log \left (\frac {6 \cdot 2^{\frac {1}{3}} {\left (x^{6} + 2 \, x^{3} + 2 \, x + 1\right )}^{\frac {1}{3}} x^{2} + 2^{\frac {2}{3}} {\left (x^{6} + 2 \, x + 1\right )} - 6 \, {\left (x^{6} + 2 \, x^{3} + 2 \, x + 1\right )}^{\frac {2}{3}} x}{x^{6} + 2 \, x + 1}\right ) - \frac {1}{12} \cdot 2^{\frac {2}{3}} \log \left (\frac {3 \cdot 2^{\frac {2}{3}} {\left (x^{7} + 6 \, x^{4} + 2 \, x^{2} + x\right )} {\left (x^{6} + 2 \, x^{3} + 2 \, x + 1\right )}^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (x^{12} + 18 \, x^{9} + 4 \, x^{7} + 38 \, x^{6} + 36 \, x^{4} + 18 \, x^{3} + 4 \, x^{2} + 4 \, x + 1\right )} + 12 \, {\left (x^{8} + 3 \, x^{5} + 2 \, x^{3} + x^{2}\right )} {\left (x^{6} + 2 \, x^{3} + 2 \, x + 1\right )}^{\frac {1}{3}}}{x^{12} + 4 \, x^{7} + 2 \, x^{6} + 4 \, x^{2} + 4 \, x + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^6-4*x-3)/(x^6+2*x+1)/(x^6+2*x^3+2*x+1)^(1/3),x, algorithm="fricas")

[Out]

-1/6*sqrt(3)*2^(2/3)*arctan(1/6*sqrt(3)*2^(1/6)*(2^(5/6)*(x^18 + 36*x^15 + 6*x^13 + 183*x^12 + 144*x^10 + 288*
x^9 + 12*x^8 + 372*x^7 + 183*x^6 + 144*x^5 + 144*x^4 + 44*x^3 + 12*x^2 + 6*x + 1) + 12*sqrt(2)*(x^14 + 18*x^11
+ 4*x^9 + 38*x^8 + 36*x^6 + 18*x^5 + 4*x^4 + 4*x^3 + x^2)*(x^6 + 2*x^3 + 2*x + 1)^(1/3) + 12*2^(1/6)*(x^13 +
6*x^10 + 4*x^8 + 2*x^7 + 12*x^5 + 6*x^4 + 4*x^3 + 4*x^2 + x)*(x^6 + 2*x^3 + 2*x + 1)^(2/3))/(x^18 + 6*x^13 - 1
05*x^12 - 216*x^9 + 12*x^8 - 204*x^7 - 105*x^6 + 8*x^3 + 12*x^2 + 6*x + 1)) + 1/6*2^(2/3)*log((6*2^(1/3)*(x^6
+ 2*x^3 + 2*x + 1)^(1/3)*x^2 + 2^(2/3)*(x^6 + 2*x + 1) - 6*(x^6 + 2*x^3 + 2*x + 1)^(2/3)*x)/(x^6 + 2*x + 1)) -
1/12*2^(2/3)*log((3*2^(2/3)*(x^7 + 6*x^4 + 2*x^2 + x)*(x^6 + 2*x^3 + 2*x + 1)^(2/3) + 2^(1/3)*(x^12 + 18*x^9
+ 4*x^7 + 38*x^6 + 36*x^4 + 18*x^3 + 4*x^2 + 4*x + 1) + 12*(x^8 + 3*x^5 + 2*x^3 + x^2)*(x^6 + 2*x^3 + 2*x + 1)
^(1/3))/(x^12 + 4*x^7 + 2*x^6 + 4*x^2 + 4*x + 1))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3 \, x^{6} - 4 \, x - 3}{{\left (x^{6} + 2 \, x^{3} + 2 \, x + 1\right )}^{\frac {1}{3}} {\left (x^{6} + 2 \, x + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^6-4*x-3)/(x^6+2*x+1)/(x^6+2*x^3+2*x+1)^(1/3),x, algorithm="giac")

[Out]

integrate((3*x^6 - 4*x - 3)/((x^6 + 2*x^3 + 2*x + 1)^(1/3)*(x^6 + 2*x + 1)), x)

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maple [C]  time = 32.71, size = 1097, normalized size = 7.57

method	result	size
trager	\(\frac {\RootOf \left (\textit {\_Z}^{3}-4\right ) \ln \left (\frac {\RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{3} x^{3}-\RootOf \left (\textit {\_Z}^{3}-4\right ) x^{6}+\left (x^{6}+2 x^{3}+2 x +1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x -\left (x^{6}+2 x^{3}+2 x +1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x^{2}-4 \left (x^{6}+2 x^{3}+2 x +1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{2}-2 \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{3}+4 \left (x^{6}+2 x^{3}+2 x +1\right )^{\frac {2}{3}} x -2 \RootOf \left (\textit {\_Z}^{3}-4\right ) x -\RootOf \left (\textit {\_Z}^{3}-4\right )}{\left (1+x \right ) \left (x^{5}-x^{4}+x^{3}-x^{2}+x +1\right )}\right )}{2}-\frac {\ln \left (-\frac {\RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{4} x^{3}-2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-4\right )^{3} x^{3}+\RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x^{6}-2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{6}+6 \left (x^{6}+2 x^{3}+2 x +1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x^{2}+4 \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x^{3}-8 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{3}-6 \RootOf \left (\textit {\_Z}^{3}-4\right ) \left (x^{6}+2 x^{3}+2 x +1\right )^{\frac {2}{3}} x -12 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \left (x^{6}+2 x^{3}+2 x +1\right )^{\frac {2}{3}} x +2 \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x -4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right ) x +\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}-2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )}{\left (1+x \right ) \left (x^{5}-x^{4}+x^{3}-x^{2}+x +1\right )}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )}{2}-\ln \left (-\frac {\RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{4} x^{3}-2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-4\right )^{3} x^{3}+\RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x^{6}-2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{6}+6 \left (x^{6}+2 x^{3}+2 x +1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x^{2}+4 \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x^{3}-8 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{3}-6 \RootOf \left (\textit {\_Z}^{3}-4\right ) \left (x^{6}+2 x^{3}+2 x +1\right )^{\frac {2}{3}} x -12 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \left (x^{6}+2 x^{3}+2 x +1\right )^{\frac {2}{3}} x +2 \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x -4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right ) x +\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}-2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )}{\left (1+x \right ) \left (x^{5}-x^{4}+x^{3}-x^{2}+x +1\right )}\right ) \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right )\)	\(1097\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^6-4*x-3)/(x^6+2*x+1)/(x^6+2*x^3+2*x+1)^(1/3),x,method=_RETURNVERBOSE)

[Out]

1/2*RootOf(_Z^3-4)*ln((RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)^3*x^3-RootOf(_Z^3-4)
*x^6+(x^6+2*x^3+2*x+1)^(2/3)*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)^2*x-(x^6+2*x^3
+2*x+1)^(1/3)*RootOf(_Z^3-4)^2*x^2-4*(x^6+2*x^3+2*x+1)^(1/3)*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^
2)*RootOf(_Z^3-4)*x^2-2*RootOf(_Z^3-4)*x^3+4*(x^6+2*x^3+2*x+1)^(2/3)*x-2*RootOf(_Z^3-4)*x-RootOf(_Z^3-4))/(1+x
)/(x^5-x^4+x^3-x^2+x+1))-1/2*ln(-(RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)^4*x^3-2*R
ootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)^2*RootOf(_Z^3-4)^3*x^3+RootOf(_Z^3-4)^2*x^6-2*RootOf(RootOf
(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)*x^6+6*(x^6+2*x^3+2*x+1)^(1/3)*RootOf(RootOf(_Z^3-4)^2+2*
_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)^2*x^2+4*RootOf(_Z^3-4)^2*x^3-8*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z
^3-4)+4*_Z^2)*RootOf(_Z^3-4)*x^3-6*RootOf(_Z^3-4)*(x^6+2*x^3+2*x+1)^(2/3)*x-12*RootOf(RootOf(_Z^3-4)^2+2*_Z*Ro
otOf(_Z^3-4)+4*_Z^2)*(x^6+2*x^3+2*x+1)^(2/3)*x+2*RootOf(_Z^3-4)^2*x-4*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3
-4)+4*_Z^2)*RootOf(_Z^3-4)*x+RootOf(_Z^3-4)^2-2*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^
3-4))/(1+x)/(x^5-x^4+x^3-x^2+x+1))*RootOf(_Z^3-4)-ln(-(RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*Roo
tOf(_Z^3-4)^4*x^3-2*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)^2*RootOf(_Z^3-4)^3*x^3+RootOf(_Z^3-4)^
2*x^6-2*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)*x^6+6*(x^6+2*x^3+2*x+1)^(1/3)*RootO
f(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)^2*x^2+4*RootOf(_Z^3-4)^2*x^3-8*RootOf(RootOf(_Z^
3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)*x^3-6*RootOf(_Z^3-4)*(x^6+2*x^3+2*x+1)^(2/3)*x-12*RootOf(Roo
tOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*(x^6+2*x^3+2*x+1)^(2/3)*x+2*RootOf(_Z^3-4)^2*x-4*RootOf(RootOf(_Z^3-
4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)*x+RootOf(_Z^3-4)^2-2*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-
4)+4*_Z^2)*RootOf(_Z^3-4))/(1+x)/(x^5-x^4+x^3-x^2+x+1))*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3 \, x^{6} - 4 \, x - 3}{{\left (x^{6} + 2 \, x^{3} + 2 \, x + 1\right )}^{\frac {1}{3}} {\left (x^{6} + 2 \, x + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^6-4*x-3)/(x^6+2*x+1)/(x^6+2*x^3+2*x+1)^(1/3),x, algorithm="maxima")

[Out]

integrate((3*x^6 - 4*x - 3)/((x^6 + 2*x^3 + 2*x + 1)^(1/3)*(x^6 + 2*x + 1)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {-3\,x^6+4\,x+3}{\left (x^6+2\,x+1\right )\,{\left (x^6+2\,x^3+2\,x+1\right )}^{1/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(4*x - 3*x^6 + 3)/((2*x + x^6 + 1)*(2*x + 2*x^3 + x^6 + 1)^(1/3)),x)

[Out]

int(-(4*x - 3*x^6 + 3)/((2*x + x^6 + 1)*(2*x + 2*x^3 + x^6 + 1)^(1/3)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3 x^{6} - 4 x - 3}{\sqrt [3]{\left (x^{2} + 1\right ) \left (x^{4} - x^{2} + 2 x + 1\right )} \left (x + 1\right ) \left (x^{5} - x^{4} + x^{3} - x^{2} + x + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x**6-4*x-3)/(x**6+2*x+1)/(x**6+2*x**3+2*x+1)**(1/3),x)

[Out]

Integral((3*x**6 - 4*x - 3)/(((x**2 + 1)*(x**4 - x**2 + 2*x + 1))**(1/3)*(x + 1)*(x**5 - x**4 + x**3 - x**2 +
x + 1)), x)

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