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

Optimal. Leaf size=117 \[ -\log \left (\sqrt [3]{x^6-x^4-x^3+1}+x\right )-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^6-x^4-x^3+1}-x}\right )+\frac {1}{2} \log \left (x^2-\sqrt [3]{x^6-x^4-x^3+1} x+\left (x^6-x^4-x^3+1\right )^{2/3}\right ) \]

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Rubi [F]  time = 0.89, 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-x^4+3 x^6}{\left (1-x^4+x^6\right ) \sqrt [3]{1-x^3-x^4+x^6}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 0.25, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-3-x^4+3 x^6}{\left (1-x^4+x^6\right ) \sqrt [3]{1-x^3-x^4+x^6}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.42, size = 117, normalized size = 1.00 \begin {gather*} -\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{1-x^3-x^4+x^6}}\right )-\log \left (x+\sqrt [3]{1-x^3-x^4+x^6}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{1-x^3-x^4+x^6}+\left (1-x^3-x^4+x^6\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[3]*ArcTan[(Sqrt[3]*x)/(-x + 2*(1 - x^3 - x^4 + x^6)^(1/3))]) - Log[x + (1 - x^3 - x^4 + x^6)^(1/3)] + L
og[x^2 - x*(1 - x^3 - x^4 + x^6)^(1/3) + (1 - x^3 - x^4 + x^6)^(2/3)]/2

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fricas [A]  time = 2.88, size = 157, normalized size = 1.34 \begin {gather*} -\sqrt {3} \arctan \left (\frac {2 \, \sqrt {3} {\left (x^{6} - x^{4} - x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 2 \, \sqrt {3} {\left (x^{6} - x^{4} - x^{3} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (x^{6} - x^{4} + 1\right )}}{3 \, {\left (x^{6} - x^{4} - 2 \, x^{3} + 1\right )}}\right ) - \frac {1}{2} \, \log \left (\frac {x^{6} - x^{4} + 3 \, {\left (x^{6} - x^{4} - x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{6} - x^{4} - x^{3} + 1\right )}^{\frac {2}{3}} x + 1}{x^{6} - x^{4} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(3)*arctan(1/3*(2*sqrt(3)*(x^6 - x^4 - x^3 + 1)^(1/3)*x^2 + 2*sqrt(3)*(x^6 - x^4 - x^3 + 1)^(2/3)*x + sqr
t(3)*(x^6 - x^4 + 1))/(x^6 - x^4 - 2*x^3 + 1)) - 1/2*log((x^6 - x^4 + 3*(x^6 - x^4 - x^3 + 1)^(1/3)*x^2 + 3*(x
^6 - x^4 - x^3 + 1)^(2/3)*x + 1)/(x^6 - x^4 + 1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 3.49, size = 340, normalized size = 2.91

method	result	size
trager	\(-\ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{6}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{4}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-x^{4}-x^{3}+1\right )^{\frac {2}{3}} x +\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-x^{4}-x^{3}+1\right )^{\frac {1}{3}} x^{2}-2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}-\left (x^{6}-x^{4}-x^{3}+1\right )^{\frac {2}{3}} x -2 \left (x^{6}-x^{4}-x^{3}+1\right )^{\frac {1}{3}} x^{2}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )}{x^{6}-x^{4}+1}\right )+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (\frac {x^{6}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-x^{4}-x^{3}+1\right )^{\frac {2}{3}} x +\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-x^{4}-x^{3}+1\right )^{\frac {1}{3}} x^{2}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}-x^{4}-\left (x^{6}-x^{4}-x^{3}+1\right )^{\frac {2}{3}} x -2 \left (x^{6}-x^{4}-x^{3}+1\right )^{\frac {1}{3}} x^{2}-x^{3}+1}{x^{6}-x^{4}+1}\right )\)	\(340\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-ln(-(RootOf(_Z^2-_Z+1)*x^6+RootOf(_Z^2-_Z+1)^2*x^3-RootOf(_Z^2-_Z+1)*x^4+2*RootOf(_Z^2-_Z+1)*(x^6-x^4-x^3+1)^
(2/3)*x+RootOf(_Z^2-_Z+1)*(x^6-x^4-x^3+1)^(1/3)*x^2-2*RootOf(_Z^2-_Z+1)*x^3-(x^6-x^4-x^3+1)^(2/3)*x-2*(x^6-x^4
-x^3+1)^(1/3)*x^2+RootOf(_Z^2-_Z+1))/(x^6-x^4+1))+RootOf(_Z^2-_Z+1)*ln((x^6+2*RootOf(_Z^2-_Z+1)*(x^6-x^4-x^3+1
)^(2/3)*x+RootOf(_Z^2-_Z+1)*(x^6-x^4-x^3+1)^(1/3)*x^2-RootOf(_Z^2-_Z+1)*x^3-x^4-(x^6-x^4-x^3+1)^(2/3)*x-2*(x^6
-x^4-x^3+1)^(1/3)*x^2-x^3+1)/(x^6-x^4+1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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