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

Optimal. Leaf size=82 \[ \log \left (\sqrt [3]{x^6+x^2}-x\right )-\frac {1}{2} \log \left (x^2+\sqrt [3]{x^6+x^2} x+\left (x^6+x^2\right )^{2/3}\right )-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^6+x^2}+x}\right ) \]

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Rubi [F]  time = 1.06, 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 {-1+3 x^4}{\left (1-x+x^4\right ) \sqrt [3]{x^2+x^6}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(9*x*(1 + x^4)^(1/3)*Hypergeometric2F1[1/12, 1/3, 13/12, -x^4])/(x^2 + x^6)^(1/3) - (12*x^(2/3)*(1 + x^4)^(1/3
)*Defer[Subst][Defer[Int][1/((1 + x^12)^(1/3)*(1 - x^3 + x^12)), x], x, x^(1/3)])/(x^2 + x^6)^(1/3) + (9*x^(2/
3)*(1 + x^4)^(1/3)*Defer[Subst][Defer[Int][x^3/((1 + x^12)^(1/3)*(1 - x^3 + x^12)), x], x, x^(1/3)])/(x^2 + x^
6)^(1/3)

Rubi steps

\begin {align*} \int \frac {-1+3 x^4}{\left (1-x+x^4\right ) \sqrt [3]{x^2+x^6}} \, dx &=\frac {\left (x^{2/3} \sqrt [3]{1+x^4}\right ) \int \frac {-1+3 x^4}{x^{2/3} \sqrt [3]{1+x^4} \left (1-x+x^4\right )} \, dx}{\sqrt [3]{x^2+x^6}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {-1+3 x^{12}}{\sqrt [3]{1+x^{12}} \left (1-x^3+x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^6}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {3}{\sqrt [3]{1+x^{12}}}-\frac {4-3 x^3}{\sqrt [3]{1+x^{12}} \left (1-x^3+x^{12}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^6}}\\ &=-\frac {\left (3 x^{2/3} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {4-3 x^3}{\sqrt [3]{1+x^{12}} \left (1-x^3+x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^6}}+\frac {\left (9 x^{2/3} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^{12}}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^6}}\\ &=\frac {9 x \sqrt [3]{1+x^4} \, _2F_1\left (\frac {1}{12},\frac {1}{3};\frac {13}{12};-x^4\right )}{\sqrt [3]{x^2+x^6}}-\frac {\left (3 x^{2/3} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {4}{\sqrt [3]{1+x^{12}} \left (1-x^3+x^{12}\right )}-\frac {3 x^3}{\sqrt [3]{1+x^{12}} \left (1-x^3+x^{12}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^6}}\\ &=\frac {9 x \sqrt [3]{1+x^4} \, _2F_1\left (\frac {1}{12},\frac {1}{3};\frac {13}{12};-x^4\right )}{\sqrt [3]{x^2+x^6}}+\frac {\left (9 x^{2/3} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\sqrt [3]{1+x^{12}} \left (1-x^3+x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^6}}-\frac {\left (12 x^{2/3} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^{12}} \left (1-x^3+x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^6}}\\ \end {align*}

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Mathematica [F]  time = 0.20, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-1+3 x^4}{\left (1-x+x^4\right ) \sqrt [3]{x^2+x^6}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 2.66, size = 82, normalized size = 1.00 \begin {gather*} -\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x^2+x^6}}\right )+\log \left (-x+\sqrt [3]{x^2+x^6}\right )-\frac {1}{2} \log \left (x^2+x \sqrt [3]{x^2+x^6}+\left (x^2+x^6\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.81, size = 112, normalized size = 1.37 \begin {gather*} -\sqrt {3} \arctan \left (-\frac {2 \, \sqrt {3} {\left (x^{6} + x^{2}\right )}^{\frac {1}{3}} x + \sqrt {3} {\left (x^{5} - x^{2} + x\right )} - 2 \, \sqrt {3} {\left (x^{6} + x^{2}\right )}^{\frac {2}{3}}}{3 \, {\left (x^{5} + x^{2} + x\right )}}\right ) + \frac {1}{2} \, \log \left (\frac {x^{5} - x^{2} + 3 \, {\left (x^{6} + x^{2}\right )}^{\frac {1}{3}} x + x - 3 \, {\left (x^{6} + x^{2}\right )}^{\frac {2}{3}}}{x^{5} - x^{2} + x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(3)*arctan(-1/3*(2*sqrt(3)*(x^6 + x^2)^(1/3)*x + sqrt(3)*(x^5 - x^2 + x) - 2*sqrt(3)*(x^6 + x^2)^(2/3))/(
x^5 + x^2 + x)) + 1/2*log((x^5 - x^2 + 3*(x^6 + x^2)^(1/3)*x + x - 3*(x^6 + x^2)^(2/3))/(x^5 - x^2 + x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 4.34, size = 511, normalized size = 6.23

method	result	size
trager	\(\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (-\frac {-1823 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{5}-1101 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{5}+9494 x^{5}+3646 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}+12039 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}+x^{2}\right )^{\frac {2}{3}}+12039 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}+x^{2}\right )^{\frac {1}{3}} x -1823 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x +14963 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+21912 \left (x^{6}+x^{2}\right )^{\frac {2}{3}}+21912 x \left (x^{6}+x^{2}\right )^{\frac {1}{3}}-1101 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x +14241 x^{2}+9494 x}{x \left (x^{4}-x +1\right )}\right )-\ln \left (\frac {1823 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{5}+2545 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{5}-8772 x^{5}-3646 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}+12039 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}+x^{2}\right )^{\frac {2}{3}}+12039 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}+x^{2}\right )^{\frac {1}{3}} x +1823 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x +7671 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-9873 \left (x^{6}+x^{2}\right )^{\frac {2}{3}}-9873 x \left (x^{6}+x^{2}\right )^{\frac {1}{3}}+2545 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x -2924 x^{2}-8772 x}{x \left (x^{4}-x +1\right )}\right ) \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-\ln \left (\frac {1823 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{5}+2545 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{5}-8772 x^{5}-3646 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}+12039 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}+x^{2}\right )^{\frac {2}{3}}+12039 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}+x^{2}\right )^{\frac {1}{3}} x +1823 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x +7671 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-9873 \left (x^{6}+x^{2}\right )^{\frac {2}{3}}-9873 x \left (x^{6}+x^{2}\right )^{\frac {1}{3}}+2545 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x -2924 x^{2}-8772 x}{x \left (x^{4}-x +1\right )}\right )\)	\(511\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootOf(_Z^2+_Z+1)*ln(-(-1823*RootOf(_Z^2+_Z+1)^2*x^5-1101*RootOf(_Z^2+_Z+1)*x^5+9494*x^5+3646*RootOf(_Z^2+_Z+1
)^2*x^2+12039*RootOf(_Z^2+_Z+1)*(x^6+x^2)^(2/3)+12039*RootOf(_Z^2+_Z+1)*(x^6+x^2)^(1/3)*x-1823*RootOf(_Z^2+_Z+
1)^2*x+14963*RootOf(_Z^2+_Z+1)*x^2+21912*(x^6+x^2)^(2/3)+21912*x*(x^6+x^2)^(1/3)-1101*RootOf(_Z^2+_Z+1)*x+1424
1*x^2+9494*x)/x/(x^4-x+1))-ln((1823*RootOf(_Z^2+_Z+1)^2*x^5+2545*RootOf(_Z^2+_Z+1)*x^5-8772*x^5-3646*RootOf(_Z
^2+_Z+1)^2*x^2+12039*RootOf(_Z^2+_Z+1)*(x^6+x^2)^(2/3)+12039*RootOf(_Z^2+_Z+1)*(x^6+x^2)^(1/3)*x+1823*RootOf(_
Z^2+_Z+1)^2*x+7671*RootOf(_Z^2+_Z+1)*x^2-9873*(x^6+x^2)^(2/3)-9873*x*(x^6+x^2)^(1/3)+2545*RootOf(_Z^2+_Z+1)*x-
2924*x^2-8772*x)/x/(x^4-x+1))*RootOf(_Z^2+_Z+1)-ln((1823*RootOf(_Z^2+_Z+1)^2*x^5+2545*RootOf(_Z^2+_Z+1)*x^5-87
72*x^5-3646*RootOf(_Z^2+_Z+1)^2*x^2+12039*RootOf(_Z^2+_Z+1)*(x^6+x^2)^(2/3)+12039*RootOf(_Z^2+_Z+1)*(x^6+x^2)^
(1/3)*x+1823*RootOf(_Z^2+_Z+1)^2*x+7671*RootOf(_Z^2+_Z+1)*x^2-9873*(x^6+x^2)^(2/3)-9873*x*(x^6+x^2)^(1/3)+2545
*RootOf(_Z^2+_Z+1)*x-2924*x^2-8772*x)/x/(x^4-x+1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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