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

Optimal. Leaf size=88 \[ -\frac {1}{6} \log \left (\sqrt [3]{x^6-1}-x^2\right )+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x^2}{2 \sqrt [3]{x^6-1}+x^2}\right )}{2 \sqrt {3}}+\frac {1}{12} \log \left (\left (x^6-1\right )^{2/3}+x^4+\sqrt [3]{x^6-1} x^2\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 53, normalized size of antiderivative = 0.60, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {275, 239} \begin {gather*} \frac {\tan ^{-1}\left (\frac {\frac {2 x^2}{\sqrt [3]{x^6-1}}+1}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{4} \log \left (x^2-\sqrt [3]{x^6-1}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(-1 + x^6)^(1/3),x]

[Out]

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

Rule 239

Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + (2*Rt[b, 3]*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sq
rt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 83, normalized size = 0.94 \begin {gather*} \frac {1}{12} \left (-2 \log \left (1-\frac {x^2}{\sqrt [3]{x^6-1}}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 x^2}{\sqrt [3]{x^6-1}}+1}{\sqrt {3}}\right )+\log \left (\frac {x^4}{\left (x^6-1\right )^{2/3}}+\frac {x^2}{\sqrt [3]{x^6-1}}+1\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/(-1 + x^6)^(1/3),x]

[Out]

(2*Sqrt[3]*ArcTan[(1 + (2*x^2)/(-1 + x^6)^(1/3))/Sqrt[3]] - 2*Log[1 - x^2/(-1 + x^6)^(1/3)] + Log[1 + x^4/(-1
+ x^6)^(2/3) + x^2/(-1 + x^6)^(1/3)])/12

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IntegrateAlgebraic [A]  time = 0.61, size = 88, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{-1+x^6}}\right )}{2 \sqrt {3}}-\frac {1}{6} \log \left (-x^2+\sqrt [3]{-1+x^6}\right )+\frac {1}{12} \log \left (x^4+x^2 \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[(Sqrt[3]*x^2)/(x^2 + 2*(-1 + x^6)^(1/3))]/(2*Sqrt[3]) - Log[-x^2 + (-1 + x^6)^(1/3)]/6 + Log[x^4 + x^2*
(-1 + x^6)^(1/3) + (-1 + x^6)^(2/3)]/12

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fricas [A]  time = 0.49, size = 82, normalized size = 0.93 \begin {gather*} -\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2} + 2 \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{3 \, x^{2}}\right ) - \frac {1}{6} \, \log \left (-\frac {x^{2} - {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}}\right ) + \frac {1}{12} \, \log \left (\frac {x^{4} + {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{2} + {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{x^{4}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*sqrt(3)*arctan(1/3*(sqrt(3)*x^2 + 2*sqrt(3)*(x^6 - 1)^(1/3))/x^2) - 1/6*log(-(x^2 - (x^6 - 1)^(1/3))/x^2)
+ 1/12*log((x^4 + (x^6 - 1)^(1/3)*x^2 + (x^6 - 1)^(2/3))/x^4)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x/(x^6 - 1)^(1/3), x)

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maple [C]  time = 3.49, size = 33, normalized size = 0.38

method	result	size
meijerg	\(\frac {\left (-\mathrm {signum}\left (x^{6}-1\right )\right )^{\frac {1}{3}} x^{2} \hypergeom \left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x^{6}\right )}{2 \mathrm {signum}\left (x^{6}-1\right )^{\frac {1}{3}}}\)	\(33\)
trager	\(\frac {\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{6}-4 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{6}-3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{4}+4 x^{6}-3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x^{2}+3 x^{4} \left (x^{6}-1\right )^{\frac {1}{3}}+3 x^{2} \left (x^{6}-1\right )^{\frac {2}{3}}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-2\right )}{6}-\frac {\ln \left (\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{6}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{6}+3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{4}+x^{6}+3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x^{2}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-1\right ) \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )}{6}+\frac {\ln \left (\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{6}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{6}+3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{4}+x^{6}+3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x^{2}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-1\right )}{6}\)	\(314\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/signum(x^6-1)^(1/3)*(-signum(x^6-1))^(1/3)*x^2*hypergeom([1/3,1/3],[4/3],x^6)

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maxima [A]  time = 0.40, size = 69, normalized size = 0.78 \begin {gather*} -\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} + 1\right )}\right ) + \frac {1}{12} \, \log \left (\frac {{\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} + \frac {{\left (x^{6} - 1\right )}^{\frac {2}{3}}}{x^{4}} + 1\right ) - \frac {1}{6} \, \log \left (\frac {{\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^6 - 1)^(1/3)/x^2 + 1)) + 1/12*log((x^6 - 1)^(1/3)/x^2 + (x^6 - 1)^(2/3)/
x^4 + 1) - 1/6*log((x^6 - 1)^(1/3)/x^2 - 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x^6 - 1)^(1/3),x)

[Out]

int(x/(x^6 - 1)^(1/3), x)

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sympy [C]  time = 0.81, size = 31, normalized size = 0.35 \begin {gather*} \frac {x^{2} e^{- \frac {i \pi }{3}} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {x^{6}} \right )}}{6 \Gamma \left (\frac {4}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x**6-1)**(1/3),x)

[Out]

x**2*exp(-I*pi/3)*gamma(1/3)*hyper((1/3, 1/3), (4/3,), x**6)/(6*gamma(4/3))

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