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

Optimal. Leaf size=76 \[ -\frac {1}{6} \log \left (\sqrt [3]{x^6-1}+1\right )+\frac {1}{12} \log \left (\left (x^6-1\right )^{2/3}-\sqrt [3]{x^6-1}+1\right )-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{x^6-1}}{\sqrt {3}}\right )}{2 \sqrt {3}} \]

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Rubi [A]  time = 0.04, antiderivative size = 52, normalized size of antiderivative = 0.68, number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {266, 56, 618, 204, 31} \begin {gather*} -\frac {1}{4} \log \left (\sqrt [3]{x^6-1}+1\right )-\frac {\tan ^{-1}\left (\frac {1-2 \sqrt [3]{x^6-1}}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {\log (x)}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*(-1 + x^6)^(1/3)),x]

[Out]

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 56

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[-((b*c - a*d)/b), 3]}, Simp
[Log[RemoveContent[a + b*x, x]]/(2*b*q), x] + (Dist[3/(2*b), Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d*x)^(
1/3)], x] - Dist[3/(2*b*q), Subst[Int[1/(q + x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && Ne
gQ[(b*c - a*d)/b]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 266

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

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rubi steps

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

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Mathematica [C]  time = 0.01, size = 28, normalized size = 0.37 \begin {gather*} \frac {1}{4} \left (x^6-1\right )^{2/3} \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};1-x^6\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(-1 + x^6)^(1/3)),x]

[Out]

((-1 + x^6)^(2/3)*Hypergeometric2F1[2/3, 1, 5/3, 1 - x^6])/4

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IntegrateAlgebraic [A]  time = 0.06, size = 76, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-1+x^6}}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{6} \log \left (1+\sqrt [3]{-1+x^6}\right )+\frac {1}{12} \log \left (1-\sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.45, size = 58, normalized size = 0.76 \begin {gather*} \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + \frac {1}{12} \, \log \left ({\left (x^{6} - 1\right )}^{\frac {2}{3}} - {\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{6} \, \log \left ({\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.26, size = 57, normalized size = 0.75 \begin {gather*} \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {1}{12} \, \log \left ({\left (x^{6} - 1\right )}^{\frac {2}{3}} - {\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{6} \, \log \left ({\left | {\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 7.24, size = 83, normalized size = 1.09

method	result	size
meijerg	\(\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (-\mathrm {signum}\left (x^{6}-1\right )\right )^{\frac {1}{3}} \left (\frac {2 \pi \sqrt {3}\, x^{6} \hypergeom \left (\left [1, 1, \frac {4}{3}\right ], \left [2, 2\right ], x^{6}\right )}{9 \Gamma \left (\frac {2}{3}\right )}+\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+6 \ln \relax (x )+i \pi \right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}\right )}{12 \pi \mathrm {signum}\left (x^{6}-1\right )^{\frac {1}{3}}}\)	\(83\)
trager	\(\frac {\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (-\frac {1505024575790858565046 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{6}-94448748446849318458973 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{6}-23171389162410581752275 x^{6}+68267310132857019576606 \left (x^{6}-1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+142296551347461340528569 \left (x^{6}-1\right )^{\frac {2}{3}}-68267310132857019576606 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}}-96321572850614948162944 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-142296551347461340528569 \left (x^{6}-1\right )^{\frac {1}{3}}+164588882983471967739550 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+45974978496846392365625}{x^{6}}\right )}{6}-\frac {\ln \left (\frac {-1505024575790858565046 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{6}-91438699295267601328881 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{6}+116115113033469041646202 x^{6}+68267310132857019576606 \left (x^{6}-1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-210563861480318360105175 \left (x^{6}-1\right )^{\frac {2}{3}}-68267310132857019576606 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}}+96321572850614948162944 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}+210563861480318360105175 \left (x^{6}-1\right )^{\frac {1}{3}}-28054262717757928586338 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-114242288629703411942231}{x^{6}}\right ) \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )}{6}+\frac {\ln \left (\frac {-1505024575790858565046 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{6}-91438699295267601328881 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{6}+116115113033469041646202 x^{6}+68267310132857019576606 \left (x^{6}-1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-210563861480318360105175 \left (x^{6}-1\right )^{\frac {2}{3}}-68267310132857019576606 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}}+96321572850614948162944 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}+210563861480318360105175 \left (x^{6}-1\right )^{\frac {1}{3}}-28054262717757928586338 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-114242288629703411942231}{x^{6}}\right )}{6}\)	\(387\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12/Pi*3^(1/2)*GAMMA(2/3)/signum(x^6-1)^(1/3)*(-signum(x^6-1))^(1/3)*(2/9*Pi*3^(1/2)/GAMMA(2/3)*x^6*hypergeom
([1,1,4/3],[2,2],x^6)+2/3*(-1/6*Pi*3^(1/2)-3/2*ln(3)+6*ln(x)+I*Pi)*Pi*3^(1/2)/GAMMA(2/3))

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maxima [A]  time = 0.43, size = 56, normalized size = 0.74 \begin {gather*} \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {1}{12} \, \log \left ({\left (x^{6} - 1\right )}^{\frac {2}{3}} - {\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{6} \, \log \left ({\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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mupad [B]  time = 1.10, size = 80, normalized size = 1.05 \begin {gather*} -\frac {\ln \left (\frac {{\left (x^6-1\right )}^{1/3}}{4}+\frac {1}{4}\right )}{6}-\ln \left (9\,{\left (-\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )}^2+\frac {{\left (x^6-1\right )}^{1/3}}{4}\right )\,\left (-\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )+\ln \left (9\,{\left (\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )}^2+\frac {{\left (x^6-1\right )}^{1/3}}{4}\right )\,\left (\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(x^6 - 1)^(1/3)),x)

[Out]

log(9*((3^(1/2)*1i)/12 + 1/12)^2 + (x^6 - 1)^(1/3)/4)*((3^(1/2)*1i)/12 + 1/12) - log(9*((3^(1/2)*1i)/12 - 1/12
)^2 + (x^6 - 1)^(1/3)/4)*((3^(1/2)*1i)/12 - 1/12) - log((x^6 - 1)^(1/3)/4 + 1/4)/6

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sympy [C]  time = 0.80, size = 32, normalized size = 0.42 \begin {gather*} - \frac {\Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{6}}} \right )}}{6 x^{2} \Gamma \left (\frac {4}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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