3.15.84 \(\int \frac {1}{x^{19} \sqrt [3]{-1+x^6}} \, dx\)

Optimal. Leaf size=104 \[ -\frac {7}{243} \log \left (\sqrt [3]{x^6-1}+1\right )+\frac {7}{486} \log \left (\left (x^6-1\right )^{2/3}-\sqrt [3]{x^6-1}+1\right )-\frac {7 \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{x^6-1}}{\sqrt {3}}\right )}{81 \sqrt {3}}+\frac {\left (x^6-1\right )^{2/3} \left (28 x^{12}+21 x^6+18\right )}{324 x^{18}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.07, antiderivative size = 100, normalized size of antiderivative = 0.96, number of steps used = 8, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {266, 51, 56, 618, 204, 31} \begin {gather*} \frac {7 \left (x^6-1\right )^{2/3}}{81 x^6}-\frac {7}{162} \log \left (\sqrt [3]{x^6-1}+1\right )-\frac {7 \tan ^{-1}\left (\frac {1-2 \sqrt [3]{x^6-1}}{\sqrt {3}}\right )}{81 \sqrt {3}}+\frac {\left (x^6-1\right )^{2/3}}{18 x^{18}}+\frac {7 \left (x^6-1\right )^{2/3}}{108 x^{12}}+\frac {7 \log (x)}{81} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^19*(-1 + x^6)^(1/3)),x]

[Out]

(-1 + x^6)^(2/3)/(18*x^18) + (7*(-1 + x^6)^(2/3))/(108*x^12) + (7*(-1 + x^6)^(2/3))/(81*x^6) - (7*ArcTan[(1 -
2*(-1 + x^6)^(1/3))/Sqrt[3]])/(81*Sqrt[3]) + (7*Log[x])/81 - (7*Log[1 + (-1 + x^6)^(1/3)])/162

Rule 31

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

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, 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^{19} \sqrt [3]{-1+x^6}} \, dx &=\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x} x^4} \, dx,x,x^6\right )\\ &=\frac {\left (-1+x^6\right )^{2/3}}{18 x^{18}}+\frac {7}{54} \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x} x^3} \, dx,x,x^6\right )\\ &=\frac {\left (-1+x^6\right )^{2/3}}{18 x^{18}}+\frac {7 \left (-1+x^6\right )^{2/3}}{108 x^{12}}+\frac {7}{81} \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x} x^2} \, dx,x,x^6\right )\\ &=\frac {\left (-1+x^6\right )^{2/3}}{18 x^{18}}+\frac {7 \left (-1+x^6\right )^{2/3}}{108 x^{12}}+\frac {7 \left (-1+x^6\right )^{2/3}}{81 x^6}+\frac {7}{243} \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x} x} \, dx,x,x^6\right )\\ &=\frac {\left (-1+x^6\right )^{2/3}}{18 x^{18}}+\frac {7 \left (-1+x^6\right )^{2/3}}{108 x^{12}}+\frac {7 \left (-1+x^6\right )^{2/3}}{81 x^6}+\frac {7 \log (x)}{81}-\frac {7}{162} \operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sqrt [3]{-1+x^6}\right )+\frac {7}{162} \operatorname {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt [3]{-1+x^6}\right )\\ &=\frac {\left (-1+x^6\right )^{2/3}}{18 x^{18}}+\frac {7 \left (-1+x^6\right )^{2/3}}{108 x^{12}}+\frac {7 \left (-1+x^6\right )^{2/3}}{81 x^6}+\frac {7 \log (x)}{81}-\frac {7}{162} \log \left (1+\sqrt [3]{-1+x^6}\right )-\frac {7}{81} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \sqrt [3]{-1+x^6}\right )\\ &=\frac {\left (-1+x^6\right )^{2/3}}{18 x^{18}}+\frac {7 \left (-1+x^6\right )^{2/3}}{108 x^{12}}+\frac {7 \left (-1+x^6\right )^{2/3}}{81 x^6}-\frac {7 \tan ^{-1}\left (\frac {1-2 \sqrt [3]{-1+x^6}}{\sqrt {3}}\right )}{81 \sqrt {3}}+\frac {7 \log (x)}{81}-\frac {7}{162} \log \left (1+\sqrt [3]{-1+x^6}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.01, size = 28, normalized size = 0.27 \begin {gather*} \frac {1}{4} \left (x^6-1\right )^{2/3} \, _2F_1\left (\frac {2}{3},4;\frac {5}{3};1-x^6\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^19*(-1 + x^6)^(1/3)),x]

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.18, size = 104, normalized size = 1.00 \begin {gather*} \frac {\left (-1+x^6\right )^{2/3} \left (18+21 x^6+28 x^{12}\right )}{324 x^{18}}-\frac {7 \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-1+x^6}}{\sqrt {3}}\right )}{81 \sqrt {3}}-\frac {7}{243} \log \left (1+\sqrt [3]{-1+x^6}\right )+\frac {7}{486} \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^19*(-1 + x^6)^(1/3)),x]

[Out]

((-1 + x^6)^(2/3)*(18 + 21*x^6 + 28*x^12))/(324*x^18) - (7*ArcTan[1/Sqrt[3] - (2*(-1 + x^6)^(1/3))/Sqrt[3]])/(
81*Sqrt[3]) - (7*Log[1 + (-1 + x^6)^(1/3)])/243 + (7*Log[1 - (-1 + x^6)^(1/3) + (-1 + x^6)^(2/3)])/486

________________________________________________________________________________________

fricas [A]  time = 0.75, size = 93, normalized size = 0.89 \begin {gather*} \frac {28 \, \sqrt {3} x^{18} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + 14 \, x^{18} \log \left ({\left (x^{6} - 1\right )}^{\frac {2}{3}} - {\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1\right ) - 28 \, x^{18} \log \left ({\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1\right ) + 3 \, {\left (28 \, x^{12} + 21 \, x^{6} + 18\right )} {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{972 \, x^{18}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^19/(x^6-1)^(1/3),x, algorithm="fricas")

[Out]

1/972*(28*sqrt(3)*x^18*arctan(2/3*sqrt(3)*(x^6 - 1)^(1/3) - 1/3*sqrt(3)) + 14*x^18*log((x^6 - 1)^(2/3) - (x^6
- 1)^(1/3) + 1) - 28*x^18*log((x^6 - 1)^(1/3) + 1) + 3*(28*x^12 + 21*x^6 + 18)*(x^6 - 1)^(2/3))/x^18

________________________________________________________________________________________

giac [A]  time = 0.35, size = 90, normalized size = 0.87 \begin {gather*} \frac {7}{243} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {28 \, {\left (x^{6} - 1\right )}^{\frac {8}{3}} + 77 \, {\left (x^{6} - 1\right )}^{\frac {5}{3}} + 67 \, {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{324 \, x^{18}} + \frac {7}{486} \, \log \left ({\left (x^{6} - 1\right )}^{\frac {2}{3}} - {\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {7}{243} \, \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^19/(x^6-1)^(1/3),x, algorithm="giac")

[Out]

7/243*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^6 - 1)^(1/3) - 1)) + 1/324*(28*(x^6 - 1)^(8/3) + 77*(x^6 - 1)^(5/3) + 6
7*(x^6 - 1)^(2/3))/x^18 + 7/486*log((x^6 - 1)^(2/3) - (x^6 - 1)^(1/3) + 1) - 7/243*log(abs((x^6 - 1)^(1/3) + 1
))

________________________________________________________________________________________

maple [C]  time = 7.85, size = 113, normalized size = 1.09

method	result	size
risch	\(\frac {28 x^{18}-7 x^{12}-3 x^{6}-18}{324 x^{18} \left (x^{6}-1\right )^{\frac {1}{3}}}+\frac {7 \sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (-\mathrm {signum}\left (x^{6}-1\right )\right )^{\frac {1}{3}} \left (\frac {2 \pi \sqrt {3}\, x^{6} \hypergeom \left (\left [1, 1, \frac {4}{3}\right ], \left [2, 2\right ], x^{6}\right )}{9 \Gamma \left (\frac {2}{3}\right )}+\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+6 \ln \relax (x )+i \pi \right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}\right )}{486 \pi \mathrm {signum}\left (x^{6}-1\right )^{\frac {1}{3}}}\)	\(113\)
meijerg	\(-\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (-\mathrm {signum}\left (x^{6}-1\right )\right )^{\frac {1}{3}} \left (-\frac {70 \pi \sqrt {3}\, x^{6} \hypergeom \left (\left [1, 1, \frac {13}{3}\right ], \left [2, 5\right ], x^{6}\right )}{729 \Gamma \left (\frac {2}{3}\right )}-\frac {28 \left (\frac {197}{84}-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+6 \ln \relax (x )+i \pi \right ) \pi \sqrt {3}}{243 \Gamma \left (\frac {2}{3}\right )}+\frac {2 \pi \sqrt {3}}{9 \Gamma \left (\frac {2}{3}\right ) x^{18}}+\frac {\pi \sqrt {3}}{9 \Gamma \left (\frac {2}{3}\right ) x^{12}}+\frac {4 \pi \sqrt {3}}{27 \Gamma \left (\frac {2}{3}\right ) x^{6}}\right )}{12 \pi \mathrm {signum}\left (x^{6}-1\right )^{\frac {1}{3}}}\)	\(123\)
trager	\(\frac {\left (x^{6}-1\right )^{\frac {2}{3}} \left (28 x^{12}+21 x^{6}+18\right )}{324 x^{18}}+\frac {12845056 \RootOf \left (272747603165184 \textit {\_Z}^{2}-16515072 \textit {\_Z} +1\right ) \ln \left (-\frac {410491845751654482455753265946558464 \RootOf \left (272747603165184 \textit {\_Z}^{2}-16515072 \textit {\_Z} +1\right )^{2} x^{6}-1559827880909604667500868141056 \RootOf \left (272747603165184 \textit {\_Z}^{2}-16515072 \textit {\_Z} +1\right ) x^{6}-23171389162410581752275 x^{6}+1127439542090463244013057605632 \left (x^{6}-1\right )^{\frac {2}{3}} \RootOf \left (272747603165184 \textit {\_Z}^{2}-16515072 \textit {\_Z} +1\right )-26271478128105886877168209020579741696 \RootOf \left (272747603165184 \textit {\_Z}^{2}-16515072 \textit {\_Z} +1\right )^{2}-1127439542090463244013057605632 \RootOf \left (272747603165184 \textit {\_Z}^{2}-16515072 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}}+142296551347461340528569 \left (x^{6}-1\right )^{\frac {2}{3}}+2718197252871614357200345497600 \RootOf \left (272747603165184 \textit {\_Z}^{2}-16515072 \textit {\_Z} +1\right )-142296551347461340528569 \left (x^{6}-1\right )^{\frac {1}{3}}+45974978496846392365625}{x^{6}}\right )}{27}+\frac {7 \ln \left (\frac {151613680100373352009382957678592 \RootOf \left (272747603165184 \textit {\_Z}^{2}-16515072 \textit {\_Z} +1\right )^{2} x^{6}+549259830886362994300944384 \RootOf \left (272747603165184 \textit {\_Z}^{2}-16515072 \textit {\_Z} +1\right ) x^{6}-74783116455558560258 x^{6}+127348568797268153299894272 \left (x^{6}-1\right )^{\frac {2}{3}} \RootOf \left (272747603165184 \textit {\_Z}^{2}-16515072 \textit {\_Z} +1\right )-9703275526423894528600509291429888 \RootOf \left (272747603165184 \textit {\_Z}^{2}-16515072 \textit {\_Z} +1\right )^{2}-127348568797268153299894272 \RootOf \left (272747603165184 \textit {\_Z}^{2}-16515072 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}}+109152962075501576931 \left (x^{6}-1\right )^{\frac {2}{3}}+714889181785446135727128576 \RootOf \left (272747603165184 \textit {\_Z}^{2}-16515072 \textit {\_Z} +1\right )-109152962075501576931 \left (x^{6}-1\right )^{\frac {1}{3}}+73576937157888260899}{x^{6}}\right )}{243}-\frac {12845056 \ln \left (\frac {151613680100373352009382957678592 \RootOf \left (272747603165184 \textit {\_Z}^{2}-16515072 \textit {\_Z} +1\right )^{2} x^{6}+549259830886362994300944384 \RootOf \left (272747603165184 \textit {\_Z}^{2}-16515072 \textit {\_Z} +1\right ) x^{6}-74783116455558560258 x^{6}+127348568797268153299894272 \left (x^{6}-1\right )^{\frac {2}{3}} \RootOf \left (272747603165184 \textit {\_Z}^{2}-16515072 \textit {\_Z} +1\right )-9703275526423894528600509291429888 \RootOf \left (272747603165184 \textit {\_Z}^{2}-16515072 \textit {\_Z} +1\right )^{2}-127348568797268153299894272 \RootOf \left (272747603165184 \textit {\_Z}^{2}-16515072 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}}+109152962075501576931 \left (x^{6}-1\right )^{\frac {2}{3}}+714889181785446135727128576 \RootOf \left (272747603165184 \textit {\_Z}^{2}-16515072 \textit {\_Z} +1\right )-109152962075501576931 \left (x^{6}-1\right )^{\frac {1}{3}}+73576937157888260899}{x^{6}}\right ) \RootOf \left (272747603165184 \textit {\_Z}^{2}-16515072 \textit {\_Z} +1\right )}{27}\)	\(451\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^19/(x^6-1)^(1/3),x,method=_RETURNVERBOSE)

[Out]

1/324*(28*x^18-7*x^12-3*x^6-18)/x^18/(x^6-1)^(1/3)+7/486/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))

________________________________________________________________________________________

maxima [A]  time = 0.43, size = 111, normalized size = 1.07 \begin {gather*} \frac {7}{243} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {28 \, {\left (x^{6} - 1\right )}^{\frac {8}{3}} + 77 \, {\left (x^{6} - 1\right )}^{\frac {5}{3}} + 67 \, {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{324 \, {\left (3 \, x^{6} + {\left (x^{6} - 1\right )}^{3} + 3 \, {\left (x^{6} - 1\right )}^{2} - 2\right )}} + \frac {7}{486} \, \log \left ({\left (x^{6} - 1\right )}^{\frac {2}{3}} - {\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {7}{243} \, \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^19/(x^6-1)^(1/3),x, algorithm="maxima")

[Out]

7/243*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^6 - 1)^(1/3) - 1)) + 1/324*(28*(x^6 - 1)^(8/3) + 77*(x^6 - 1)^(5/3) + 6
7*(x^6 - 1)^(2/3))/(3*x^6 + (x^6 - 1)^3 + 3*(x^6 - 1)^2 - 2) + 7/486*log((x^6 - 1)^(2/3) - (x^6 - 1)^(1/3) + 1
) - 7/243*log((x^6 - 1)^(1/3) + 1)

________________________________________________________________________________________

mupad [B]  time = 1.06, size = 134, normalized size = 1.29 \begin {gather*} \frac {\frac {67\,{\left (x^6-1\right )}^{2/3}}{324}+\frac {77\,{\left (x^6-1\right )}^{5/3}}{324}+\frac {7\,{\left (x^6-1\right )}^{8/3}}{81}}{3\,{\left (x^6-1\right )}^2+{\left (x^6-1\right )}^3+3\,x^6-2}-\ln \left (9\,{\left (-\frac {7}{486}+\frac {\sqrt {3}\,7{}\mathrm {i}}{486}\right )}^2+\frac {49\,{\left (x^6-1\right )}^{1/3}}{6561}\right )\,\left (-\frac {7}{486}+\frac {\sqrt {3}\,7{}\mathrm {i}}{486}\right )+\ln \left (9\,{\left (\frac {7}{486}+\frac {\sqrt {3}\,7{}\mathrm {i}}{486}\right )}^2+\frac {49\,{\left (x^6-1\right )}^{1/3}}{6561}\right )\,\left (\frac {7}{486}+\frac {\sqrt {3}\,7{}\mathrm {i}}{486}\right )-\frac {7\,\ln \left (\frac {49\,{\left (x^6-1\right )}^{1/3}}{6561}+\frac {49}{6561}\right )}{243} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^19*(x^6 - 1)^(1/3)),x)

[Out]

log(9*((3^(1/2)*7i)/486 + 7/486)^2 + (49*(x^6 - 1)^(1/3))/6561)*((3^(1/2)*7i)/486 + 7/486) - log(9*((3^(1/2)*7
i)/486 - 7/486)^2 + (49*(x^6 - 1)^(1/3))/6561)*((3^(1/2)*7i)/486 - 7/486) - (7*log((49*(x^6 - 1)^(1/3))/6561 +
49/6561))/243 + ((67*(x^6 - 1)^(2/3))/324 + (77*(x^6 - 1)^(5/3))/324 + (7*(x^6 - 1)^(8/3))/81)/(3*(x^6 - 1)^2
+ (x^6 - 1)^3 + 3*x^6 - 2)

________________________________________________________________________________________

sympy [C]  time = 2.41, size = 32, normalized size = 0.31 \begin {gather*} - \frac {\Gamma \left (\frac {10}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {10}{3} \\ \frac {13}{3} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{6}}} \right )}}{6 x^{20} \Gamma \left (\frac {13}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**19/(x**6-1)**(1/3),x)

[Out]

-gamma(10/3)*hyper((1/3, 10/3), (13/3,), exp_polar(2*I*pi)/x**6)/(6*x**20*gamma(13/3))

________________________________________________________________________________________