3.15.70 \(\int \frac {\sqrt [3]{-1+x^3}}{x^{10}} \, dx\)

Optimal. Leaf size=104 \[ \frac {5}{243} \log \left (\sqrt [3]{x^3-1}+1\right )-\frac {5}{486} \log \left (\left (x^3-1\right )^{2/3}-\sqrt [3]{x^3-1}+1\right )-\frac {5 \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{x^3-1}}{\sqrt {3}}\right )}{81 \sqrt {3}}+\frac {\sqrt [3]{x^3-1} \left (5 x^6+3 x^3-18\right )}{162 x^9} \]

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Rubi [A]  time = 0.11, antiderivative size = 100, normalized size of antiderivative = 0.96, number of steps used = 8, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {266, 47, 51, 58, 618, 204, 31} \begin {gather*} \frac {5 \sqrt [3]{x^3-1}}{162 x^3}+\frac {5}{162} \log \left (\sqrt [3]{x^3-1}+1\right )-\frac {5 \tan ^{-1}\left (\frac {1-2 \sqrt [3]{x^3-1}}{\sqrt {3}}\right )}{81 \sqrt {3}}-\frac {\sqrt [3]{x^3-1}}{9 x^9}+\frac {\sqrt [3]{x^3-1}}{54 x^6}-\frac {5 \log (x)}{162} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + x^3)^(1/3)/x^10,x]

[Out]

-1/9*(-1 + x^3)^(1/3)/x^9 + (-1 + x^3)^(1/3)/(54*x^6) + (5*(-1 + x^3)^(1/3))/(162*x^3) - (5*ArcTan[(1 - 2*(-1
+ x^3)^(1/3))/Sqrt[3]])/(81*Sqrt[3]) - (5*Log[x])/162 + (5*Log[1 + (-1 + x^3)^(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 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, 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 58

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[-((b*c - a*d)/b), 3]}, -Sim
p[Log[RemoveContent[a + b*x, x]]/(2*b*q^2), x] + (Dist[3/(2*b*q), Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d
*x)^(1/3)], x] + Dist[3/(2*b*q^2), Subst[Int[1/(q + x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x
] && NegQ[(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 {\sqrt [3]{-1+x^3}}{x^{10}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1+x}}{x^4} \, dx,x,x^3\right )\\ &=-\frac {\sqrt [3]{-1+x^3}}{9 x^9}+\frac {1}{27} \operatorname {Subst}\left (\int \frac {1}{(-1+x)^{2/3} x^3} \, dx,x,x^3\right )\\ &=-\frac {\sqrt [3]{-1+x^3}}{9 x^9}+\frac {\sqrt [3]{-1+x^3}}{54 x^6}+\frac {5}{162} \operatorname {Subst}\left (\int \frac {1}{(-1+x)^{2/3} x^2} \, dx,x,x^3\right )\\ &=-\frac {\sqrt [3]{-1+x^3}}{9 x^9}+\frac {\sqrt [3]{-1+x^3}}{54 x^6}+\frac {5 \sqrt [3]{-1+x^3}}{162 x^3}+\frac {5}{243} \operatorname {Subst}\left (\int \frac {1}{(-1+x)^{2/3} x} \, dx,x,x^3\right )\\ &=-\frac {\sqrt [3]{-1+x^3}}{9 x^9}+\frac {\sqrt [3]{-1+x^3}}{54 x^6}+\frac {5 \sqrt [3]{-1+x^3}}{162 x^3}-\frac {5 \log (x)}{162}+\frac {5}{162} \operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sqrt [3]{-1+x^3}\right )+\frac {5}{162} \operatorname {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt [3]{-1+x^3}\right )\\ &=-\frac {\sqrt [3]{-1+x^3}}{9 x^9}+\frac {\sqrt [3]{-1+x^3}}{54 x^6}+\frac {5 \sqrt [3]{-1+x^3}}{162 x^3}-\frac {5 \log (x)}{162}+\frac {5}{162} \log \left (1+\sqrt [3]{-1+x^3}\right )-\frac {5}{81} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \sqrt [3]{-1+x^3}\right )\\ &=-\frac {\sqrt [3]{-1+x^3}}{9 x^9}+\frac {\sqrt [3]{-1+x^3}}{54 x^6}+\frac {5 \sqrt [3]{-1+x^3}}{162 x^3}-\frac {5 \tan ^{-1}\left (\frac {1-2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )}{81 \sqrt {3}}-\frac {5 \log (x)}{162}+\frac {5}{162} \log \left (1+\sqrt [3]{-1+x^3}\right )\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 28, normalized size = 0.27 \begin {gather*} \frac {1}{4} \left (x^3-1\right )^{4/3} \, _2F_1\left (\frac {4}{3},4;\frac {7}{3};1-x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + x^3)^(1/3)/x^10,x]

[Out]

((-1 + x^3)^(4/3)*Hypergeometric2F1[4/3, 4, 7/3, 1 - x^3])/4

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IntegrateAlgebraic [A]  time = 0.17, size = 104, normalized size = 1.00 \begin {gather*} \frac {\sqrt [3]{-1+x^3} \left (-18+3 x^3+5 x^6\right )}{162 x^9}-\frac {5 \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )}{81 \sqrt {3}}+\frac {5}{243} \log \left (1+\sqrt [3]{-1+x^3}\right )-\frac {5}{486} \log \left (1-\sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-1 + x^3)^(1/3)/x^10,x]

[Out]

((-1 + x^3)^(1/3)*(-18 + 3*x^3 + 5*x^6))/(162*x^9) - (5*ArcTan[1/Sqrt[3] - (2*(-1 + x^3)^(1/3))/Sqrt[3]])/(81*
Sqrt[3]) + (5*Log[1 + (-1 + x^3)^(1/3)])/243 - (5*Log[1 - (-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)])/486

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fricas [A]  time = 1.07, size = 93, normalized size = 0.89 \begin {gather*} \frac {10 \, \sqrt {3} x^{9} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) - 5 \, x^{9} \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + 10 \, x^{9} \log \left ({\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + 3 \, {\left (5 \, x^{6} + 3 \, x^{3} - 18\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{486 \, x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-1)^(1/3)/x^10,x, algorithm="fricas")

[Out]

1/486*(10*sqrt(3)*x^9*arctan(2/3*sqrt(3)*(x^3 - 1)^(1/3) - 1/3*sqrt(3)) - 5*x^9*log((x^3 - 1)^(2/3) - (x^3 - 1
)^(1/3) + 1) + 10*x^9*log((x^3 - 1)^(1/3) + 1) + 3*(5*x^6 + 3*x^3 - 18)*(x^3 - 1)^(1/3))/x^9

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giac [A]  time = 0.24, size = 90, normalized size = 0.87 \begin {gather*} \frac {5}{243} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {5 \, {\left (x^{3} - 1\right )}^{\frac {7}{3}} + 13 \, {\left (x^{3} - 1\right )}^{\frac {4}{3}} - 10 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{162 \, x^{9}} - \frac {5}{486} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {5}{243} \, \log \left ({\left | {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-1)^(1/3)/x^10,x, algorithm="giac")

[Out]

5/243*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^3 - 1)^(1/3) - 1)) + 1/162*(5*(x^3 - 1)^(7/3) + 13*(x^3 - 1)^(4/3) - 10
*(x^3 - 1)^(1/3))/x^9 - 5/486*log((x^3 - 1)^(2/3) - (x^3 - 1)^(1/3) + 1) + 5/243*log(abs((x^3 - 1)^(1/3) + 1))

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maple [C]  time = 2.76, size = 89, normalized size = 0.86

method	result	size
meijerg	\(\frac {\mathrm {signum}\left (x^{3}-1\right )^{\frac {1}{3}} \left (-\frac {10 \Gamma \left (\frac {2}{3}\right ) x^{3} \hypergeom \left (\left [1, 1, \frac {11}{3}\right ], \left [2, 5\right ], x^{3}\right )}{81}-\frac {5 \left (\frac {4}{15}+\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+3 \ln \relax (x )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )}{27}-\frac {\Gamma \left (\frac {2}{3}\right )}{x^{9}}+\frac {\Gamma \left (\frac {2}{3}\right )}{2 x^{6}}+\frac {\Gamma \left (\frac {2}{3}\right )}{3 x^{3}}\right )}{9 \Gamma \left (\frac {2}{3}\right ) \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {1}{3}}}\)	\(89\)
risch	\(\frac {5 x^{9}-2 x^{6}-21 x^{3}+18}{162 x^{9} \left (x^{3}-1\right )^{\frac {2}{3}}}+\frac {5 \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {2}{3}} \left (\frac {2 \Gamma \left (\frac {2}{3}\right ) x^{3} \hypergeom \left (\left [1, 1, \frac {5}{3}\right ], \left [2, 2\right ], x^{3}\right )}{3}+\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+3 \ln \relax (x )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )\right )}{243 \Gamma \left (\frac {2}{3}\right ) \mathrm {signum}\left (x^{3}-1\right )^{\frac {2}{3}}}\)	\(96\)
trager	\(\frac {\left (x^{3}-1\right )^{\frac {1}{3}} \left (5 x^{6}+3 x^{3}-18\right )}{162 x^{9}}-\frac {5 \ln \left (\frac {-376963072 \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )^{2} x^{3}-5045760 \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right ) x^{3}+2817024 \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}+1477 x^{3}-14247 \left (x^{3}-1\right )^{\frac {2}{3}}-2817024 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )[phone] \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )^{2}+14247 \left (x^{3}-1\right )^{\frac {1}{3}}+8707072 \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )-2743}{x^{3}}\right )}{243}-\frac {2560 \ln \left (\frac {-376963072 \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )^{2} x^{3}-5045760 \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right ) x^{3}+2817024 \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}+1477 x^{3}-14247 \left (x^{3}-1\right )^{\frac {2}{3}}-2817024 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )[phone] \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )^{2}+14247 \left (x^{3}-1\right )^{\frac {1}{3}}+8707072 \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )-2743}{x^{3}}\right ) \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )}{243}+\frac {2560 \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right ) \ln \left (-\frac {376963072 \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )^{2} x^{3}-3573248 \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right ) x^{3}+2817024 \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}-9894 x^{3}+19749 \left (x^{3}-1\right )^{\frac {2}{3}}-2817024 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )-3015704576 \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )^{2}-19749 \left (x^{3}-1\right )^{\frac {1}{3}}-3073024 \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )+8245}{x^{3}}\right )}{243}\)	\(451\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3-1)^(1/3)/x^10,x,method=_RETURNVERBOSE)

[Out]

1/9/GAMMA(2/3)*signum(x^3-1)^(1/3)/(-signum(x^3-1))^(1/3)*(-10/81*GAMMA(2/3)*x^3*hypergeom([1,1,11/3],[2,5],x^
3)-5/27*(4/15+1/6*Pi*3^(1/2)-3/2*ln(3)+3*ln(x)+I*Pi)*GAMMA(2/3)-GAMMA(2/3)/x^9+1/2*GAMMA(2/3)/x^6+1/3*GAMMA(2/
3)/x^3)

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maxima [A]  time = 0.41, size = 111, normalized size = 1.07 \begin {gather*} \frac {5}{243} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {5 \, {\left (x^{3} - 1\right )}^{\frac {7}{3}} + 13 \, {\left (x^{3} - 1\right )}^{\frac {4}{3}} - 10 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{162 \, {\left ({\left (x^{3} - 1\right )}^{3} + 3 \, x^{3} + 3 \, {\left (x^{3} - 1\right )}^{2} - 2\right )}} - \frac {5}{486} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {5}{243} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-1)^(1/3)/x^10,x, algorithm="maxima")

[Out]

5/243*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^3 - 1)^(1/3) - 1)) + 1/162*(5*(x^3 - 1)^(7/3) + 13*(x^3 - 1)^(4/3) - 10
*(x^3 - 1)^(1/3))/((x^3 - 1)^3 + 3*x^3 + 3*(x^3 - 1)^2 - 2) - 5/486*log((x^3 - 1)^(2/3) - (x^3 - 1)^(1/3) + 1)
+ 5/243*log((x^3 - 1)^(1/3) + 1)

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mupad [B]  time = 0.99, size = 124, normalized size = 1.19 \begin {gather*} \frac {5\,\ln \left (\frac {25\,{\left (x^3-1\right )}^{1/3}}{6561}+\frac {25}{6561}\right )}{243}+\frac {\frac {13\,{\left (x^3-1\right )}^{4/3}}{162}-\frac {5\,{\left (x^3-1\right )}^{1/3}}{81}+\frac {5\,{\left (x^3-1\right )}^{7/3}}{162}}{3\,{\left (x^3-1\right )}^2+{\left (x^3-1\right )}^3+3\,x^3-2}-\ln \left (\frac {5}{54}-\frac {5\,{\left (x^3-1\right )}^{1/3}}{27}+\frac {\sqrt {3}\,5{}\mathrm {i}}{54}\right )\,\left (\frac {5}{486}+\frac {\sqrt {3}\,5{}\mathrm {i}}{486}\right )+\ln \left (\frac {5\,{\left (x^3-1\right )}^{1/3}}{27}-\frac {5}{54}+\frac {\sqrt {3}\,5{}\mathrm {i}}{54}\right )\,\left (-\frac {5}{486}+\frac {\sqrt {3}\,5{}\mathrm {i}}{486}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3 - 1)^(1/3)/x^10,x)

[Out]

(5*log((25*(x^3 - 1)^(1/3))/6561 + 25/6561))/243 + ((13*(x^3 - 1)^(4/3))/162 - (5*(x^3 - 1)^(1/3))/81 + (5*(x^
3 - 1)^(7/3))/162)/(3*(x^3 - 1)^2 + (x^3 - 1)^3 + 3*x^3 - 2) - log((3^(1/2)*5i)/54 - (5*(x^3 - 1)^(1/3))/27 +
5/54)*((3^(1/2)*5i)/486 + 5/486) + log((3^(1/2)*5i)/54 + (5*(x^3 - 1)^(1/3))/27 - 5/54)*((3^(1/2)*5i)/486 - 5/
486)

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sympy [C]  time = 1.42, size = 34, normalized size = 0.33 \begin {gather*} - \frac {\Gamma \left (\frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {8}{3} \\ \frac {11}{3} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{3}}} \right )}}{3 x^{8} \Gamma \left (\frac {11}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**3-1)**(1/3)/x**10,x)

[Out]

-gamma(8/3)*hyper((-1/3, 8/3), (11/3,), exp_polar(2*I*pi)/x**3)/(3*x**8*gamma(11/3))

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