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

Optimal. Leaf size=115 \[ \frac {14}{243} \log \left (\sqrt [3]{x^3+1}-1\right )-\frac {7}{243} \log \left (\left (x^3+1\right )^{2/3}+\sqrt [3]{x^3+1}+1\right )+\frac {14 \tan ^{-1}\left (\frac {2 \sqrt [3]{x^3+1}}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{81 \sqrt {3}}+\frac {\left (x^3+1\right )^{2/3} \left (-729 x^7+560 x^6+486 x^4-420 x^3-405 x+360\right )}{3240 x^9} \]

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Rubi [A]  time = 0.10, antiderivative size = 150, normalized size of antiderivative = 1.30, number of steps used = 13, number of rules used = 9, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {1844, 266, 51, 55, 618, 204, 31, 271, 264} \begin {gather*} \frac {14 \left (x^3+1\right )^{2/3}}{81 x^3}+\frac {7}{81} \log \left (1-\sqrt [3]{x^3+1}\right )+\frac {14 \tan ^{-1}\left (\frac {2 \sqrt [3]{x^3+1}+1}{\sqrt {3}}\right )}{81 \sqrt {3}}+\frac {\left (x^3+1\right )^{2/3}}{9 x^9}-\frac {\left (x^3+1\right )^{2/3}}{8 x^8}-\frac {7 \left (x^3+1\right )^{2/3}}{54 x^6}+\frac {3 \left (x^3+1\right )^{2/3}}{20 x^5}-\frac {9 \left (x^3+1\right )^{2/3}}{40 x^2}-\frac {7 \log (x)}{81} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(1 + x^3)^(2/3)/(9*x^9) - (1 + x^3)^(2/3)/(8*x^8) - (7*(1 + x^3)^(2/3))/(54*x^6) + (3*(1 + x^3)^(2/3))/(20*x^5
) + (14*(1 + x^3)^(2/3))/(81*x^3) - (9*(1 + x^3)^(2/3))/(40*x^2) + (14*ArcTan[(1 + 2*(1 + x^3)^(1/3))/Sqrt[3]]
)/(81*Sqrt[3]) - (7*Log[x])/81 + (7*Log[1 - (1 + x^3)^(1/3)])/81

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 55

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, -Simp[L
og[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] && PosQ
[(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 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

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 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

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]

Rule 1844

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a +
b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] && (PolyQ[Pq, x] || PolyQ[Pq, x^n]) &&  !IGtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {-1+x}{x^{10} \sqrt [3]{1+x^3}} \, dx &=\int \left (-\frac {1}{x^{10} \sqrt [3]{1+x^3}}+\frac {1}{x^9 \sqrt [3]{1+x^3}}\right ) \, dx\\ &=-\int \frac {1}{x^{10} \sqrt [3]{1+x^3}} \, dx+\int \frac {1}{x^9 \sqrt [3]{1+x^3}} \, dx\\ &=-\frac {\left (1+x^3\right )^{2/3}}{8 x^8}-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x^4 \sqrt [3]{1+x}} \, dx,x,x^3\right )-\frac {3}{4} \int \frac {1}{x^6 \sqrt [3]{1+x^3}} \, dx\\ &=\frac {\left (1+x^3\right )^{2/3}}{9 x^9}-\frac {\left (1+x^3\right )^{2/3}}{8 x^8}+\frac {3 \left (1+x^3\right )^{2/3}}{20 x^5}+\frac {7}{27} \operatorname {Subst}\left (\int \frac {1}{x^3 \sqrt [3]{1+x}} \, dx,x,x^3\right )+\frac {9}{20} \int \frac {1}{x^3 \sqrt [3]{1+x^3}} \, dx\\ &=\frac {\left (1+x^3\right )^{2/3}}{9 x^9}-\frac {\left (1+x^3\right )^{2/3}}{8 x^8}-\frac {7 \left (1+x^3\right )^{2/3}}{54 x^6}+\frac {3 \left (1+x^3\right )^{2/3}}{20 x^5}-\frac {9 \left (1+x^3\right )^{2/3}}{40 x^2}-\frac {14}{81} \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt [3]{1+x}} \, dx,x,x^3\right )\\ &=\frac {\left (1+x^3\right )^{2/3}}{9 x^9}-\frac {\left (1+x^3\right )^{2/3}}{8 x^8}-\frac {7 \left (1+x^3\right )^{2/3}}{54 x^6}+\frac {3 \left (1+x^3\right )^{2/3}}{20 x^5}+\frac {14 \left (1+x^3\right )^{2/3}}{81 x^3}-\frac {9 \left (1+x^3\right )^{2/3}}{40 x^2}+\frac {14}{243} \operatorname {Subst}\left (\int \frac {1}{x \sqrt [3]{1+x}} \, dx,x,x^3\right )\\ &=\frac {\left (1+x^3\right )^{2/3}}{9 x^9}-\frac {\left (1+x^3\right )^{2/3}}{8 x^8}-\frac {7 \left (1+x^3\right )^{2/3}}{54 x^6}+\frac {3 \left (1+x^3\right )^{2/3}}{20 x^5}+\frac {14 \left (1+x^3\right )^{2/3}}{81 x^3}-\frac {9 \left (1+x^3\right )^{2/3}}{40 x^2}-\frac {7 \log (x)}{81}-\frac {7}{81} \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [3]{1+x^3}\right )+\frac {7}{81} \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt [3]{1+x^3}\right )\\ &=\frac {\left (1+x^3\right )^{2/3}}{9 x^9}-\frac {\left (1+x^3\right )^{2/3}}{8 x^8}-\frac {7 \left (1+x^3\right )^{2/3}}{54 x^6}+\frac {3 \left (1+x^3\right )^{2/3}}{20 x^5}+\frac {14 \left (1+x^3\right )^{2/3}}{81 x^3}-\frac {9 \left (1+x^3\right )^{2/3}}{40 x^2}-\frac {7 \log (x)}{81}+\frac {7}{81} \log \left (1-\sqrt [3]{1+x^3}\right )-\frac {14}{81} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{1+x^3}\right )\\ &=\frac {\left (1+x^3\right )^{2/3}}{9 x^9}-\frac {\left (1+x^3\right )^{2/3}}{8 x^8}-\frac {7 \left (1+x^3\right )^{2/3}}{54 x^6}+\frac {3 \left (1+x^3\right )^{2/3}}{20 x^5}+\frac {14 \left (1+x^3\right )^{2/3}}{81 x^3}-\frac {9 \left (1+x^3\right )^{2/3}}{40 x^2}+\frac {14 \tan ^{-1}\left (\frac {1+2 \sqrt [3]{1+x^3}}{\sqrt {3}}\right )}{81 \sqrt {3}}-\frac {7 \log (x)}{81}+\frac {7}{81} \log \left (1-\sqrt [3]{1+x^3}\right )\\ \end {align*}

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Mathematica [C]  time = 0.03, size = 46, normalized size = 0.40 \begin {gather*} -\frac {\left (x^3+1\right )^{2/3} \left (20 x^8 \, _2F_1\left (\frac {2}{3},4;\frac {5}{3};x^3+1\right )+9 x^6-6 x^3+5\right )}{40 x^8} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/40*((1 + x^3)^(2/3)*(5 - 6*x^3 + 9*x^6 + 20*x^8*Hypergeometric2F1[2/3, 4, 5/3, 1 + x^3]))/x^8

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IntegrateAlgebraic [A]  time = 17.29, size = 115, normalized size = 1.00 \begin {gather*} \frac {\left (1+x^3\right )^{2/3} \left (360-405 x-420 x^3+486 x^4+560 x^6-729 x^7\right )}{3240 x^9}+\frac {14 \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x^3}}{\sqrt {3}}\right )}{81 \sqrt {3}}+\frac {14}{243} \log \left (-1+\sqrt [3]{1+x^3}\right )-\frac {7}{243} \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)/(x^10*(1 + x^3)^(1/3)),x]

[Out]

((1 + x^3)^(2/3)*(360 - 405*x - 420*x^3 + 486*x^4 + 560*x^6 - 729*x^7))/(3240*x^9) + (14*ArcTan[1/Sqrt[3] + (2
*(1 + x^3)^(1/3))/Sqrt[3]])/(81*Sqrt[3]) + (14*Log[-1 + (1 + x^3)^(1/3)])/243 - (7*Log[1 + (1 + x^3)^(1/3) + (
1 + x^3)^(2/3)])/243

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fricas [A]  time = 0.86, size = 124, normalized size = 1.08 \begin {gather*} -\frac {560 \, \sqrt {3} x^{9} \arctan \left (-\frac {\sqrt {3} {\left (x^{3} + 1\right )} - 2 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {2}{3}} + 4 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x^{3} + 9}\right ) - 280 \, x^{9} \log \left (\frac {x^{3} - 3 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} + 3 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x^{3}}\right ) + 3 \, {\left (729 \, x^{7} - 560 \, x^{6} - 486 \, x^{4} + 420 \, x^{3} + 405 \, x - 360\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{9720 \, x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9720*(560*sqrt(3)*x^9*arctan(-(sqrt(3)*(x^3 + 1) - 2*sqrt(3)*(x^3 + 1)^(2/3) + 4*sqrt(3)*(x^3 + 1)^(1/3))/(
x^3 + 9)) - 280*x^9*log((x^3 - 3*(x^3 + 1)^(2/3) + 3*(x^3 + 1)^(1/3))/x^3) + 3*(729*x^7 - 560*x^6 - 486*x^4 +
420*x^3 + 405*x - 360)*(x^3 + 1)^(2/3))/x^9

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x - 1}{{\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{10}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 2.65, size = 111, normalized size = 0.97

method	result	size
risch	\(-\frac {729 x^{10}-560 x^{9}+243 x^{7}-140 x^{6}-81 x^{4}+60 x^{3}+405 x -360}{3240 x^{9} \left (x^{3}+1\right )^{\frac {1}{3}}}+\frac {7 \sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (-\frac {2 \pi \sqrt {3}\, x^{3} \hypergeom \left (\left [1, 1, \frac {4}{3}\right ], \left [2, 2\right ], -x^{3}\right )}{9 \Gamma \left (\frac {2}{3}\right )}+\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+3 \ln \relax (x )\right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}\right )}{243 \pi }\)	\(111\)
meijerg	\(-\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (\frac {70 \pi \sqrt {3}\, x^{3} \hypergeom \left (\left [1, 1, \frac {13}{3}\right ], \left [2, 5\right ], -x^{3}\right )}{729 \Gamma \left (\frac {2}{3}\right )}-\frac {28 \left (\frac {197}{84}-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+3 \ln \relax (x )\right ) \pi \sqrt {3}}{243 \Gamma \left (\frac {2}{3}\right )}-\frac {2 \pi \sqrt {3}}{9 \Gamma \left (\frac {2}{3}\right ) x^{9}}+\frac {\pi \sqrt {3}}{9 \Gamma \left (\frac {2}{3}\right ) x^{6}}-\frac {4 \pi \sqrt {3}}{27 \Gamma \left (\frac {2}{3}\right ) x^{3}}\right )}{6 \pi }-\frac {\left (\frac {9}{5} x^{6}-\frac {6}{5} x^{3}+1\right ) \left (x^{3}+1\right )^{\frac {2}{3}}}{8 x^{8}}\)	\(128\)
trager	\(-\frac {\left (729 x^{7}-560 x^{6}-486 x^{4}+420 x^{3}+405 x -360\right ) \left (x^{3}+1\right )^{\frac {2}{3}}}{3240 x^{9}}+\frac {28 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \ln \left (-\frac {16 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{3}-18 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{3}-30 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}}+2 x^{3}-16 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2}-30 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}}+9 \left (x^{3}+1\right )^{\frac {2}{3}}-38 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+9 \left (x^{3}+1\right )^{\frac {1}{3}}+5}{x^{3}}\right )}{243}-\frac {14 \ln \left (-\frac {16 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{3}+34 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{3}+30 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}}+15 x^{3}-16 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2}+30 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}}+24 \left (x^{3}+1\right )^{\frac {2}{3}}+22 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+24 \left (x^{3}+1\right )^{\frac {1}{3}}+20}{x^{3}}\right )}{243}-\frac {28 \ln \left (-\frac {16 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{3}+34 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{3}+30 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}}+15 x^{3}-16 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2}+30 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}}+24 \left (x^{3}+1\right )^{\frac {2}{3}}+22 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+24 \left (x^{3}+1\right )^{\frac {1}{3}}+20}{x^{3}}\right ) \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )}{243}\)	\(466\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3240*(729*x^10-560*x^9+243*x^7-140*x^6-81*x^4+60*x^3+405*x-360)/x^9/(x^3+1)^(1/3)+7/243/Pi*3^(1/2)*GAMMA(2/
3)*(-2/9*Pi*3^(1/2)/GAMMA(2/3)*x^3*hypergeom([1,1,4/3],[2,2],-x^3)+2/3*(-1/6*Pi*3^(1/2)-3/2*ln(3)+3*ln(x))*Pi*
3^(1/2)/GAMMA(2/3))

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maxima [A]  time = 0.43, size = 145, normalized size = 1.26 \begin {gather*} \frac {14}{243} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {28 \, {\left (x^{3} + 1\right )}^{\frac {8}{3}} - 77 \, {\left (x^{3} + 1\right )}^{\frac {5}{3}} + 67 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{162 \, {\left ({\left (x^{3} + 1\right )}^{3} + 3 \, x^{3} - 3 \, {\left (x^{3} + 1\right )}^{2} + 2\right )}} - \frac {{\left (x^{3} + 1\right )}^{\frac {2}{3}}}{2 \, x^{2}} + \frac {2 \, {\left (x^{3} + 1\right )}^{\frac {5}{3}}}{5 \, x^{5}} - \frac {{\left (x^{3} + 1\right )}^{\frac {8}{3}}}{8 \, x^{8}} - \frac {7}{243} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {2}{3}} + {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {14}{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((-1+x)/x^10/(x^3+1)^(1/3),x, algorithm="maxima")

[Out]

14/243*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^3 + 1)^(1/3) + 1)) + 1/162*(28*(x^3 + 1)^(8/3) - 77*(x^3 + 1)^(5/3) +
67*(x^3 + 1)^(2/3))/((x^3 + 1)^3 + 3*x^3 - 3*(x^3 + 1)^2 + 2) - 1/2*(x^3 + 1)^(2/3)/x^2 + 2/5*(x^3 + 1)^(5/3)/
x^5 - 1/8*(x^3 + 1)^(8/3)/x^8 - 7/243*log((x^3 + 1)^(2/3) + (x^3 + 1)^(1/3) + 1) + 14/243*log((x^3 + 1)^(1/3)
- 1)

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mupad [B]  time = 1.29, size = 158, normalized size = 1.37 \begin {gather*} \frac {14\,\ln \left (\frac {196\,{\left (x^3+1\right )}^{1/3}}{6561}-\frac {196}{6561}\right )}{243}+\ln \left (\frac {196\,{\left (x^3+1\right )}^{1/3}}{6561}-9\,{\left (-\frac {7}{243}+\frac {\sqrt {3}\,7{}\mathrm {i}}{243}\right )}^2\right )\,\left (-\frac {7}{243}+\frac {\sqrt {3}\,7{}\mathrm {i}}{243}\right )-\ln \left (\frac {196\,{\left (x^3+1\right )}^{1/3}}{6561}-9\,{\left (\frac {7}{243}+\frac {\sqrt {3}\,7{}\mathrm {i}}{243}\right )}^2\right )\,\left (\frac {7}{243}+\frac {\sqrt {3}\,7{}\mathrm {i}}{243}\right )+\frac {\frac {67\,{\left (x^3+1\right )}^{2/3}}{162}-\frac {77\,{\left (x^3+1\right )}^{5/3}}{162}+\frac {14\,{\left (x^3+1\right )}^{8/3}}{81}}{{\left (x^3+1\right )}^3-3\,{\left (x^3+1\right )}^2+3\,x^3+2}-\frac {{\left (x^3+1\right )}^{2/3}\,\left (9\,x^6-6\,x^3+5\right )}{40\,x^8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x - 1)/(x^10*(x^3 + 1)^(1/3)),x)

[Out]

(14*log((196*(x^3 + 1)^(1/3))/6561 - 196/6561))/243 + log((196*(x^3 + 1)^(1/3))/6561 - 9*((3^(1/2)*7i)/243 - 7
/243)^2)*((3^(1/2)*7i)/243 - 7/243) - log((196*(x^3 + 1)^(1/3))/6561 - 9*((3^(1/2)*7i)/243 + 7/243)^2)*((3^(1/
2)*7i)/243 + 7/243) + ((67*(x^3 + 1)^(2/3))/162 - (77*(x^3 + 1)^(5/3))/162 + (14*(x^3 + 1)^(8/3))/81)/((x^3 +
1)^3 - 3*(x^3 + 1)^2 + 3*x^3 + 2) - ((x^3 + 1)^(2/3)*(9*x^6 - 6*x^3 + 5))/(40*x^8)

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sympy [C]  time = 2.85, size = 112, normalized size = 0.97 \begin {gather*} \frac {2 \left (1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{3 \Gamma \left (\frac {1}{3}\right )} - \frac {4 \left (1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{3} \Gamma \left (\frac {1}{3}\right )} + \frac {10 \left (1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{27 x^{6} \Gamma \left (\frac {1}{3}\right )} + \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^{i \pi }}{x^{3}}} \right )}}{3 x^{10} \Gamma \left (\frac {13}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(1 + x**(-3))**(2/3)*gamma(-8/3)/(3*gamma(1/3)) - 4*(1 + x**(-3))**(2/3)*gamma(-8/3)/(9*x**3*gamma(1/3)) + 1
0*(1 + x**(-3))**(2/3)*gamma(-8/3)/(27*x**6*gamma(1/3)) + gamma(10/3)*hyper((1/3, 10/3), (13/3,), exp_polar(I*
pi)/x**3)/(3*x**10*gamma(13/3))

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