3.15.73 \(\int \frac {\sqrt [3]{-1+x^3} (-1+2 x^3)}{x^{10}} \, dx\) [1473]

Optimal. Leaf size=104 \[ \frac {\sqrt [3]{-1+x^3} \left (18-57 x^3+13 x^6\right )}{162 x^9}-\frac {13 \text {ArcTan}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )}{81 \sqrt {3}}+\frac {13}{243} \log \left (1+\sqrt [3]{-1+x^3}\right )-\frac {13}{486} \log \left (1-\sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \]

[Out]

1/162*(x^3-1)^(1/3)*(13*x^6-57*x^3+18)/x^9+13/243*arctan(-1/3*3^(1/2)+2/3*(x^3-1)^(1/3)*3^(1/2))*3^(1/2)+13/24
3*ln(1+(x^3-1)^(1/3))-13/486*ln(1-(x^3-1)^(1/3)+(x^3-1)^(2/3))

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Rubi [A]
time = 0.04, antiderivative size = 100, normalized size of antiderivative = 0.96, number of steps used = 8, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {457, 79, 43, 44, 60, 632, 210, 31} \begin {gather*} -\frac {13 \text {ArcTan}\left (\frac {1-2 \sqrt [3]{x^3-1}}{\sqrt {3}}\right )}{81 \sqrt {3}}+\frac {13 \sqrt [3]{x^3-1}}{162 x^3}+\frac {13}{162} \log \left (\sqrt [3]{x^3-1}+1\right )-\frac {\left (x^3-1\right )^{4/3}}{9 x^9}-\frac {13 \sqrt [3]{x^3-1}}{54 x^6}-\frac {13 \log (x)}{162} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-1 + x^3)^(1/3)*(-1 + 2*x^3))/x^10,x]

[Out]

(-13*(-1 + x^3)^(1/3))/(54*x^6) + (13*(-1 + x^3)^(1/3))/(162*x^3) - (-1 + x^3)^(4/3)/(9*x^9) - (13*ArcTan[(1 -
2*(-1 + x^3)^(1/3))/Sqrt[3]])/(81*Sqrt[3]) - (13*Log[x])/162 + (13*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 43

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, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

Rule 44

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] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 60

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[-(b*c - a*d)/b, 3]}, Simp[-
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 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rule 210

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

Rule 457

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 632

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} \left (-1+2 x^3\right )}{x^{10}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {\sqrt [3]{-1+x} (-1+2 x)}{x^4} \, dx,x,x^3\right )\\ &=-\frac {\left (-1+x^3\right )^{4/3}}{9 x^9}+\frac {13}{27} \text {Subst}\left (\int \frac {\sqrt [3]{-1+x}}{x^3} \, dx,x,x^3\right )\\ &=-\frac {13 \sqrt [3]{-1+x^3}}{54 x^6}-\frac {\left (-1+x^3\right )^{4/3}}{9 x^9}+\frac {13}{162} \text {Subst}\left (\int \frac {1}{(-1+x)^{2/3} x^2} \, dx,x,x^3\right )\\ &=-\frac {13 \sqrt [3]{-1+x^3}}{54 x^6}+\frac {13 \sqrt [3]{-1+x^3}}{162 x^3}-\frac {\left (-1+x^3\right )^{4/3}}{9 x^9}+\frac {13}{243} \text {Subst}\left (\int \frac {1}{(-1+x)^{2/3} x} \, dx,x,x^3\right )\\ &=-\frac {13 \sqrt [3]{-1+x^3}}{54 x^6}+\frac {13 \sqrt [3]{-1+x^3}}{162 x^3}-\frac {\left (-1+x^3\right )^{4/3}}{9 x^9}-\frac {13 \log (x)}{162}+\frac {13}{162} \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sqrt [3]{-1+x^3}\right )+\frac {13}{162} \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt [3]{-1+x^3}\right )\\ &=-\frac {13 \sqrt [3]{-1+x^3}}{54 x^6}+\frac {13 \sqrt [3]{-1+x^3}}{162 x^3}-\frac {\left (-1+x^3\right )^{4/3}}{9 x^9}-\frac {13 \log (x)}{162}+\frac {13}{162} \log \left (1+\sqrt [3]{-1+x^3}\right )-\frac {13}{81} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \sqrt [3]{-1+x^3}\right )\\ &=-\frac {13 \sqrt [3]{-1+x^3}}{54 x^6}+\frac {13 \sqrt [3]{-1+x^3}}{162 x^3}-\frac {\left (-1+x^3\right )^{4/3}}{9 x^9}-\frac {13 \tan ^{-1}\left (\frac {1-2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )}{81 \sqrt {3}}-\frac {13 \log (x)}{162}+\frac {13}{162} \log \left (1+\sqrt [3]{-1+x^3}\right )\\ \end {align*}

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Mathematica [A]
time = 0.15, size = 97, normalized size = 0.93 \begin {gather*} \frac {1}{486} \left (\frac {3 \sqrt [3]{-1+x^3} \left (18-57 x^3+13 x^6\right )}{x^9}-26 \sqrt {3} \text {ArcTan}\left (\frac {1-2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )+26 \log \left (1+\sqrt [3]{-1+x^3}\right )-13 \log \left (1-\sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-1 + x^3)^(1/3)*(-1 + 2*x^3))/x^10,x]

[Out]

((3*(-1 + x^3)^(1/3)*(18 - 57*x^3 + 13*x^6))/x^9 - 26*Sqrt[3]*ArcTan[(1 - 2*(-1 + x^3)^(1/3))/Sqrt[3]] + 26*Lo
g[1 + (-1 + x^3)^(1/3)] - 13*Log[1 - (-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)])/486

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 2.53, size = 96, normalized size = 0.92

method	result	size
risch	\(\frac {13 x^{9}-70 x^{6}+75 x^{3}-18}{162 x^{9} \left (x^{3}-1\right )^{\frac {2}{3}}}+\frac {13 \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {2}{3}} \left (\frac {2 \Gamma \left (\frac {2}{3}\right ) x^{3} \hypergeom \left (\left [1, 1, \frac {5}{3}\right ], \left [2, 2\right ], x^{3}\right )}{3}+\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )\right )}{243 \Gamma \left (\frac {2}{3}\right ) \mathrm {signum}\left (x^{3}-1\right )^{\frac {2}{3}}}\)	\(96\)
meijerg	\(-\frac {2 \mathrm {signum}\left (x^{3}-1\right )^{\frac {1}{3}} \left (\frac {5 \Gamma \left (\frac {2}{3}\right ) x^{3} \hypergeom \left (\left [1, 1, \frac {8}{3}\right ], \left [2, 4\right ], x^{3}\right )}{27}+\frac {\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )}{3}+\frac {3 \Gamma \left (\frac {2}{3}\right )}{2 x^{6}}-\frac {\Gamma \left (\frac {2}{3}\right )}{x^{3}}\right )}{9 \Gamma \left (\frac {2}{3}\right ) \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {1}{3}}}-\frac {\mathrm {signum}\left (x^{3}-1\right )^{\frac {1}{3}} \left (-\frac {10 \Gamma \left (\frac {2}{3}\right ) x^{3} \hypergeom \left (\left [1, 1, \frac {11}{3}\right ], \left [2, 5\right ], x^{3}\right )}{81}-\frac {5 \left (\frac {4}{15}+\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )}{27}-\frac {\Gamma \left (\frac {2}{3}\right )}{x^{9}}+\frac {\Gamma \left (\frac {2}{3}\right )}{2 x^{6}}+\frac {\Gamma \left (\frac {2}{3}\right )}{3 x^{3}}\right )}{9 \Gamma \left (\frac {2}{3}\right ) \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {1}{3}}}\)	\(170\)
trager	\(\frac {\left (x^{3}-1\right )^{\frac {1}{3}} \left (13 x^{6}-57 x^{3}+18\right )}{162 x^{9}}-\frac {13 \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}-3015704576 \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )^{2}+2817024 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )+14247 \left (x^{3}-1\right )^{\frac {2}{3}}-8707072 \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )-14247 \left (x^{3}-1\right )^{\frac {1}{3}}+2743}{x^{3}}\right )}{243}-\frac {6656 \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}-3015704576 \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )^{2}+2817024 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )+14247 \left (x^{3}-1\right )^{\frac {2}{3}}-8707072 \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )-14247 \left (x^{3}-1\right )^{\frac {1}{3}}+2743}{x^{3}}\right ) \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )}{243}+\frac {6656 \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}-3015704576 \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )^{2}-2817024 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )+19749 \left (x^{3}-1\right )^{\frac {2}{3}}-3073024 \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )-19749 \left (x^{3}-1\right )^{\frac {1}{3}}+8245}{x^{3}}\right )}{243}\)	\(453\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3-1)^(1/3)*(2*x^3-1)/x^10,x,method=_RETURNVERBOSE)

[Out]

1/162*(13*x^9-70*x^6+75*x^3-18)/x^9/(x^3-1)^(2/3)+13/243/GAMMA(2/3)/signum(x^3-1)^(2/3)*(-signum(x^3-1))^(2/3)
*(2/3*GAMMA(2/3)*x^3*hypergeom([1,1,5/3],[2,2],x^3)+(1/6*Pi*3^(1/2)-3/2*ln(3)+3*ln(x)+I*Pi)*GAMMA(2/3))

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Maxima [A]
time = 0.47, size = 146, normalized size = 1.40 \begin {gather*} \frac {13}{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 {{\left (x^{3} - 1\right )}^{\frac {4}{3}} - 2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{9 \, {\left (2 \, x^{3} + {\left (x^{3} - 1\right )}^{2} - 1\right )}} - \frac {13}{486} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {13}{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)*(2*x^3-1)/x^10,x, algorithm="maxima")

[Out]

13/243*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) - 1
0*(x^3 - 1)^(1/3))/((x^3 - 1)^3 + 3*x^3 + 3*(x^3 - 1)^2 - 2) + 1/9*((x^3 - 1)^(4/3) - 2*(x^3 - 1)^(1/3))/(2*x^
3 + (x^3 - 1)^2 - 1) - 13/486*log((x^3 - 1)^(2/3) - (x^3 - 1)^(1/3) + 1) + 13/243*log((x^3 - 1)^(1/3) + 1)

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Fricas [A]
time = 0.35, size = 93, normalized size = 0.89 \begin {gather*} \frac {26 \, \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 ) - 13 \, x^{9} \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + 26 \, x^{9} \log \left ({\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + 3 \, {\left (13 \, x^{6} - 57 \, 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)*(2*x^3-1)/x^10,x, algorithm="fricas")

[Out]

1/486*(26*sqrt(3)*x^9*arctan(2/3*sqrt(3)*(x^3 - 1)^(1/3) - 1/3*sqrt(3)) - 13*x^9*log((x^3 - 1)^(2/3) - (x^3 -
1)^(1/3) + 1) + 26*x^9*log((x^3 - 1)^(1/3) + 1) + 3*(13*x^6 - 57*x^3 + 18)*(x^3 - 1)^(1/3))/x^9

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Sympy [C] Result contains complex when optimal does not.
time = 129.22, size = 68, normalized size = 0.65 \begin {gather*} - \frac {2 \Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {5}{3} \\ \frac {8}{3} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{3}}} \right )}}{3 x^{5} \Gamma \left (\frac {8}{3}\right )} + \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)*(2*x**3-1)/x**10,x)

[Out]

-2*gamma(5/3)*hyper((-1/3, 5/3), (8/3,), exp_polar(2*I*pi)/x**3)/(3*x**5*gamma(8/3)) + 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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Giac [A]
time = 0.41, size = 90, normalized size = 0.87 \begin {gather*} \frac {13}{243} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {13 \, {\left (x^{3} - 1\right )}^{\frac {7}{3}} - 31 \, {\left (x^{3} - 1\right )}^{\frac {4}{3}} - 26 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{162 \, x^{9}} - \frac {13}{486} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {13}{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)*(2*x^3-1)/x^10,x, algorithm="giac")

[Out]

13/243*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^3 - 1)^(1/3) - 1)) + 1/162*(13*(x^3 - 1)^(7/3) - 31*(x^3 - 1)^(4/3) -
26*(x^3 - 1)^(1/3))/x^9 - 13/486*log((x^3 - 1)^(2/3) - (x^3 - 1)^(1/3) + 1) + 13/243*log(abs((x^3 - 1)^(1/3) +
1))

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Mupad [B]
time = 1.38, size = 231, normalized size = 2.22 \begin {gather*} \frac {2\,\ln \left (\frac {4\,{\left (x^3-1\right )}^{1/3}}{81}+\frac {4}{81}\right )}{27}-\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}-\frac {\frac {2\,{\left (x^3-1\right )}^{1/3}}{9}-\frac {{\left (x^3-1\right )}^{4/3}}{9}}{{\left (x^3-1\right )}^2+2\,x^3-1}-\ln \left (\frac {1}{3}-\frac {2\,{\left (x^3-1\right )}^{1/3}}{3}+\frac {\sqrt {3}\,1{}\mathrm {i}}{3}\right )\,\left (\frac {1}{27}+\frac {\sqrt {3}\,1{}\mathrm {i}}{27}\right )+\ln \left (\frac {2\,{\left (x^3-1\right )}^{1/3}}{3}-\frac {1}{3}+\frac {\sqrt {3}\,1{}\mathrm {i}}{3}\right )\,\left (-\frac {1}{27}+\frac {\sqrt {3}\,1{}\mathrm {i}}{27}\right )+\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)*(2*x^3 - 1))/x^10,x)

[Out]

(2*log((4*(x^3 - 1)^(1/3))/81 + 4/81))/27 - (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) - ((2
*(x^3 - 1)^(1/3))/9 - (x^3 - 1)^(4/3)/9)/((x^3 - 1)^2 + 2*x^3 - 1) - log((3^(1/2)*1i)/3 - (2*(x^3 - 1)^(1/3))/
3 + 1/3)*((3^(1/2)*1i)/27 + 1/27) + log((3^(1/2)*1i)/3 + (2*(x^3 - 1)^(1/3))/3 - 1/3)*((3^(1/2)*1i)/27 - 1/27)
+ 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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