3.16.87 \(\int \frac {\sqrt [3]{-1+x^3} (1+x^3)}{x^{13}} \, dx\)

Optimal. Leaf size=109 \[ \frac {25}{729} \log \left (\sqrt [3]{x^3-1}+1\right )-\frac {25 \log \left (\left (x^3-1\right )^{2/3}-\sqrt [3]{x^3-1}+1\right )}{1458}-\frac {25 \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{x^3-1}}{\sqrt {3}}\right )}{243 \sqrt {3}}+\frac {\sqrt [3]{x^3-1} \left (50 x^9+30 x^6-99 x^3-81\right )}{972 x^{12}} \]

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Rubi [A]  time = 0.07, antiderivative size = 116, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 8, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {446, 78, 47, 51, 58, 618, 204, 31} \begin {gather*} \frac {25 \sqrt [3]{x^3-1}}{486 x^3}+\frac {25}{486} \log \left (\sqrt [3]{x^3-1}+1\right )-\frac {25 \tan ^{-1}\left (\frac {1-2 \sqrt [3]{x^3-1}}{\sqrt {3}}\right )}{243 \sqrt {3}}+\frac {\left (x^3-1\right )^{4/3}}{12 x^{12}}-\frac {5 \sqrt [3]{x^3-1}}{27 x^9}+\frac {5 \sqrt [3]{x^3-1}}{162 x^6}-\frac {25 \log (x)}{486} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-1 + x^3)^(1/3)*(1 + x^3))/x^13,x]

[Out]

(-5*(-1 + x^3)^(1/3))/(27*x^9) + (5*(-1 + x^3)^(1/3))/(162*x^6) + (25*(-1 + x^3)^(1/3))/(486*x^3) + (-1 + x^3)
^(4/3)/(12*x^12) - (25*ArcTan[(1 - 2*(-1 + x^3)^(1/3))/Sqrt[3]])/(243*Sqrt[3]) - (25*Log[x])/486 + (25*Log[1 +
(-1 + x^3)^(1/3)])/486

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 78

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] || LtQ
[p, n]))))

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 446

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 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} \left (1+x^3\right )}{x^{13}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1+x} (1+x)}{x^5} \, dx,x,x^3\right )\\ &=\frac {\left (-1+x^3\right )^{4/3}}{12 x^{12}}+\frac {5}{9} \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1+x}}{x^4} \, dx,x,x^3\right )\\ &=-\frac {5 \sqrt [3]{-1+x^3}}{27 x^9}+\frac {\left (-1+x^3\right )^{4/3}}{12 x^{12}}+\frac {5}{81} \operatorname {Subst}\left (\int \frac {1}{(-1+x)^{2/3} x^3} \, dx,x,x^3\right )\\ &=-\frac {5 \sqrt [3]{-1+x^3}}{27 x^9}+\frac {5 \sqrt [3]{-1+x^3}}{162 x^6}+\frac {\left (-1+x^3\right )^{4/3}}{12 x^{12}}+\frac {25}{486} \operatorname {Subst}\left (\int \frac {1}{(-1+x)^{2/3} x^2} \, dx,x,x^3\right )\\ &=-\frac {5 \sqrt [3]{-1+x^3}}{27 x^9}+\frac {5 \sqrt [3]{-1+x^3}}{162 x^6}+\frac {25 \sqrt [3]{-1+x^3}}{486 x^3}+\frac {\left (-1+x^3\right )^{4/3}}{12 x^{12}}+\frac {25}{729} \operatorname {Subst}\left (\int \frac {1}{(-1+x)^{2/3} x} \, dx,x,x^3\right )\\ &=-\frac {5 \sqrt [3]{-1+x^3}}{27 x^9}+\frac {5 \sqrt [3]{-1+x^3}}{162 x^6}+\frac {25 \sqrt [3]{-1+x^3}}{486 x^3}+\frac {\left (-1+x^3\right )^{4/3}}{12 x^{12}}-\frac {25 \log (x)}{486}+\frac {25}{486} \operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sqrt [3]{-1+x^3}\right )+\frac {25}{486} \operatorname {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt [3]{-1+x^3}\right )\\ &=-\frac {5 \sqrt [3]{-1+x^3}}{27 x^9}+\frac {5 \sqrt [3]{-1+x^3}}{162 x^6}+\frac {25 \sqrt [3]{-1+x^3}}{486 x^3}+\frac {\left (-1+x^3\right )^{4/3}}{12 x^{12}}-\frac {25 \log (x)}{486}+\frac {25}{486} \log \left (1+\sqrt [3]{-1+x^3}\right )-\frac {25}{243} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \sqrt [3]{-1+x^3}\right )\\ &=-\frac {5 \sqrt [3]{-1+x^3}}{27 x^9}+\frac {5 \sqrt [3]{-1+x^3}}{162 x^6}+\frac {25 \sqrt [3]{-1+x^3}}{486 x^3}+\frac {\left (-1+x^3\right )^{4/3}}{12 x^{12}}-\frac {25 \tan ^{-1}\left (\frac {1-2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )}{243 \sqrt {3}}-\frac {25 \log (x)}{486}+\frac {25}{486} \log \left (1+\sqrt [3]{-1+x^3}\right )\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 38, normalized size = 0.35 \begin {gather*} \frac {\left (x^3-1\right )^{4/3} \left (5 x^{12} \, _2F_1\left (\frac {4}{3},4;\frac {7}{3};1-x^3\right )+1\right )}{12 x^{12}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-1 + x^3)^(1/3)*(1 + x^3))/x^13,x]

[Out]

((-1 + x^3)^(4/3)*(1 + 5*x^12*Hypergeometric2F1[4/3, 4, 7/3, 1 - x^3]))/(12*x^12)

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IntegrateAlgebraic [A]  time = 0.18, size = 109, normalized size = 1.00 \begin {gather*} \frac {\sqrt [3]{-1+x^3} \left (-81-99 x^3+30 x^6+50 x^9\right )}{972 x^{12}}-\frac {25 \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )}{243 \sqrt {3}}+\frac {25}{729} \log \left (1+\sqrt [3]{-1+x^3}\right )-\frac {25 \log \left (1-\sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )}{1458} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-1 + x^3)^(1/3)*(1 + x^3))/x^13,x]

[Out]

((-1 + x^3)^(1/3)*(-81 - 99*x^3 + 30*x^6 + 50*x^9))/(972*x^12) - (25*ArcTan[1/Sqrt[3] - (2*(-1 + x^3)^(1/3))/S
qrt[3]])/(243*Sqrt[3]) + (25*Log[1 + (-1 + x^3)^(1/3)])/729 - (25*Log[1 - (-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)]
)/1458

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fricas [A]  time = 0.47, size = 98, normalized size = 0.90 \begin {gather*} \frac {100 \, \sqrt {3} x^{12} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) - 50 \, x^{12} \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + 100 \, x^{12} \log \left ({\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + 3 \, {\left (50 \, x^{9} + 30 \, x^{6} - 99 \, x^{3} - 81\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{2916 \, x^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-1)^(1/3)*(x^3+1)/x^13,x, algorithm="fricas")

[Out]

1/2916*(100*sqrt(3)*x^12*arctan(2/3*sqrt(3)*(x^3 - 1)^(1/3) - 1/3*sqrt(3)) - 50*x^12*log((x^3 - 1)^(2/3) - (x^
3 - 1)^(1/3) + 1) + 100*x^12*log((x^3 - 1)^(1/3) + 1) + 3*(50*x^9 + 30*x^6 - 99*x^3 - 81)*(x^3 - 1)^(1/3))/x^1
2

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giac [A]  time = 0.35, size = 99, normalized size = 0.91 \begin {gather*} \frac {25}{729} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {50 \, {\left (x^{3} - 1\right )}^{\frac {10}{3}} + 180 \, {\left (x^{3} - 1\right )}^{\frac {7}{3}} + 111 \, {\left (x^{3} - 1\right )}^{\frac {4}{3}} - 100 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{972 \, x^{12}} - \frac {25}{1458} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {25}{729} \, \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^3+1)/x^13,x, algorithm="giac")

[Out]

25/729*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^3 - 1)^(1/3) - 1)) + 1/972*(50*(x^3 - 1)^(10/3) + 180*(x^3 - 1)^(7/3)
+ 111*(x^3 - 1)^(4/3) - 100*(x^3 - 1)^(1/3))/x^12 - 25/1458*log((x^3 - 1)^(2/3) - (x^3 - 1)^(1/3) + 1) + 25/72
9*log(abs((x^3 - 1)^(1/3) + 1))

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maple [C]  time = 3.04, size = 101, normalized size = 0.93

method	result	size
risch	\(\frac {50 x^{12}-20 x^{9}-129 x^{6}+18 x^{3}+81}{972 x^{12} \left (x^{3}-1\right )^{\frac {2}{3}}}+\frac {25 \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {2}{3}} \left (\frac {2 \Gamma \left (\frac {2}{3}\right ) x^{3} \hypergeom \left (\left [1, 1, \frac {5}{3}\right ], \left [2, 2\right ], x^{3}\right )}{3}+\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+3 \ln \relax (x )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )\right )}{729 \Gamma \left (\frac {2}{3}\right ) \mathrm {signum}\left (x^{3}-1\right )^{\frac {2}{3}}}\)	\(101\)
meijerg	\(\frac {\mathrm {signum}\left (x^{3}-1\right )^{\frac {1}{3}} \left (-\frac {10 \Gamma \left (\frac {2}{3}\right ) x^{3} \hypergeom \left (\left [1, 1, \frac {11}{3}\right ], \left [2, 5\right ], x^{3}\right )}{81}-\frac {5 \left (\frac {4}{15}+\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+3 \ln \relax (x )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )}{27}-\frac {\Gamma \left (\frac {2}{3}\right )}{x^{9}}+\frac {\Gamma \left (\frac {2}{3}\right )}{2 x^{6}}+\frac {\Gamma \left (\frac {2}{3}\right )}{3 x^{3}}\right )}{9 \Gamma \left (\frac {2}{3}\right ) \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {1}{3}}}-\frac {\mathrm {signum}\left (x^{3}-1\right )^{\frac {1}{3}} \left (\frac {22 \Gamma \left (\frac {2}{3}\right ) x^{3} \hypergeom \left (\left [1, 1, \frac {14}{3}\right ], \left [2, 6\right ], x^{3}\right )}{243}+\frac {10 \left (\frac {47}{120}+\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+3 \ln \relax (x )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )}{81}+\frac {3 \Gamma \left (\frac {2}{3}\right )}{4 x^{12}}-\frac {\Gamma \left (\frac {2}{3}\right )}{3 x^{9}}-\frac {\Gamma \left (\frac {2}{3}\right )}{6 x^{6}}-\frac {5 \Gamma \left (\frac {2}{3}\right )}{27 x^{3}}\right )}{9 \Gamma \left (\frac {2}{3}\right ) \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {1}{3}}}\)	\(185\)
trager	\(\frac {\left (x^{3}-1\right )^{\frac {1}{3}} \left (50 x^{9}+30 x^{6}-99 x^{3}-81\right )}{972 x^{12}}+\frac {102400 \RootOf \left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right ) \ln \left (-\frac {24125636608 \RootOf \left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right )^{2} x^{3}-28585984 \RootOf \left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right ) x^{3}[phone] \RootOf \left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}-9894 x^{3}-193005092864 \RootOf \left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right )^{2}-22536192 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right )+19749 \left (x^{3}-1\right )^{\frac {2}{3}}-24584192 \RootOf \left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right )-19749 \left (x^{3}-1\right )^{\frac {1}{3}}+8245}{x^{3}}\right )}{729}-\frac {25 \ln \left (-\frac {24125636608 \RootOf \left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right )^{2} x^{3}[phone] \RootOf \left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right ) x^{3}-22536192 \RootOf \left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}-1477 x^{3}-193005092864 \RootOf \left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right )^{2}[phone] \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right )+14247 \left (x^{3}-1\right )^{\frac {2}{3}}-69656576 \RootOf \left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right )-14247 \left (x^{3}-1\right )^{\frac {1}{3}}+2743}{x^{3}}\right )}{729}-\frac {102400 \ln \left (-\frac {24125636608 \RootOf \left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right )^{2} x^{3}[phone] \RootOf \left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right ) x^{3}-22536192 \RootOf \left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}-1477 x^{3}-193005092864 \RootOf \left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right )^{2}[phone] \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right )+14247 \left (x^{3}-1\right )^{\frac {2}{3}}-69656576 \RootOf \left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right )-14247 \left (x^{3}-1\right )^{\frac {1}{3}}+2743}{x^{3}}\right ) \RootOf \left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right )}{729}\)	\(458\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3-1)^(1/3)*(x^3+1)/x^13,x,method=_RETURNVERBOSE)

[Out]

1/972*(50*x^12-20*x^9-129*x^6+18*x^3+81)/x^12/(x^3-1)^(2/3)+25/729/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 [B]  time = 0.42, size = 184, normalized size = 1.69 \begin {gather*} \frac {25}{729} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {20 \, {\left (x^{3} - 1\right )}^{\frac {10}{3}} + 72 \, {\left (x^{3} - 1\right )}^{\frac {7}{3}} + 93 \, {\left (x^{3} - 1\right )}^{\frac {4}{3}} - 40 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{972 \, {\left ({\left (x^{3} - 1\right )}^{4} + 4 \, {\left (x^{3} - 1\right )}^{3} + 4 \, x^{3} + 6 \, {\left (x^{3} - 1\right )}^{2} - 3\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 {25}{1458} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {25}{729} \, \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^3+1)/x^13,x, algorithm="maxima")

[Out]

25/729*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^3 - 1)^(1/3) - 1)) + 1/972*(20*(x^3 - 1)^(10/3) + 72*(x^3 - 1)^(7/3) +
93*(x^3 - 1)^(4/3) - 40*(x^3 - 1)^(1/3))/((x^3 - 1)^4 + 4*(x^3 - 1)^3 + 4*x^3 + 6*(x^3 - 1)^2 - 3) + 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) - 25/145
8*log((x^3 - 1)^(2/3) - (x^3 - 1)^(1/3) + 1) + 25/729*log((x^3 - 1)^(1/3) + 1)

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mupad [B]  time = 1.61, size = 265, normalized size = 2.43 \begin {gather*} \frac {5\,\ln \left (\frac {25\,{\left (x^3-1\right )}^{1/3}}{6561}+\frac {25}{6561}\right )}{243}+\frac {10\,\ln \left (\frac {100\,{\left (x^3-1\right )}^{1/3}}{59049}+\frac {100}{59049}\right )}{729}+\frac {\frac {31\,{\left (x^3-1\right )}^{4/3}}{324}-\frac {10\,{\left (x^3-1\right )}^{1/3}}{243}+\frac {2\,{\left (x^3-1\right )}^{7/3}}{27}+\frac {5\,{\left (x^3-1\right )}^{10/3}}{243}}{6\,{\left (x^3-1\right )}^2+4\,{\left (x^3-1\right )}^3+{\left (x^3-1\right )}^4+4\,x^3-3}+\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 )-\ln \left (\frac {5}{81}-\frac {10\,{\left (x^3-1\right )}^{1/3}}{81}+\frac {\sqrt {3}\,5{}\mathrm {i}}{81}\right )\,\left (\frac {5}{729}+\frac {\sqrt {3}\,5{}\mathrm {i}}{729}\right )+\ln \left (\frac {10\,{\left (x^3-1\right )}^{1/3}}{81}-\frac {5}{81}+\frac {\sqrt {3}\,5{}\mathrm {i}}{81}\right )\,\left (-\frac {5}{729}+\frac {\sqrt {3}\,5{}\mathrm {i}}{729}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^3 - 1)^(1/3)*(x^3 + 1))/x^13,x)

[Out]

(5*log((25*(x^3 - 1)^(1/3))/6561 + 25/6561))/243 + (10*log((100*(x^3 - 1)^(1/3))/59049 + 100/59049))/729 + ((3
1*(x^3 - 1)^(4/3))/324 - (10*(x^3 - 1)^(1/3))/243 + (2*(x^3 - 1)^(7/3))/27 + (5*(x^3 - 1)^(10/3))/243)/(6*(x^3
- 1)^2 + 4*(x^3 - 1)^3 + (x^3 - 1)^4 + 4*x^3 - 3) + ((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) - log((3^(1/2)*5i)/81 - (10*(x^3 - 1)^(1/3))/81 + 5/81)*((3^(1/2)*5i)/729 + 5/729) + log((3^(1/2)*5i)/8
1 + (10*(x^3 - 1)^(1/3))/81 - 5/81)*((3^(1/2)*5i)/729 - 5/729)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**3-1)**(1/3)*(x**3+1)/x**13,x)

[Out]

Timed out

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