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

Optimal. Leaf size=117 \[ -\frac {7}{243} \log \left (\sqrt [3]{x^6-1}-x^2\right )+\frac {7 \tan ^{-1}\left (\frac {\sqrt {3} x^2}{2 \sqrt [3]{x^6-1}+x^2}\right )}{81 \sqrt {3}}+\frac {7}{486} \log \left (\left (x^6-1\right )^{2/3}+x^4+\sqrt [3]{x^6-1} x^2\right )+\frac {1}{324} \left (x^6-1\right )^{2/3} \left (18 x^{14}+21 x^8+28 x^2\right ) \]

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Rubi [A]  time = 0.05, antiderivative size = 101, normalized size of antiderivative = 0.86, number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {275, 321, 239} \begin {gather*} \frac {1}{18} \left (x^6-1\right )^{2/3} x^{14}+\frac {7}{108} \left (x^6-1\right )^{2/3} x^8+\frac {7}{81} \left (x^6-1\right )^{2/3} x^2-\frac {7}{162} \log \left (x^2-\sqrt [3]{x^6-1}\right )+\frac {7 \tan ^{-1}\left (\frac {\frac {2 x^2}{\sqrt [3]{x^6-1}}+1}{\sqrt {3}}\right )}{81 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^19/(-1 + x^6)^(1/3),x]

[Out]

(7*x^2*(-1 + x^6)^(2/3))/81 + (7*x^8*(-1 + x^6)^(2/3))/108 + (x^14*(-1 + x^6)^(2/3))/18 + (7*ArcTan[(1 + (2*x^
2)/(-1 + x^6)^(1/3))/Sqrt[3]])/(81*Sqrt[3]) - (7*Log[x^2 - (-1 + x^6)^(1/3)])/162

Rule 239

Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + (2*Rt[b, 3]*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sq
rt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 321

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

Rubi steps

\begin {align*} \int \frac {x^{19}}{\sqrt [3]{-1+x^6}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^9}{\sqrt [3]{-1+x^3}} \, dx,x,x^2\right )\\ &=\frac {1}{18} x^{14} \left (-1+x^6\right )^{2/3}+\frac {7}{18} \operatorname {Subst}\left (\int \frac {x^6}{\sqrt [3]{-1+x^3}} \, dx,x,x^2\right )\\ &=\frac {7}{108} x^8 \left (-1+x^6\right )^{2/3}+\frac {1}{18} x^{14} \left (-1+x^6\right )^{2/3}+\frac {7}{27} \operatorname {Subst}\left (\int \frac {x^3}{\sqrt [3]{-1+x^3}} \, dx,x,x^2\right )\\ &=\frac {7}{81} x^2 \left (-1+x^6\right )^{2/3}+\frac {7}{108} x^8 \left (-1+x^6\right )^{2/3}+\frac {1}{18} x^{14} \left (-1+x^6\right )^{2/3}+\frac {7}{81} \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^3}} \, dx,x,x^2\right )\\ &=\frac {7}{81} x^2 \left (-1+x^6\right )^{2/3}+\frac {7}{108} x^8 \left (-1+x^6\right )^{2/3}+\frac {1}{18} x^{14} \left (-1+x^6\right )^{2/3}+\frac {7 \tan ^{-1}\left (\frac {1+\frac {2 x^2}{\sqrt [3]{-1+x^6}}}{\sqrt {3}}\right )}{81 \sqrt {3}}-\frac {7}{162} \log \left (x^2-\sqrt [3]{-1+x^6}\right )\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 127, normalized size = 1.09 \begin {gather*} \frac {1}{972} \left (54 \left (x^6-1\right )^{2/3} x^{14}+63 \left (x^6-1\right )^{2/3} x^8+84 \left (x^6-1\right )^{2/3} x^2-28 \log \left (1-\frac {x^2}{\sqrt [3]{x^6-1}}\right )+28 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 x^2}{\sqrt [3]{x^6-1}}+1}{\sqrt {3}}\right )+14 \log \left (\frac {x^4}{\left (x^6-1\right )^{2/3}}+\frac {x^2}{\sqrt [3]{x^6-1}}+1\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^19/(-1 + x^6)^(1/3),x]

[Out]

(84*x^2*(-1 + x^6)^(2/3) + 63*x^8*(-1 + x^6)^(2/3) + 54*x^14*(-1 + x^6)^(2/3) + 28*Sqrt[3]*ArcTan[(1 + (2*x^2)
/(-1 + x^6)^(1/3))/Sqrt[3]] - 28*Log[1 - x^2/(-1 + x^6)^(1/3)] + 14*Log[1 + x^4/(-1 + x^6)^(2/3) + x^2/(-1 + x
^6)^(1/3)])/972

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IntegrateAlgebraic [A]  time = 7.50, size = 117, normalized size = 1.00 \begin {gather*} \frac {1}{324} \left (-1+x^6\right )^{2/3} \left (28 x^2+21 x^8+18 x^{14}\right )+\frac {7 \tan ^{-1}\left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{-1+x^6}}\right )}{81 \sqrt {3}}-\frac {7}{243} \log \left (-x^2+\sqrt [3]{-1+x^6}\right )+\frac {7}{486} \log \left (x^4+x^2 \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-1 + x^6)^(2/3)*(28*x^2 + 21*x^8 + 18*x^14))/324 + (7*ArcTan[(Sqrt[3]*x^2)/(x^2 + 2*(-1 + x^6)^(1/3))])/(81*
Sqrt[3]) - (7*Log[-x^2 + (-1 + x^6)^(1/3)])/243 + (7*Log[x^4 + x^2*(-1 + x^6)^(1/3) + (-1 + x^6)^(2/3)])/486

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fricas [A]  time = 0.44, size = 107, normalized size = 0.91 \begin {gather*} \frac {1}{324} \, {\left (18 \, x^{14} + 21 \, x^{8} + 28 \, x^{2}\right )} {\left (x^{6} - 1\right )}^{\frac {2}{3}} - \frac {7}{243} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2} + 2 \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{3 \, x^{2}}\right ) - \frac {7}{243} \, \log \left (-\frac {x^{2} - {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}}\right ) + \frac {7}{486} \, \log \left (\frac {x^{4} + {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{2} + {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{x^{4}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/324*(18*x^14 + 21*x^8 + 28*x^2)*(x^6 - 1)^(2/3) - 7/243*sqrt(3)*arctan(1/3*(sqrt(3)*x^2 + 2*sqrt(3)*(x^6 - 1
)^(1/3))/x^2) - 7/243*log(-(x^2 - (x^6 - 1)^(1/3))/x^2) + 7/486*log((x^4 + (x^6 - 1)^(1/3)*x^2 + (x^6 - 1)^(2/
3))/x^4)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{19}}{{\left (x^{6} - 1\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^19/(x^6 - 1)^(1/3), x)

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maple [C]  time = 3.35, size = 33, normalized size = 0.28

method	result	size
meijerg	\(\frac {\left (-\mathrm {signum}\left (x^{6}-1\right )\right )^{\frac {1}{3}} x^{20} \hypergeom \left (\left [\frac {1}{3}, \frac {10}{3}\right ], \left [\frac {13}{3}\right ], x^{6}\right )}{20 \mathrm {signum}\left (x^{6}-1\right )^{\frac {1}{3}}}\)	\(33\)
risch	\(\frac {x^{2} \left (18 x^{12}+21 x^{6}+28\right ) \left (x^{6}-1\right )^{\frac {2}{3}}}{324}+\frac {7 \left (-\mathrm {signum}\left (x^{6}-1\right )\right )^{\frac {1}{3}} x^{2} \hypergeom \left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x^{6}\right )}{81 \mathrm {signum}\left (x^{6}-1\right )^{\frac {1}{3}}}\)	\(58\)
trager	\(\frac {x^{2} \left (18 x^{12}+21 x^{6}+28\right ) \left (x^{6}-1\right )^{\frac {2}{3}}}{324}+\frac {7 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{6}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{6}+2 x^{6}+3 x^{4} \left (x^{6}-1\right )^{\frac {1}{3}}+3 x^{2} \left (x^{6}-1\right )^{\frac {2}{3}}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-1\right )}{243}-\frac {7 \ln \left (\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{6}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{6}+3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{4}+x^{6}+3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x^{2}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-1\right ) \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )}{243}+\frac {7 \ln \left (\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{6}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{6}+3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{4}+x^{6}+3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x^{2}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-1\right )}{243}\)	\(298\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20/signum(x^6-1)^(1/3)*(-signum(x^6-1))^(1/3)*x^20*hypergeom([1/3,10/3],[13/3],x^6)

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maxima [A]  time = 0.41, size = 145, normalized size = 1.24 \begin {gather*} -\frac {7}{243} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} + 1\right )}\right ) - \frac {\frac {67 \, {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{x^{4}} - \frac {77 \, {\left (x^{6} - 1\right )}^{\frac {5}{3}}}{x^{10}} + \frac {28 \, {\left (x^{6} - 1\right )}^{\frac {8}{3}}}{x^{16}}}{324 \, {\left (\frac {3 \, {\left (x^{6} - 1\right )}}{x^{6}} - \frac {3 \, {\left (x^{6} - 1\right )}^{2}}{x^{12}} + \frac {{\left (x^{6} - 1\right )}^{3}}{x^{18}} - 1\right )}} + \frac {7}{486} \, \log \left (\frac {{\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} + \frac {{\left (x^{6} - 1\right )}^{\frac {2}{3}}}{x^{4}} + 1\right ) - \frac {7}{243} \, \log \left (\frac {{\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(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)/x^2 + 1)) - 1/324*(67*(x^6 - 1)^(2/3)/x^4 - 77*(x^6 - 1)^
(5/3)/x^10 + 28*(x^6 - 1)^(8/3)/x^16)/(3*(x^6 - 1)/x^6 - 3*(x^6 - 1)^2/x^12 + (x^6 - 1)^3/x^18 - 1) + 7/486*lo
g((x^6 - 1)^(1/3)/x^2 + (x^6 - 1)^(2/3)/x^4 + 1) - 7/243*log((x^6 - 1)^(1/3)/x^2 - 1)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^{19}}{{\left (x^6-1\right )}^{1/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^19/(x^6 - 1)^(1/3),x)

[Out]

int(x^19/(x^6 - 1)^(1/3), x)

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sympy [C]  time = 2.38, size = 34, normalized size = 0.29 \begin {gather*} - \frac {x^{20} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {10}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {10}{3} \\ \frac {13}{3} \end {matrix}\middle | {x^{6}} \right )}}{6 \Gamma \left (\frac {13}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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