3.14.80 \(\int \frac {1}{x^{13} \sqrt [3]{-1+x^6}} \, dx\)

Optimal. Leaf size=99 \[ -\frac {1}{27} \log \left (\sqrt [3]{x^6-1}+1\right )+\frac {1}{54} \log \left (\left (x^6-1\right )^{2/3}-\sqrt [3]{x^6-1}+1\right )-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{x^6-1}}{\sqrt {3}}\right )}{9 \sqrt {3}}+\frac {\left (x^6-1\right )^{2/3} \left (4 x^6+3\right )}{36 x^{12}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.06, antiderivative size = 84, normalized size of antiderivative = 0.85, number of steps used = 7, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {266, 51, 56, 618, 204, 31} \begin {gather*} \frac {\left (x^6-1\right )^{2/3}}{9 x^6}-\frac {1}{18} \log \left (\sqrt [3]{x^6-1}+1\right )-\frac {\tan ^{-1}\left (\frac {1-2 \sqrt [3]{x^6-1}}{\sqrt {3}}\right )}{9 \sqrt {3}}+\frac {\left (x^6-1\right )^{2/3}}{12 x^{12}}+\frac {\log (x)}{9} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^13*(-1 + x^6)^(1/3)),x]

[Out]

(-1 + x^6)^(2/3)/(12*x^12) + (-1 + x^6)^(2/3)/(9*x^6) - ArcTan[(1 - 2*(-1 + x^6)^(1/3))/Sqrt[3]]/(9*Sqrt[3]) +
Log[x]/9 - Log[1 + (-1 + x^6)^(1/3)]/18

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 56

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[-((b*c - a*d)/b), 3]}, Simp
[Log[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] && Ne
gQ[(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 {1}{x^{13} \sqrt [3]{-1+x^6}} \, dx &=\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x} x^3} \, dx,x,x^6\right )\\ &=\frac {\left (-1+x^6\right )^{2/3}}{12 x^{12}}+\frac {1}{9} \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x} x^2} \, dx,x,x^6\right )\\ &=\frac {\left (-1+x^6\right )^{2/3}}{12 x^{12}}+\frac {\left (-1+x^6\right )^{2/3}}{9 x^6}+\frac {1}{27} \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x} x} \, dx,x,x^6\right )\\ &=\frac {\left (-1+x^6\right )^{2/3}}{12 x^{12}}+\frac {\left (-1+x^6\right )^{2/3}}{9 x^6}+\frac {\log (x)}{9}-\frac {1}{18} \operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sqrt [3]{-1+x^6}\right )+\frac {1}{18} \operatorname {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt [3]{-1+x^6}\right )\\ &=\frac {\left (-1+x^6\right )^{2/3}}{12 x^{12}}+\frac {\left (-1+x^6\right )^{2/3}}{9 x^6}+\frac {\log (x)}{9}-\frac {1}{18} \log \left (1+\sqrt [3]{-1+x^6}\right )-\frac {1}{9} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \sqrt [3]{-1+x^6}\right )\\ &=\frac {\left (-1+x^6\right )^{2/3}}{12 x^{12}}+\frac {\left (-1+x^6\right )^{2/3}}{9 x^6}-\frac {\tan ^{-1}\left (\frac {1-2 \sqrt [3]{-1+x^6}}{\sqrt {3}}\right )}{9 \sqrt {3}}+\frac {\log (x)}{9}-\frac {1}{18} \log \left (1+\sqrt [3]{-1+x^6}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.00, size = 28, normalized size = 0.28 \begin {gather*} \frac {1}{4} \left (x^6-1\right )^{2/3} \, _2F_1\left (\frac {2}{3},3;\frac {5}{3};1-x^6\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^13*(-1 + x^6)^(1/3)),x]

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.10, size = 99, normalized size = 1.00 \begin {gather*} \frac {\left (-1+x^6\right )^{2/3} \left (3+4 x^6\right )}{36 x^{12}}-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-1+x^6}}{\sqrt {3}}\right )}{9 \sqrt {3}}-\frac {1}{27} \log \left (1+\sqrt [3]{-1+x^6}\right )+\frac {1}{54} \log \left (1-\sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x^13*(-1 + x^6)^(1/3)),x]

[Out]

((-1 + x^6)^(2/3)*(3 + 4*x^6))/(36*x^12) - ArcTan[1/Sqrt[3] - (2*(-1 + x^6)^(1/3))/Sqrt[3]]/(9*Sqrt[3]) - Log[
1 + (-1 + x^6)^(1/3)]/27 + Log[1 - (-1 + x^6)^(1/3) + (-1 + x^6)^(2/3)]/54

________________________________________________________________________________________

fricas [A]  time = 0.68, size = 88, normalized size = 0.89 \begin {gather*} \frac {4 \, \sqrt {3} x^{12} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + 2 \, x^{12} \log \left ({\left (x^{6} - 1\right )}^{\frac {2}{3}} - {\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1\right ) - 4 \, x^{12} \log \left ({\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1\right ) + 3 \, {\left (4 \, x^{6} + 3\right )} {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{108 \, x^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^13/(x^6-1)^(1/3),x, algorithm="fricas")

[Out]

1/108*(4*sqrt(3)*x^12*arctan(2/3*sqrt(3)*(x^6 - 1)^(1/3) - 1/3*sqrt(3)) + 2*x^12*log((x^6 - 1)^(2/3) - (x^6 -
1)^(1/3) + 1) - 4*x^12*log((x^6 - 1)^(1/3) + 1) + 3*(4*x^6 + 3)*(x^6 - 1)^(2/3))/x^12

________________________________________________________________________________________

giac [A]  time = 0.16, size = 81, normalized size = 0.82 \begin {gather*} \frac {1}{27} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {4 \, {\left (x^{6} - 1\right )}^{\frac {5}{3}} + 7 \, {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{36 \, x^{12}} + \frac {1}{54} \, \log \left ({\left (x^{6} - 1\right )}^{\frac {2}{3}} - {\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{27} \, \log \left ({\left | {\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^13/(x^6-1)^(1/3),x, algorithm="giac")

[Out]

1/27*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^6 - 1)^(1/3) - 1)) + 1/36*(4*(x^6 - 1)^(5/3) + 7*(x^6 - 1)^(2/3))/x^12 +
1/54*log((x^6 - 1)^(2/3) - (x^6 - 1)^(1/3) + 1) - 1/27*log(abs((x^6 - 1)^(1/3) + 1))

________________________________________________________________________________________

maple [C]  time = 8.24, size = 108, normalized size = 1.09

method	result	size
risch	\(\frac {4 x^{12}-x^{6}-3}{36 x^{12} \left (x^{6}-1\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (-\mathrm {signum}\left (x^{6}-1\right )\right )^{\frac {1}{3}} \left (\frac {2 \pi \sqrt {3}\, x^{6} \hypergeom \left (\left [1, 1, \frac {4}{3}\right ], \left [2, 2\right ], x^{6}\right )}{9 \Gamma \left (\frac {2}{3}\right )}+\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+6 \ln \relax (x )+i \pi \right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}\right )}{54 \pi \mathrm {signum}\left (x^{6}-1\right )^{\frac {1}{3}}}\)	\(108\)
meijerg	\(\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (-\mathrm {signum}\left (x^{6}-1\right )\right )^{\frac {1}{3}} \left (\frac {28 \pi \sqrt {3}\, x^{6} \hypergeom \left (\left [1, 1, \frac {10}{3}\right ], \left [2, 4\right ], x^{6}\right )}{243 \Gamma \left (\frac {2}{3}\right )}+\frac {4 \left (\frac {9}{4}-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+6 \ln \relax (x )+i \pi \right ) \pi \sqrt {3}}{27 \Gamma \left (\frac {2}{3}\right )}-\frac {\pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right ) x^{12}}-\frac {2 \pi \sqrt {3}}{9 \Gamma \left (\frac {2}{3}\right ) x^{6}}\right )}{12 \pi \mathrm {signum}\left (x^{6}-1\right )^{\frac {1}{3}}}\)	\(110\)
trager	\(\frac {\left (x^{6}-1\right )^{\frac {2}{3}} \left (4 x^{6}+3\right )}{36 x^{12}}+\frac {\ln \left (-\frac {100217735779212520130799137193984 \RootOf \left (66588770304 \textit {\_Z}^{2}-258048 \textit {\_Z} +1\right )^{2} x^{6}+23595573475745213987715084288 \RootOf \left (66588770304 \textit {\_Z}^{2}-258048 \textit {\_Z} +1\right ) x^{6}-116115113033469041646202 x^{6}-17616242845163488187704025088 \left (x^{6}-1\right )^{\frac {2}{3}} \RootOf \left (66588770304 \textit {\_Z}^{2}-258048 \textit {\_Z} +1\right )-6413935089869601288371144780414976 \RootOf \left (66588770304 \textit {\_Z}^{2}-258048 \textit {\_Z} +1\right )^{2}+17616242845163488187704025088 \RootOf \left (66588770304 \textit {\_Z}^{2}-258048 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}}+210563861480318360105175 \left (x^{6}-1\right )^{\frac {2}{3}}+7239346385791997955847348224 \RootOf \left (66588770304 \textit {\_Z}^{2}-258048 \textit {\_Z} +1\right )-210563861480318360105175 \left (x^{6}-1\right )^{\frac {1}{3}}+114242288629703411942231}{x^{6}}\right )}{27}-\frac {28672 \ln \left (-\frac {100217735779212520130799137193984 \RootOf \left (66588770304 \textit {\_Z}^{2}-258048 \textit {\_Z} +1\right )^{2} x^{6}+23595573475745213987715084288 \RootOf \left (66588770304 \textit {\_Z}^{2}-258048 \textit {\_Z} +1\right ) x^{6}-116115113033469041646202 x^{6}-17616242845163488187704025088 \left (x^{6}-1\right )^{\frac {2}{3}} \RootOf \left (66588770304 \textit {\_Z}^{2}-258048 \textit {\_Z} +1\right )-6413935089869601288371144780414976 \RootOf \left (66588770304 \textit {\_Z}^{2}-258048 \textit {\_Z} +1\right )^{2}+17616242845163488187704025088 \RootOf \left (66588770304 \textit {\_Z}^{2}-258048 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}}+210563861480318360105175 \left (x^{6}-1\right )^{\frac {2}{3}}+7239346385791997955847348224 \RootOf \left (66588770304 \textit {\_Z}^{2}-258048 \textit {\_Z} +1\right )-210563861480318360105175 \left (x^{6}-1\right )^{\frac {1}{3}}+114242288629703411942231}{x^{6}}\right ) \RootOf \left (66588770304 \textit {\_Z}^{2}-258048 \textit {\_Z} +1\right )}{3}+\frac {28672 \RootOf \left (66588770304 \textit {\_Z}^{2}-258048 \textit {\_Z} +1\right ) \ln \left (-\frac {100217735779212520130799137193984 \RootOf \left (66588770304 \textit {\_Z}^{2}-258048 \textit {\_Z} +1\right )^{2} x^{6}-24372310639212572929701064704 \RootOf \left (66588770304 \textit {\_Z}^{2}-258048 \textit {\_Z} +1\right ) x^{6}-23171389162410581752275 x^{6}+17616242845163488187704025088 \left (x^{6}-1\right )^{\frac {2}{3}} \RootOf \left (66588770304 \textit {\_Z}^{2}-258048 \textit {\_Z} +1\right )-6413935089869601288371144780414976 \RootOf \left (66588770304 \textit {\_Z}^{2}-258048 \textit {\_Z} +1\right )^{2}-17616242845163488187704025088 \RootOf \left (66588770304 \textit {\_Z}^{2}-258048 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}}+142296551347461340528569 \left (x^{6}-1\right )^{\frac {2}{3}}+42471832076118974331255398400 \RootOf \left (66588770304 \textit {\_Z}^{2}-258048 \textit {\_Z} +1\right )-142296551347461340528569 \left (x^{6}-1\right )^{\frac {1}{3}}+45974978496846392365625}{x^{6}}\right )}{3}\)	\(448\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^13/(x^6-1)^(1/3),x,method=_RETURNVERBOSE)

[Out]

1/36*(4*x^12-x^6-3)/x^12/(x^6-1)^(1/3)+1/54/Pi*3^(1/2)*GAMMA(2/3)/signum(x^6-1)^(1/3)*(-signum(x^6-1))^(1/3)*(
2/9*Pi*3^(1/2)/GAMMA(2/3)*x^6*hypergeom([1,1,4/3],[2,2],x^6)+2/3*(-1/6*Pi*3^(1/2)-3/2*ln(3)+6*ln(x)+I*Pi)*Pi*3
^(1/2)/GAMMA(2/3))

________________________________________________________________________________________

maxima [A]  time = 0.42, size = 93, normalized size = 0.94 \begin {gather*} \frac {1}{27} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {4 \, {\left (x^{6} - 1\right )}^{\frac {5}{3}} + 7 \, {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{36 \, {\left (2 \, x^{6} + {\left (x^{6} - 1\right )}^{2} - 1\right )}} + \frac {1}{54} \, \log \left ({\left (x^{6} - 1\right )}^{\frac {2}{3}} - {\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{27} \, \log \left ({\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^13/(x^6-1)^(1/3),x, algorithm="maxima")

[Out]

1/27*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^6 - 1)^(1/3) - 1)) + 1/36*(4*(x^6 - 1)^(5/3) + 7*(x^6 - 1)^(2/3))/(2*x^6
+ (x^6 - 1)^2 - 1) + 1/54*log((x^6 - 1)^(2/3) - (x^6 - 1)^(1/3) + 1) - 1/27*log((x^6 - 1)^(1/3) + 1)

________________________________________________________________________________________

mupad [B]  time = 0.99, size = 116, normalized size = 1.17 \begin {gather*} \frac {\frac {7\,{\left (x^6-1\right )}^{2/3}}{36}+\frac {{\left (x^6-1\right )}^{5/3}}{9}}{{\left (x^6-1\right )}^2+2\,x^6-1}-\ln \left (9\,{\left (-\frac {1}{54}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54}\right )}^2+\frac {{\left (x^6-1\right )}^{1/3}}{81}\right )\,\left (-\frac {1}{54}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54}\right )+\ln \left (9\,{\left (\frac {1}{54}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54}\right )}^2+\frac {{\left (x^6-1\right )}^{1/3}}{81}\right )\,\left (\frac {1}{54}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54}\right )-\frac {\ln \left (\frac {{\left (x^6-1\right )}^{1/3}}{81}+\frac {1}{81}\right )}{27} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^13*(x^6 - 1)^(1/3)),x)

[Out]

log(9*((3^(1/2)*1i)/54 + 1/54)^2 + (x^6 - 1)^(1/3)/81)*((3^(1/2)*1i)/54 + 1/54) - log(9*((3^(1/2)*1i)/54 - 1/5
4)^2 + (x^6 - 1)^(1/3)/81)*((3^(1/2)*1i)/54 - 1/54) - log((x^6 - 1)^(1/3)/81 + 1/81)/27 + ((7*(x^6 - 1)^(2/3))
/36 + (x^6 - 1)^(5/3)/9)/((x^6 - 1)^2 + 2*x^6 - 1)

________________________________________________________________________________________

sympy [C]  time = 1.60, size = 32, normalized size = 0.32 \begin {gather*} - \frac {\Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{6}}} \right )}}{6 x^{14} \Gamma \left (\frac {10}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**13/(x**6-1)**(1/3),x)

[Out]

-gamma(7/3)*hyper((1/3, 7/3), (10/3,), exp_polar(2*I*pi)/x**6)/(6*x**14*gamma(10/3))

________________________________________________________________________________________