3.17.45 \(\int \frac {\sqrt [3]{-1+x^6} (1+x^6)}{x^9} \, dx\)

Optimal. Leaf size=111 \[ \frac {\sqrt [3]{x^6-1} \left (-3 x^6-1\right )}{8 x^8}-\frac {1}{6} \log \left (\sqrt [3]{x^6-1}-x^2\right )-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x^2}{2 \sqrt [3]{x^6-1}+x^2}\right )}{2 \sqrt {3}}+\frac {1}{12} \log \left (\left (x^6-1\right )^{2/3}+x^4+\sqrt [3]{x^6-1} x^2\right ) \]

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Rubi [A]  time = 0.07, antiderivative size = 119, normalized size of antiderivative = 1.07, number of steps used = 10, number of rules used = 10, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {451, 275, 277, 331, 292, 31, 634, 618, 204, 628} \begin {gather*} \frac {\left (x^6-1\right )^{4/3}}{8 x^8}-\frac {\sqrt [3]{x^6-1}}{2 x^2}-\frac {1}{6} \log \left (1-\frac {x^2}{\sqrt [3]{x^6-1}}\right )-\frac {\tan ^{-1}\left (\frac {\frac {2 x^2}{\sqrt [3]{x^6-1}}+1}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {1}{12} \log \left (\frac {x^4}{\left (x^6-1\right )^{2/3}}+\frac {x^2}{\sqrt [3]{x^6-1}}+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-1 + x^6)^(1/3)*(1 + x^6))/x^9,x]

[Out]

-1/2*(-1 + x^6)^(1/3)/x^2 + (-1 + x^6)^(4/3)/(8*x^8) - ArcTan[(1 + (2*x^2)/(-1 + x^6)^(1/3))/Sqrt[3]]/(2*Sqrt[
3]) - Log[1 - x^2/(-1 + x^6)^(1/3)]/6 + Log[1 + x^4/(-1 + x^6)^(2/3) + x^2/(-1 + x^6)^(1/3)]/12

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

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 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 277

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

Rule 292

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

Rule 331

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + (m + 1)/n), Subst[Int[x^m/(1 - b*x^n)^(
p + (m + 1)/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[
p, -2^(-1)] && IntegersQ[m, p + (m + 1)/n]

Rule 451

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] + Dist[d/e^n, Int[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a,
b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n*(p + 1) + 1, 0] && (IntegerQ[n] || GtQ[e, 0]) && (
(GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -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 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rubi steps

\begin {align*} \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^9} \, dx &=\frac {\left (-1+x^6\right )^{4/3}}{8 x^8}+\int \frac {\sqrt [3]{-1+x^6}}{x^3} \, dx\\ &=\frac {\left (-1+x^6\right )^{4/3}}{8 x^8}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1+x^3}}{x^2} \, dx,x,x^2\right )\\ &=-\frac {\sqrt [3]{-1+x^6}}{2 x^2}+\frac {\left (-1+x^6\right )^{4/3}}{8 x^8}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {x}{\left (-1+x^3\right )^{2/3}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt [3]{-1+x^6}}{2 x^2}+\frac {\left (-1+x^6\right )^{4/3}}{8 x^8}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {x}{1-x^3} \, dx,x,\frac {x^2}{\sqrt [3]{-1+x^6}}\right )\\ &=-\frac {\sqrt [3]{-1+x^6}}{2 x^2}+\frac {\left (-1+x^6\right )^{4/3}}{8 x^8}+\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\frac {x^2}{\sqrt [3]{-1+x^6}}\right )-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1-x}{1+x+x^2} \, dx,x,\frac {x^2}{\sqrt [3]{-1+x^6}}\right )\\ &=-\frac {\sqrt [3]{-1+x^6}}{2 x^2}+\frac {\left (-1+x^6\right )^{4/3}}{8 x^8}-\frac {1}{6} \log \left (1-\frac {x^2}{\sqrt [3]{-1+x^6}}\right )+\frac {1}{12} \operatorname {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\frac {x^2}{\sqrt [3]{-1+x^6}}\right )-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\frac {x^2}{\sqrt [3]{-1+x^6}}\right )\\ &=-\frac {\sqrt [3]{-1+x^6}}{2 x^2}+\frac {\left (-1+x^6\right )^{4/3}}{8 x^8}-\frac {1}{6} \log \left (1-\frac {x^2}{\sqrt [3]{-1+x^6}}\right )+\frac {1}{12} \log \left (1+\frac {x^4}{\left (-1+x^6\right )^{2/3}}+\frac {x^2}{\sqrt [3]{-1+x^6}}\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 x^2}{\sqrt [3]{-1+x^6}}\right )\\ &=-\frac {\sqrt [3]{-1+x^6}}{2 x^2}+\frac {\left (-1+x^6\right )^{4/3}}{8 x^8}-\frac {\tan ^{-1}\left (\frac {1+\frac {2 x^2}{\sqrt [3]{-1+x^6}}}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{6} \log \left (1-\frac {x^2}{\sqrt [3]{-1+x^6}}\right )+\frac {1}{12} \log \left (1+\frac {x^4}{\left (-1+x^6\right )^{2/3}}+\frac {x^2}{\sqrt [3]{-1+x^6}}\right )\\ \end {align*}

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Mathematica [C]  time = 0.02, size = 50, normalized size = 0.45 \begin {gather*} \frac {\sqrt [3]{x^6-1} \left (-\frac {4 x^6 \, _2F_1\left (-\frac {1}{3},-\frac {1}{3};\frac {2}{3};x^6\right )}{\sqrt [3]{1-x^6}}+x^6-1\right )}{8 x^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-1 + x^6)^(1/3)*(1 + x^6))/x^9,x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.50, size = 111, normalized size = 1.00 \begin {gather*} \frac {\left (-1-3 x^6\right ) \sqrt [3]{-1+x^6}}{8 x^8}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{-1+x^6}}\right )}{2 \sqrt {3}}-\frac {1}{6} \log \left (-x^2+\sqrt [3]{-1+x^6}\right )+\frac {1}{12} \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[((-1 + x^6)^(1/3)*(1 + x^6))/x^9,x]

[Out]

((-1 - 3*x^6)*(-1 + x^6)^(1/3))/(8*x^8) - ArcTan[(Sqrt[3]*x^2)/(x^2 + 2*(-1 + x^6)^(1/3))]/(2*Sqrt[3]) - Log[-
x^2 + (-1 + x^6)^(1/3)]/6 + Log[x^4 + x^2*(-1 + x^6)^(1/3) + (-1 + x^6)^(2/3)]/12

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fricas [A]  time = 0.71, size = 116, normalized size = 1.05 \begin {gather*} -\frac {4 \, \sqrt {3} x^{8} \arctan \left (-\frac {25382 \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{4} - 13720 \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {2}{3}} x^{2} + \sqrt {3} {\left (5831 \, x^{6} - 7200\right )}}{58653 \, x^{6} - 8000}\right ) + 2 \, x^{8} \log \left (-3 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{4} + 3 \, {\left (x^{6} - 1\right )}^{\frac {2}{3}} x^{2} + 1\right ) + 3 \, {\left (3 \, x^{6} + 1\right )} {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{24 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6-1)^(1/3)*(x^6+1)/x^9,x, algorithm="fricas")

[Out]

-1/24*(4*sqrt(3)*x^8*arctan(-(25382*sqrt(3)*(x^6 - 1)^(1/3)*x^4 - 13720*sqrt(3)*(x^6 - 1)^(2/3)*x^2 + sqrt(3)*
(5831*x^6 - 7200))/(58653*x^6 - 8000)) + 2*x^8*log(-3*(x^6 - 1)^(1/3)*x^4 + 3*(x^6 - 1)^(2/3)*x^2 + 1) + 3*(3*
x^6 + 1)*(x^6 - 1)^(1/3))/x^8

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{6} + 1\right )} {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{9}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6-1)^(1/3)*(x^6+1)/x^9,x, algorithm="giac")

[Out]

integrate((x^6 + 1)*(x^6 - 1)^(1/3)/x^9, x)

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maple [C]  time = 7.69, size = 58, normalized size = 0.52

method	result	size
risch	\(-\frac {3 x^{12}-2 x^{6}-1}{8 x^{8} \left (x^{6}-1\right )^{\frac {2}{3}}}+\frac {\left (-\mathrm {signum}\left (x^{6}-1\right )\right )^{\frac {2}{3}} x^{4} \hypergeom \left (\left [\frac {2}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x^{6}\right )}{4 \mathrm {signum}\left (x^{6}-1\right )^{\frac {2}{3}}}\)	\(58\)
meijerg	\(-\frac {\mathrm {signum}\left (x^{6}-1\right )^{\frac {1}{3}} \hypergeom \left (\left [-\frac {1}{3}, -\frac {1}{3}\right ], \left [\frac {2}{3}\right ], x^{6}\right )}{2 \left (-\mathrm {signum}\left (x^{6}-1\right )\right )^{\frac {1}{3}} x^{2}}-\frac {\mathrm {signum}\left (x^{6}-1\right )^{\frac {1}{3}} \left (-x^{6}+1\right )^{\frac {4}{3}}}{8 \left (-\mathrm {signum}\left (x^{6}-1\right )\right )^{\frac {1}{3}} x^{8}}\)	\(66\)
trager	\(-\frac {\left (3 x^{6}+1\right ) \left (x^{6}-1\right )^{\frac {1}{3}}}{8 x^{8}}-\frac {\ln \left (-95839020187648 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )^{2} x^{6}[phone] \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) x^{6}-147783054114048 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{4}[phone] x^{6}[phone] \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x^{2}[phone] x^{4} \left (x^{6}-1\right )^{\frac {1}{3}}-577277555133 x^{2} \left (x^{6}-1\right )^{\frac {2}{3}}+6133697292009472 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )^{2}-21851442872064 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )-199163781631\right )}{6}+\frac {128 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) \ln \left (198613804384256 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )^{2} x^{6}[phone] \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) x^{6}-70729058794752 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{4}-297961563070 x^{6}-77053995319296 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x^{2}[phone] x^{4} \left (x^{6}-1\right )^{\frac {1}{3}}-276285385917 x^{2} \left (x^{6}-1\right )^{\frac {2}{3}}-12711283480592384 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )^{2}-23183919873536 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )[phone]\right )}{3}\)	\(316\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6-1)^(1/3)*(x^6+1)/x^9,x,method=_RETURNVERBOSE)

[Out]

-1/8*(3*x^12-2*x^6-1)/x^8/(x^6-1)^(2/3)+1/4/signum(x^6-1)^(2/3)*(-signum(x^6-1))^(2/3)*x^4*hypergeom([2/3,2/3]
,[5/3],x^6)

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maxima [A]  time = 0.45, size = 93, normalized size = 0.84 \begin {gather*} \frac {1}{6} \, \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 {{\left (x^{6} - 1\right )}^{\frac {1}{3}}}{2 \, x^{2}} + \frac {{\left (x^{6} - 1\right )}^{\frac {4}{3}}}{8 \, x^{8}} + \frac {1}{12} \, \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 {1}{6} \, \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^6-1)^(1/3)*(x^6+1)/x^9,x, algorithm="maxima")

[Out]

1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^6 - 1)^(1/3)/x^2 + 1)) - 1/2*(x^6 - 1)^(1/3)/x^2 + 1/8*(x^6 - 1)^(4/3)/x^
8 + 1/12*log((x^6 - 1)^(1/3)/x^2 + (x^6 - 1)^(2/3)/x^4 + 1) - 1/6*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 {{\left (x^6-1\right )}^{1/3}\,\left (x^6+1\right )}{x^9} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^6 - 1)^(1/3)*(x^6 + 1))/x^9,x)

[Out]

int(((x^6 - 1)^(1/3)*(x^6 + 1))/x^9, x)

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sympy [C]  time = 3.37, size = 167, normalized size = 1.50 \begin {gather*} \begin {cases} \frac {\sqrt [3]{-1 + \frac {1}{x^{6}}} e^{- \frac {2 i \pi }{3}} \Gamma \left (- \frac {4}{3}\right )}{6 \Gamma \left (- \frac {1}{3}\right )} - \frac {\sqrt [3]{-1 + \frac {1}{x^{6}}} e^{- \frac {2 i \pi }{3}} \Gamma \left (- \frac {4}{3}\right )}{6 x^{6} \Gamma \left (- \frac {1}{3}\right )} & \text {for}\: \frac {1}{\left |{x^{6}}\right |} > 1 \\- \frac {\sqrt [3]{1 - \frac {1}{x^{6}}} \Gamma \left (- \frac {4}{3}\right )}{6 \Gamma \left (- \frac {1}{3}\right )} + \frac {\sqrt [3]{1 - \frac {1}{x^{6}}} \Gamma \left (- \frac {4}{3}\right )}{6 x^{6} \Gamma \left (- \frac {1}{3}\right )} & \text {otherwise} \end {cases} + \frac {e^{\frac {i \pi }{3}} \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, - \frac {1}{3} \\ \frac {2}{3} \end {matrix}\middle | {x^{6}} \right )}}{6 x^{2} \Gamma \left (\frac {2}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**6-1)**(1/3)*(x**6+1)/x**9,x)

[Out]

Piecewise(((-1 + x**(-6))**(1/3)*exp(-2*I*pi/3)*gamma(-4/3)/(6*gamma(-1/3)) - (-1 + x**(-6))**(1/3)*exp(-2*I*p
i/3)*gamma(-4/3)/(6*x**6*gamma(-1/3)), 1/Abs(x**6) > 1), (-(1 - 1/x**6)**(1/3)*gamma(-4/3)/(6*gamma(-1/3)) + (
1 - 1/x**6)**(1/3)*gamma(-4/3)/(6*x**6*gamma(-1/3)), True)) + exp(I*pi/3)*gamma(-1/3)*hyper((-1/3, -1/3), (2/3
,), x**6)/(6*x**2*gamma(2/3))

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