3.14.73 \(\int \frac {\sqrt [3]{-1+x^3} (1+x^3)}{x^7} \, dx\)

Optimal. Leaf size=99 \[ \frac {4}{27} \log \left (\sqrt [3]{x^3-1}+1\right )-\frac {2}{27} \log \left (\left (x^3-1\right )^{2/3}-\sqrt [3]{x^3-1}+1\right )-\frac {4 \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{x^3-1}}{\sqrt {3}}\right )}{9 \sqrt {3}}+\frac {\sqrt [3]{x^3-1} \left (-5 x^3-3\right )}{18 x^6} \]

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Rubi [A]  time = 0.06, antiderivative size = 84, normalized size of antiderivative = 0.85, number of steps used = 7, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {446, 78, 47, 58, 618, 204, 31} \begin {gather*} -\frac {4 \sqrt [3]{x^3-1}}{9 x^3}+\frac {2}{9} \log \left (\sqrt [3]{x^3-1}+1\right )-\frac {4 \tan ^{-1}\left (\frac {1-2 \sqrt [3]{x^3-1}}{\sqrt {3}}\right )}{9 \sqrt {3}}+\frac {\left (x^3-1\right )^{4/3}}{6 x^6}-\frac {2 \log (x)}{9} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-1 + x^3)^(1/3)*(1 + x^3))/x^7,x]

[Out]

(-4*(-1 + x^3)^(1/3))/(9*x^3) + (-1 + x^3)^(4/3)/(6*x^6) - (4*ArcTan[(1 - 2*(-1 + x^3)^(1/3))/Sqrt[3]])/(9*Sqr
t[3]) - (2*Log[x])/9 + (2*Log[1 + (-1 + x^3)^(1/3)])/9

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 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^7} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1+x} (1+x)}{x^3} \, dx,x,x^3\right )\\ &=\frac {\left (-1+x^3\right )^{4/3}}{6 x^6}+\frac {4}{9} \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1+x}}{x^2} \, dx,x,x^3\right )\\ &=-\frac {4 \sqrt [3]{-1+x^3}}{9 x^3}+\frac {\left (-1+x^3\right )^{4/3}}{6 x^6}+\frac {4}{27} \operatorname {Subst}\left (\int \frac {1}{(-1+x)^{2/3} x} \, dx,x,x^3\right )\\ &=-\frac {4 \sqrt [3]{-1+x^3}}{9 x^3}+\frac {\left (-1+x^3\right )^{4/3}}{6 x^6}-\frac {2 \log (x)}{9}+\frac {2}{9} \operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sqrt [3]{-1+x^3}\right )+\frac {2}{9} \operatorname {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt [3]{-1+x^3}\right )\\ &=-\frac {4 \sqrt [3]{-1+x^3}}{9 x^3}+\frac {\left (-1+x^3\right )^{4/3}}{6 x^6}-\frac {2 \log (x)}{9}+\frac {2}{9} \log \left (1+\sqrt [3]{-1+x^3}\right )-\frac {4}{9} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \sqrt [3]{-1+x^3}\right )\\ &=-\frac {4 \sqrt [3]{-1+x^3}}{9 x^3}+\frac {\left (-1+x^3\right )^{4/3}}{6 x^6}-\frac {4 \tan ^{-1}\left (\frac {1-2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )}{9 \sqrt {3}}-\frac {2 \log (x)}{9}+\frac {2}{9} \log \left (1+\sqrt [3]{-1+x^3}\right )\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 38, normalized size = 0.38 \begin {gather*} \frac {\left (x^3-1\right )^{4/3} \left (2 x^6 \, _2F_1\left (\frac {4}{3},2;\frac {7}{3};1-x^3\right )+1\right )}{6 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-1 + x^3)^(1/3)*(1 + x^3))/x^7,x]

[Out]

((-1 + x^3)^(4/3)*(1 + 2*x^6*Hypergeometric2F1[4/3, 2, 7/3, 1 - x^3]))/(6*x^6)

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IntegrateAlgebraic [A]  time = 0.11, size = 99, normalized size = 1.00 \begin {gather*} \frac {\left (-3-5 x^3\right ) \sqrt [3]{-1+x^3}}{18 x^6}-\frac {4 \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )}{9 \sqrt {3}}+\frac {4}{27} \log \left (1+\sqrt [3]{-1+x^3}\right )-\frac {2}{27} \log \left (1-\sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-1 + x^3)^(1/3)*(1 + x^3))/x^7,x]

[Out]

((-3 - 5*x^3)*(-1 + x^3)^(1/3))/(18*x^6) - (4*ArcTan[1/Sqrt[3] - (2*(-1 + x^3)^(1/3))/Sqrt[3]])/(9*Sqrt[3]) +
(4*Log[1 + (-1 + x^3)^(1/3)])/27 - (2*Log[1 - (-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)])/27

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fricas [A]  time = 0.43, size = 88, normalized size = 0.89 \begin {gather*} \frac {8 \, \sqrt {3} x^{6} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) - 4 \, x^{6} \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + 8 \, x^{6} \log \left ({\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) - 3 \, {\left (5 \, x^{3} + 3\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{54 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-1)^(1/3)*(x^3+1)/x^7,x, algorithm="fricas")

[Out]

1/54*(8*sqrt(3)*x^6*arctan(2/3*sqrt(3)*(x^3 - 1)^(1/3) - 1/3*sqrt(3)) - 4*x^6*log((x^3 - 1)^(2/3) - (x^3 - 1)^
(1/3) + 1) + 8*x^6*log((x^3 - 1)^(1/3) + 1) - 3*(5*x^3 + 3)*(x^3 - 1)^(1/3))/x^6

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giac [A]  time = 0.18, size = 81, normalized size = 0.82 \begin {gather*} \frac {4}{27} \, \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 {4}{3}} + 8 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{18 \, x^{6}} - \frac {2}{27} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {4}{27} \, \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^7,x, algorithm="giac")

[Out]

4/27*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^3 - 1)^(1/3) - 1)) - 1/18*(5*(x^3 - 1)^(4/3) + 8*(x^3 - 1)^(1/3))/x^6 -
2/27*log((x^3 - 1)^(2/3) - (x^3 - 1)^(1/3) + 1) + 4/27*log(abs((x^3 - 1)^(1/3) + 1))

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maple [C]  time = 3.00, size = 91, normalized size = 0.92

method	result	size
risch	\(-\frac {5 x^{6}-2 x^{3}-3}{18 x^{6} \left (x^{3}-1\right )^{\frac {2}{3}}}+\frac {4 \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {2}{3}} \left (\frac {2 \Gamma \left (\frac {2}{3}\right ) x^{3} \hypergeom \left (\left [1, 1, \frac {5}{3}\right ], \left [2, 2\right ], x^{3}\right )}{3}+\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+3 \ln \relax (x )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )\right )}{27 \Gamma \left (\frac {2}{3}\right ) \mathrm {signum}\left (x^{3}-1\right )^{\frac {2}{3}}}\)	\(91\)
meijerg	\(\frac {\mathrm {signum}\left (x^{3}-1\right )^{\frac {1}{3}} \left (-\frac {\Gamma \left (\frac {2}{3}\right ) x^{3} \hypergeom \left (\left [1, 1, \frac {5}{3}\right ], \left [2, 3\right ], x^{3}\right )}{3}-\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}-1+3 \ln \relax (x )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )-\frac {3 \Gamma \left (\frac {2}{3}\right )}{x^{3}}\right )}{9 \Gamma \left (\frac {2}{3}\right ) \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {1}{3}}}-\frac {\mathrm {signum}\left (x^{3}-1\right )^{\frac {1}{3}} \left (\frac {5 \Gamma \left (\frac {2}{3}\right ) x^{3} \hypergeom \left (\left [1, 1, \frac {8}{3}\right ], \left [2, 4\right ], x^{3}\right )}{27}+\frac {\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+3 \ln \relax (x )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )}{3}+\frac {3 \Gamma \left (\frac {2}{3}\right )}{2 x^{6}}-\frac {\Gamma \left (\frac {2}{3}\right )}{x^{3}}\right )}{9 \Gamma \left (\frac {2}{3}\right ) \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {1}{3}}}\)	\(156\)
trager	\(-\frac {\left (5 x^{3}+3\right ) \left (x^{3}-1\right )^{\frac {1}{3}}}{18 x^{6}}+\frac {4 \ln \left (-\frac {32768 \RootOf \left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )^{2} x^{3}+4416 \RootOf \left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right ) x^{3}+2496 \RootOf \left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}+91 x^{3}-262144 \RootOf \left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )^{2}-6720 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )+144 \left (x^{3}-1\right )^{\frac {2}{3}}-13312 \RootOf \left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )+39 \left (x^{3}-1\right )^{\frac {1}{3}}-169}{x^{3}}\right )}{27}-\frac {4 \ln \left (\frac {6754304 \RootOf \left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )^{2} x^{3}-619712 \RootOf \left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right ) x^{3}-352128 \RootOf \left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}-1266 x^{3}-54034432 \RootOf \left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )^{2}+911808 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )-19749 \left (x^{3}-1\right )^{\frac {2}{3}}+419648 \RootOf \left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )-5502 \left (x^{3}-1\right )^{\frac {1}{3}}+1055}{x^{3}}\right )}{27}-\frac {256 \ln \left (\frac {6754304 \RootOf \left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )^{2} x^{3}-619712 \RootOf \left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right ) x^{3}-352128 \RootOf \left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}-1266 x^{3}-54034432 \RootOf \left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )^{2}+911808 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )-19749 \left (x^{3}-1\right )^{\frac {2}{3}}+419648 \RootOf \left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )-5502 \left (x^{3}-1\right )^{\frac {1}{3}}+1055}{x^{3}}\right ) \RootOf \left (4096 \textit {\_Z}^{2}+64 \textit {\_Z} +1\right )}{27}\)	\(435\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3-1)^(1/3)*(x^3+1)/x^7,x,method=_RETURNVERBOSE)

[Out]

-1/18*(5*x^6-2*x^3-3)/x^6/(x^3-1)^(2/3)+4/27/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 = 0.48, size = 103, normalized size = 1.04 \begin {gather*} \frac {4}{27} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {{\left (x^{3} - 1\right )}^{\frac {4}{3}} - 2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{18 \, {\left (2 \, x^{3} + {\left (x^{3} - 1\right )}^{2} - 1\right )}} - \frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{3 \, x^{3}} - \frac {2}{27} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {4}{27} \, \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^7,x, algorithm="maxima")

[Out]

4/27*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^3 - 1)^(1/3) - 1)) + 1/18*((x^3 - 1)^(4/3) - 2*(x^3 - 1)^(1/3))/(2*x^3 +
(x^3 - 1)^2 - 1) - 1/3*(x^3 - 1)^(1/3)/x^3 - 2/27*log((x^3 - 1)^(2/3) - (x^3 - 1)^(1/3) + 1) + 4/27*log((x^3
- 1)^(1/3) + 1)

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mupad [B]  time = 1.04, size = 186, normalized size = 1.88 \begin {gather*} \frac {\ln \left (\frac {{\left (x^3-1\right )}^{1/3}}{9}+\frac {1}{9}\right )}{9}+\frac {\ln \left (\frac {{\left (x^3-1\right )}^{1/3}}{81}+\frac {1}{81}\right )}{27}-\frac {\frac {{\left (x^3-1\right )}^{1/3}}{9}-\frac {{\left (x^3-1\right )}^{4/3}}{18}}{{\left (x^3-1\right )}^2+2\,x^3-1}+\ln \left ({\left (x^3-1\right )}^{1/3}-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{18}\right )-\frac {{\left (x^3-1\right )}^{1/3}}{3\,x^3}-\ln \left (\frac {1}{2}-{\left (x^3-1\right )}^{1/3}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{18}\right )-\ln \left (\frac {1}{6}-\frac {{\left (x^3-1\right )}^{1/3}}{3}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )\,\left (\frac {1}{54}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54}\right )+\ln \left (\frac {{\left (x^3-1\right )}^{1/3}}{3}-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )\,\left (-\frac {1}{54}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^3 - 1)^(1/3)*(x^3 + 1))/x^7,x)

[Out]

log((x^3 - 1)^(1/3)/9 + 1/9)/9 + log((x^3 - 1)^(1/3)/81 + 1/81)/27 - ((x^3 - 1)^(1/3)/9 - (x^3 - 1)^(4/3)/18)/
((x^3 - 1)^2 + 2*x^3 - 1) + log((3^(1/2)*1i)/2 + (x^3 - 1)^(1/3) - 1/2)*((3^(1/2)*1i)/18 - 1/18) - (x^3 - 1)^(
1/3)/(3*x^3) - log((3^(1/2)*1i)/2 - (x^3 - 1)^(1/3) + 1/2)*((3^(1/2)*1i)/18 + 1/18) - log((3^(1/2)*1i)/6 - (x^
3 - 1)^(1/3)/3 + 1/6)*((3^(1/2)*1i)/54 + 1/54) + log((3^(1/2)*1i)/6 + (x^3 - 1)^(1/3)/3 - 1/6)*((3^(1/2)*1i)/5
4 - 1/54)

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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**7,x)

[Out]

Timed out

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