3.16.14 \(\int \frac {(-1+x^3)^{2/3} (-2+2 x^3+x^6)}{x^9} \, dx\)

Optimal. Leaf size=105 \[ -\frac {1}{3} \log \left (\sqrt [3]{x^3-1}-x\right )+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3-1}+x}\right )}{\sqrt {3}}+\frac {1}{6} \log \left (\sqrt [3]{x^3-1} x+\left (x^3-1\right )^{2/3}+x^2\right )+\frac {\left (x^3-1\right )^{2/3} \left (-x^6-2 x^3+1\right )}{4 x^8} \]

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Rubi [A]  time = 0.05, antiderivative size = 94, normalized size of antiderivative = 0.90, number of steps used = 7, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {1488, 271, 264, 277, 239} \begin {gather*} -\frac {1}{2} \log \left (\sqrt [3]{x^3-1}-x\right )+\frac {\tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\left (x^3-1\right )^{5/3}}{4 x^8}+\frac {\left (x^3-1\right )^{5/3}}{4 x^5}-\frac {\left (x^3-1\right )^{2/3}}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-1 + x^3)^(2/3)*(-2 + 2*x^3 + x^6))/x^9,x]

[Out]

-1/2*(-1 + x^3)^(2/3)/x^2 - (-1 + x^3)^(5/3)/(4*x^8) + (-1 + x^3)^(5/3)/(4*x^5) + ArcTan[(1 + (2*x)/(-1 + x^3)
^(1/3))/Sqrt[3]]/Sqrt[3] - Log[-x + (-1 + x^3)^(1/3)]/2

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 264

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

Rule 271

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

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 1488

Int[((f_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Sy
mbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, e,
f, m, q}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int \frac {\left (-1+x^3\right )^{2/3} \left (-2+2 x^3+x^6\right )}{x^9} \, dx &=\int \left (-\frac {2 \left (-1+x^3\right )^{2/3}}{x^9}+\frac {2 \left (-1+x^3\right )^{2/3}}{x^6}+\frac {\left (-1+x^3\right )^{2/3}}{x^3}\right ) \, dx\\ &=-\left (2 \int \frac {\left (-1+x^3\right )^{2/3}}{x^9} \, dx\right )+2 \int \frac {\left (-1+x^3\right )^{2/3}}{x^6} \, dx+\int \frac {\left (-1+x^3\right )^{2/3}}{x^3} \, dx\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{2 x^2}-\frac {\left (-1+x^3\right )^{5/3}}{4 x^8}+\frac {2 \left (-1+x^3\right )^{5/3}}{5 x^5}-\frac {3}{4} \int \frac {\left (-1+x^3\right )^{2/3}}{x^6} \, dx+\int \frac {1}{\sqrt [3]{-1+x^3}} \, dx\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{2 x^2}-\frac {\left (-1+x^3\right )^{5/3}}{4 x^8}+\frac {\left (-1+x^3\right )^{5/3}}{4 x^5}+\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (-x+\sqrt [3]{-1+x^3}\right )\\ \end {align*}

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Mathematica [C]  time = 0.03, size = 57, normalized size = 0.54 \begin {gather*} \frac {\left (x^3-1\right )^{2/3} \left (\left (1-x^3\right )^{8/3}-2 x^6 \, _2F_1\left (-\frac {2}{3},-\frac {2}{3};\frac {1}{3};x^3\right )\right )}{4 x^8 \left (1-x^3\right )^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-1 + x^3)^(2/3)*(-2 + 2*x^3 + x^6))/x^9,x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.17, size = 105, normalized size = 1.00 \begin {gather*} \frac {\left (-1+x^3\right )^{2/3} \left (1-2 x^3-x^6\right )}{4 x^8}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )}{\sqrt {3}}-\frac {1}{3} \log \left (-x+\sqrt [3]{-1+x^3}\right )+\frac {1}{6} \log \left (x^2+x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-1 + x^3)^(2/3)*(-2 + 2*x^3 + x^6))/x^9,x]

[Out]

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

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fricas [A]  time = 0.78, size = 115, normalized size = 1.10 \begin {gather*} \frac {4 \, \sqrt {3} x^{8} \arctan \left (-\frac {25382 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} - 13720 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (5831 \, x^{3} - 7200\right )}}{58653 \, x^{3} - 8000}\right ) - 2 \, x^{8} \log \left (-3 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x + 1\right ) - 3 \, {\left (x^{6} + 2 \, x^{3} - 1\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{12 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-1)^(2/3)*(x^6+2*x^3-2)/x^9,x, algorithm="fricas")

[Out]

1/12*(4*sqrt(3)*x^8*arctan(-(25382*sqrt(3)*(x^3 - 1)^(1/3)*x^2 - 13720*sqrt(3)*(x^3 - 1)^(2/3)*x + sqrt(3)*(58
31*x^3 - 7200))/(58653*x^3 - 8000)) - 2*x^8*log(-3*(x^3 - 1)^(1/3)*x^2 + 3*(x^3 - 1)^(2/3)*x + 1) - 3*(x^6 + 2
*x^3 - 1)*(x^3 - 1)^(2/3))/x^8

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{6} + 2 \, x^{3} - 2\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x^{9}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-1)^(2/3)*(x^6+2*x^3-2)/x^9,x, algorithm="giac")

[Out]

integrate((x^6 + 2*x^3 - 2)*(x^3 - 1)^(2/3)/x^9, x)

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maple [C]  time = 2.65, size = 56, normalized size = 0.53

method	result	size
risch	\(-\frac {x^{9}+x^{6}-3 x^{3}+1}{4 x^{8} \left (x^{3}-1\right )^{\frac {1}{3}}}+\frac {\left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {1}{3}} x \hypergeom \left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x^{3}\right )}{\mathrm {signum}\left (x^{3}-1\right )^{\frac {1}{3}}}\)	\(56\)
meijerg	\(-\frac {\mathrm {signum}\left (x^{3}-1\right )^{\frac {2}{3}} \hypergeom \left (\left [-\frac {2}{3}, -\frac {2}{3}\right ], \left [\frac {1}{3}\right ], x^{3}\right )}{2 \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {2}{3}} x^{2}}-\frac {2 \mathrm {signum}\left (x^{3}-1\right )^{\frac {2}{3}} \left (-x^{3}+1\right )^{\frac {5}{3}}}{5 \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {2}{3}} x^{5}}+\frac {\mathrm {signum}\left (x^{3}-1\right )^{\frac {2}{3}} \left (-\frac {3}{5} x^{6}-\frac {2}{5} x^{3}+1\right ) \left (-x^{3}+1\right )^{\frac {2}{3}}}{4 \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {2}{3}} x^{8}}\)	\(110\)
trager	\(-\frac {\left (x^{6}+2 x^{3}-1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}}{4 x^{8}}+\frac {\ln \left (746584997888 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )^{2} x^{3}-49654493184 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -49654493184 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}-52570840832 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) x^{3}-1314589509 x \left (x^{3}-1\right )^{\frac {2}{3}}-1314589509 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}-1303197526 x^{3}-5972679983104 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )^{2}-72296025856 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )[phone]\right )}{3}-\frac {256 \ln \left (746584997888 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )^{2} x^{3}-49654493184 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -49654493184 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}-52570840832 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) x^{3}-1314589509 x \left (x^{3}-1\right )^{\frac {2}{3}}-1314589509 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}-1303197526 x^{3}-5972679983104 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )^{2}-72296025856 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )[phone]\right ) \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )}{3}+\frac {256 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) \ln \left (746584997888 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )^{2} x^{3}[phone] \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x [phone] \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}[phone] \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) x^{3}-1508552373 x \left (x^{3}-1\right )^{\frac {2}{3}}-1508552373 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}-1497160390 x^{3}-5972679983104 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )^{2}[phone] \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )[phone]\right )}{3}\)	\(460\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3-1)^(2/3)*(x^6+2*x^3-2)/x^9,x,method=_RETURNVERBOSE)

[Out]

-1/4*(x^9+x^6-3*x^3+1)/x^8/(x^3-1)^(1/3)+1/signum(x^3-1)^(1/3)*(-signum(x^3-1))^(1/3)*x*hypergeom([1/3,1/3],[4
/3],x^3)

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maxima [A]  time = 0.42, size = 93, normalized size = 0.89 \begin {gather*} -\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) - \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}}}{2 \, x^{2}} + \frac {{\left (x^{3} - 1\right )}^{\frac {8}{3}}}{4 \, x^{8}} + \frac {1}{6} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) - \frac {1}{3} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-1)^(2/3)*(x^6+2*x^3-2)/x^9,x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (x^3-1\right )}^{2/3}\,\left (x^6+2\,x^3-2\right )}{x^9} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^3 - 1)^(2/3)*(2*x^3 + x^6 - 2))/x^9,x)

[Out]

int(((x^3 - 1)^(2/3)*(2*x^3 + x^6 - 2))/x^9, x)

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sympy [C]  time = 3.58, size = 461, normalized size = 4.39 \begin {gather*} 2 \left (\begin {cases} \frac {\left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{- \frac {i \pi }{3}} \Gamma \left (- \frac {5}{3}\right )}{3 \Gamma \left (- \frac {2}{3}\right )} - \frac {\left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{- \frac {i \pi }{3}} \Gamma \left (- \frac {5}{3}\right )}{3 x^{3} \Gamma \left (- \frac {2}{3}\right )} & \text {for}\: \frac {1}{\left |{x^{3}}\right |} > 1 \\- \frac {\left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {5}{3}\right )}{3 \Gamma \left (- \frac {2}{3}\right )} + \frac {\left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {5}{3}\right )}{3 x^{3} \Gamma \left (- \frac {2}{3}\right )} & \text {otherwise} \end {cases}\right ) - 2 \left (\begin {cases} \frac {\left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (- \frac {8}{3}\right )}{3 \Gamma \left (- \frac {2}{3}\right )} + \frac {2 \left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{3} \Gamma \left (- \frac {2}{3}\right )} - \frac {5 \left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{6} \Gamma \left (- \frac {2}{3}\right )} & \text {for}\: \frac {1}{\left |{x^{3}}\right |} > 1 \\\frac {3 x^{6} \left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{6} \Gamma \left (- \frac {2}{3}\right ) - 9 x^{3} \Gamma \left (- \frac {2}{3}\right )} - \frac {x^{3} \left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{6} \Gamma \left (- \frac {2}{3}\right ) - 9 x^{3} \Gamma \left (- \frac {2}{3}\right )} + \frac {5 \left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{9} \Gamma \left (- \frac {2}{3}\right ) - 9 x^{6} \Gamma \left (- \frac {2}{3}\right )} - \frac {7 \left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{6} \Gamma \left (- \frac {2}{3}\right ) - 9 x^{3} \Gamma \left (- \frac {2}{3}\right )} & \text {otherwise} \end {cases}\right ) + \frac {e^{\frac {2 i \pi }{3}} \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {2}{3} \\ \frac {1}{3} \end {matrix}\middle | {x^{3}} \right )}}{3 x^{2} \Gamma \left (\frac {1}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**3-1)**(2/3)*(x**6+2*x**3-2)/x**9,x)

[Out]

2*Piecewise(((-1 + x**(-3))**(2/3)*exp(-I*pi/3)*gamma(-5/3)/(3*gamma(-2/3)) - (-1 + x**(-3))**(2/3)*exp(-I*pi/
3)*gamma(-5/3)/(3*x**3*gamma(-2/3)), 1/Abs(x**3) > 1), (-(1 - 1/x**3)**(2/3)*gamma(-5/3)/(3*gamma(-2/3)) + (1
- 1/x**3)**(2/3)*gamma(-5/3)/(3*x**3*gamma(-2/3)), True)) - 2*Piecewise(((-1 + x**(-3))**(2/3)*exp(2*I*pi/3)*g
amma(-8/3)/(3*gamma(-2/3)) + 2*(-1 + x**(-3))**(2/3)*exp(2*I*pi/3)*gamma(-8/3)/(9*x**3*gamma(-2/3)) - 5*(-1 +
x**(-3))**(2/3)*exp(2*I*pi/3)*gamma(-8/3)/(9*x**6*gamma(-2/3)), 1/Abs(x**3) > 1), (3*x**6*(1 - 1/x**3)**(2/3)*
gamma(-8/3)/(9*x**6*gamma(-2/3) - 9*x**3*gamma(-2/3)) - x**3*(1 - 1/x**3)**(2/3)*gamma(-8/3)/(9*x**6*gamma(-2/
3) - 9*x**3*gamma(-2/3)) + 5*(1 - 1/x**3)**(2/3)*gamma(-8/3)/(9*x**9*gamma(-2/3) - 9*x**6*gamma(-2/3)) - 7*(1
- 1/x**3)**(2/3)*gamma(-8/3)/(9*x**6*gamma(-2/3) - 9*x**3*gamma(-2/3)), True)) + exp(2*I*pi/3)*gamma(-2/3)*hyp
er((-2/3, -2/3), (1/3,), x**3)/(3*x**2*gamma(1/3))

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