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

Optimal. Leaf size=109 \[ \frac {1}{12} \text {RootSum}\left [2 \text {$\#$1}^6-3 \text {$\#$1}^3-1\& ,\frac {-\text {$\#$1}^3 \log \left (\sqrt [3]{x^3-1}-\text {$\#$1} x\right )+\text {$\#$1}^3 \log (x)-\log \left (\sqrt [3]{x^3-1}-\text {$\#$1} x\right )+\log (x)}{4 \text {$\#$1}^4-3 \text {$\#$1}}\& \right ]+\frac {\left (x^3-1\right )^{2/3} \left (-x^3-4\right )}{40 x^5} \]

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Rubi [B]  time = 0.62, antiderivative size = 225, normalized size of antiderivative = 2.06, number of steps used = 11, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6728, 264, 277, 239, 430, 429} \begin {gather*} -\frac {\left (17+\sqrt {17}\right ) x \left (x^3-1\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,-\frac {4 x^3}{1-\sqrt {17}}\right )}{34 \left (1-\sqrt {17}\right ) \left (1-x^3\right )^{2/3}}-\frac {\left (17-\sqrt {17}\right ) x \left (x^3-1\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,-\frac {4 x^3}{1+\sqrt {17}}\right )}{34 \left (1+\sqrt {17}\right ) \left (1-x^3\right )^{2/3}}-\frac {1}{8} \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 )}{4 \sqrt {3}}+\frac {\left (x^3-1\right )^{5/3}}{10 x^5}-\frac {\left (x^3-1\right )^{2/3}}{8 x^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/8*(-1 + x^3)^(2/3)/x^2 + (-1 + x^3)^(5/3)/(10*x^5) - ((17 + Sqrt[17])*x*(-1 + x^3)^(2/3)*AppellF1[1/3, -2/3
, 1, 4/3, x^3, (-4*x^3)/(1 - Sqrt[17])])/(34*(1 - Sqrt[17])*(1 - x^3)^(2/3)) - ((17 - Sqrt[17])*x*(-1 + x^3)^(
2/3)*AppellF1[1/3, -2/3, 1, 4/3, x^3, (-4*x^3)/(1 + Sqrt[17])])/(34*(1 + Sqrt[17])*(1 - x^3)^(2/3)) + ArcTan[(
1 + (2*x)/(-1 + x^3)^(1/3))/Sqrt[3]]/(4*Sqrt[3]) - Log[-x + (-1 + x^3)^(1/3)]/8

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 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 429

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p,
-q, 1 + 1/n, -((b*x^n)/a), -((d*x^n)/c)], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n
, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 430

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^F
racPart[p])/(1 + (b*x^n)/a)^FracPart[p], Int[(1 + (b*x^n)/a)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n,
p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] &&  !(IntegerQ[p] || GtQ[a, 0])

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {\left (-1+x^3\right )^{2/3} \left (-1+x^6\right )}{x^6 \left (-2+x^3+2 x^6\right )} \, dx &=\int \left (\frac {\left (-1+x^3\right )^{2/3}}{2 x^6}+\frac {\left (-1+x^3\right )^{2/3}}{4 x^3}+\frac {\left (-1-2 x^3\right ) \left (-1+x^3\right )^{2/3}}{4 \left (-2+x^3+2 x^6\right )}\right ) \, dx\\ &=\frac {1}{4} \int \frac {\left (-1+x^3\right )^{2/3}}{x^3} \, dx+\frac {1}{4} \int \frac {\left (-1-2 x^3\right ) \left (-1+x^3\right )^{2/3}}{-2+x^3+2 x^6} \, dx+\frac {1}{2} \int \frac {\left (-1+x^3\right )^{2/3}}{x^6} \, dx\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{8 x^2}+\frac {\left (-1+x^3\right )^{5/3}}{10 x^5}+\frac {1}{4} \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx+\frac {1}{4} \int \left (\frac {\left (-2-\frac {2}{\sqrt {17}}\right ) \left (-1+x^3\right )^{2/3}}{1-\sqrt {17}+4 x^3}+\frac {\left (-2+\frac {2}{\sqrt {17}}\right ) \left (-1+x^3\right )^{2/3}}{1+\sqrt {17}+4 x^3}\right ) \, dx\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{8 x^2}+\frac {\left (-1+x^3\right )^{5/3}}{10 x^5}+\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{4 \sqrt {3}}-\frac {1}{8} \log \left (-x+\sqrt [3]{-1+x^3}\right )+\frac {1}{34} \left (-17+\sqrt {17}\right ) \int \frac {\left (-1+x^3\right )^{2/3}}{1+\sqrt {17}+4 x^3} \, dx-\frac {1}{34} \left (17+\sqrt {17}\right ) \int \frac {\left (-1+x^3\right )^{2/3}}{1-\sqrt {17}+4 x^3} \, dx\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{8 x^2}+\frac {\left (-1+x^3\right )^{5/3}}{10 x^5}+\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{4 \sqrt {3}}-\frac {1}{8} \log \left (-x+\sqrt [3]{-1+x^3}\right )+\frac {\left (\left (-17+\sqrt {17}\right ) \left (-1+x^3\right )^{2/3}\right ) \int \frac {\left (1-x^3\right )^{2/3}}{1+\sqrt {17}+4 x^3} \, dx}{34 \left (1-x^3\right )^{2/3}}-\frac {\left (\left (17+\sqrt {17}\right ) \left (-1+x^3\right )^{2/3}\right ) \int \frac {\left (1-x^3\right )^{2/3}}{1-\sqrt {17}+4 x^3} \, dx}{34 \left (1-x^3\right )^{2/3}}\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{8 x^2}+\frac {\left (-1+x^3\right )^{5/3}}{10 x^5}-\frac {\left (17+\sqrt {17}\right ) x \left (-1+x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,-\frac {4 x^3}{1-\sqrt {17}}\right )}{34 \left (1-\sqrt {17}\right ) \left (1-x^3\right )^{2/3}}-\frac {\left (17-\sqrt {17}\right ) x \left (-1+x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,-\frac {4 x^3}{1+\sqrt {17}}\right )}{34 \left (1+\sqrt {17}\right ) \left (1-x^3\right )^{2/3}}+\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{4 \sqrt {3}}-\frac {1}{8} \log \left (-x+\sqrt [3]{-1+x^3}\right )\\ \end {align*}

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Mathematica [B]  time = 0.70, size = 372, normalized size = 3.41 \begin {gather*} \frac {-2 \sqrt [3]{43+13 \sqrt {17}} \log \left (\sqrt [3]{\sqrt {17}-3}-\frac {\sqrt [3]{2} x}{\sqrt [3]{x^3-1}}\right )-2 \sqrt [3]{13 \sqrt {17}-43} \log \left (\frac {\sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+\sqrt [3]{3+\sqrt {17}}\right )+2 \sqrt {3} \sqrt [3]{43+13 \sqrt {17}} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{\frac {2}{\sqrt {17}-3}} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )+2 \sqrt {3} \sqrt [3]{13 \sqrt {17}-43} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{\frac {2}{3+\sqrt {17}}} x}{\sqrt [3]{x^3-1}}}{\sqrt {3}}\right )+\sqrt [3]{43+13 \sqrt {17}} \log \left (\frac {\sqrt [3]{2 \left (\sqrt {17}-3\right )} x}{\sqrt [3]{x^3-1}}+\frac {2^{2/3} x^2}{\left (x^3-1\right )^{2/3}}+\left (\sqrt {17}-3\right )^{2/3}\right )+\sqrt [3]{13 \sqrt {17}-43} \log \left (-\frac {\sqrt [3]{2 \left (3+\sqrt {17}\right )} x}{\sqrt [3]{x^3-1}}+\frac {2^{2/3} x^2}{\left (x^3-1\right )^{2/3}}+\left (3+\sqrt {17}\right )^{2/3}\right )}{48 \sqrt {17}}-\frac {\left (x^3-1\right )^{2/3} \left (x^3+4\right )}{40 x^5} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/40*((-1 + x^3)^(2/3)*(4 + x^3))/x^5 + (2*Sqrt[3]*(43 + 13*Sqrt[17])^(1/3)*ArcTan[(1 + (2*(2/(-3 + Sqrt[17])
)^(1/3)*x)/(-1 + x^3)^(1/3))/Sqrt[3]] + 2*Sqrt[3]*(-43 + 13*Sqrt[17])^(1/3)*ArcTan[(1 - (2*(2/(3 + Sqrt[17]))^
(1/3)*x)/(-1 + x^3)^(1/3))/Sqrt[3]] - 2*(43 + 13*Sqrt[17])^(1/3)*Log[(-3 + Sqrt[17])^(1/3) - (2^(1/3)*x)/(-1 +
x^3)^(1/3)] - 2*(-43 + 13*Sqrt[17])^(1/3)*Log[(3 + Sqrt[17])^(1/3) + (2^(1/3)*x)/(-1 + x^3)^(1/3)] + (43 + 13
*Sqrt[17])^(1/3)*Log[(-3 + Sqrt[17])^(2/3) + (2^(2/3)*x^2)/(-1 + x^3)^(2/3) + ((2*(-3 + Sqrt[17]))^(1/3)*x)/(-
1 + x^3)^(1/3)] + (-43 + 13*Sqrt[17])^(1/3)*Log[(3 + Sqrt[17])^(2/3) + (2^(2/3)*x^2)/(-1 + x^3)^(2/3) - ((2*(3
+ Sqrt[17]))^(1/3)*x)/(-1 + x^3)^(1/3)])/(48*Sqrt[17])

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IntegrateAlgebraic [A]  time = 0.22, size = 109, normalized size = 1.00 \begin {gather*} \frac {\left (-4-x^3\right ) \left (-1+x^3\right )^{2/3}}{40 x^5}+\frac {1}{12} \text {RootSum}\left [-1-3 \text {$\#$1}^3+2 \text {$\#$1}^6\&,\frac {\log (x)-\log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^3-\log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-3 \text {$\#$1}+4 \text {$\#$1}^4}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-4 - x^3)*(-1 + x^3)^(2/3))/(40*x^5) + RootSum[-1 - 3*#1^3 + 2*#1^6 & , (Log[x] - Log[(-1 + x^3)^(1/3) - x*#
1] + Log[x]*#1^3 - Log[(-1 + x^3)^(1/3) - x*#1]*#1^3)/(-3*#1 + 4*#1^4) & ]/12

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (tr
ace 0)

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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^{3} - 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{6} + x^{3} - 2\right )} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 38.12, size = 7065, normalized size = 64.82 \[\text {output too large to display}\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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^6-1\right )}{x^6\,\left (2\,x^6+x^3-2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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