\(\int \frac {(-1+x^3)^{2/3} (-1+x^6)}{x^6 (-2+x^3+2 x^6)} \, dx\) [1598]

Optimal result

Integrand size = 30, antiderivative size = 109 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-1+x^6\right )}{x^6 \left (-2+x^3+2 x^6\right )} \, dx=\frac {\left (-4-x^3\right ) \left (-1+x^3\right )^{2/3}}{40 x^5}+\frac {1}{12} \text {RootSum}\left [-1-3 \text {$\#$1}^3+2 \text {$\#$1}^6\&,\frac {\log (x)-\log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^3-\log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-3 \text {$\#$1}+4 \text {$\#$1}^4}\&\right ] \]

[Out]

Unintegrable

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(499\) vs. \(2(109)=218\).

Time = 0.55 (sec) , antiderivative size = 499, normalized size of antiderivative = 4.58, number of steps used = 13, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6860, 270, 283, 245, 399, 384} \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-1+x^6\right )}{x^6 \left (-2+x^3+2 x^6\right )} \, dx=-\frac {\left (1+\sqrt {17}\right ) \arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{8 \sqrt {51}}+\frac {\left (1-\sqrt {17}\right ) \arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{8 \sqrt {51}}+\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{4 \sqrt {3}}+\frac {\sqrt [3]{13 \sqrt {17}-43} \arctan \left (\frac {1-\frac {\sqrt [3]{2 \left (\sqrt {17}-3\right )} x}{\sqrt [3]{x^3-1}}}{\sqrt {3}}\right )}{8 \sqrt {51}}+\frac {\sqrt [3]{43+13 \sqrt {17}} \arctan \left (\frac {\frac {\sqrt [3]{2 \left (3+\sqrt {17}\right )} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{8 \sqrt {51}}+\frac {\sqrt [3]{13 \sqrt {17}-43} \log \left (4 x^3-\sqrt {17}+1\right )}{48 \sqrt {17}}+\frac {\sqrt [3]{43+13 \sqrt {17}} \log \left (4 x^3+\sqrt {17}+1\right )}{48 \sqrt {17}}-\frac {\sqrt [3]{13 \sqrt {17}-43} \log \left (-\sqrt [3]{x^3-1}-\frac {\sqrt [3]{\sqrt {17}-3} x}{2^{2/3}}\right )}{16 \sqrt {17}}-\frac {\sqrt [3]{43+13 \sqrt {17}} \log \left (\frac {\sqrt [3]{3+\sqrt {17}} x}{2^{2/3}}-\sqrt [3]{x^3-1}\right )}{16 \sqrt {17}}+\frac {1}{272} \left (17+\sqrt {17}\right ) \log \left (\sqrt [3]{x^3-1}-x\right )+\frac {1}{272} \left (17-\sqrt {17}\right ) \log \left (\sqrt [3]{x^3-1}-x\right )-\frac {1}{8} \log \left (\sqrt [3]{x^3-1}-x\right )+\frac {\left (x^3-1\right )^{5/3}}{10 x^5}-\frac {\left (x^3-1\right )^{2/3}}{8 x^2} \]

[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) + ArcTan[(1 + (2*x)/(-1 + x^3)^(1/3))/Sqrt[3]]/(4*Sqrt[3
]) + ((1 - Sqrt[17])*ArcTan[(1 + (2*x)/(-1 + x^3)^(1/3))/Sqrt[3]])/(8*Sqrt[51]) - ((1 + Sqrt[17])*ArcTan[(1 +
(2*x)/(-1 + x^3)^(1/3))/Sqrt[3]])/(8*Sqrt[51]) + ((-43 + 13*Sqrt[17])^(1/3)*ArcTan[(1 - ((2*(-3 + Sqrt[17]))^(
1/3)*x)/(-1 + x^3)^(1/3))/Sqrt[3]])/(8*Sqrt[51]) + ((43 + 13*Sqrt[17])^(1/3)*ArcTan[(1 + ((2*(3 + Sqrt[17]))^(
1/3)*x)/(-1 + x^3)^(1/3))/Sqrt[3]])/(8*Sqrt[51]) + ((-43 + 13*Sqrt[17])^(1/3)*Log[1 - Sqrt[17] + 4*x^3])/(48*S
qrt[17]) + ((43 + 13*Sqrt[17])^(1/3)*Log[1 + Sqrt[17] + 4*x^3])/(48*Sqrt[17]) - ((-43 + 13*Sqrt[17])^(1/3)*Log
[-(((-3 + Sqrt[17])^(1/3)*x)/2^(2/3)) - (-1 + x^3)^(1/3)])/(16*Sqrt[17]) - ((43 + 13*Sqrt[17])^(1/3)*Log[((3 +
Sqrt[17])^(1/3)*x)/2^(2/3) - (-1 + x^3)^(1/3)])/(16*Sqrt[17]) - Log[-x + (-1 + x^3)^(1/3)]/8 + ((17 - Sqrt[17
])*Log[-x + (-1 + x^3)^(1/3)])/272 + ((17 + Sqrt[17])*Log[-x + (-1 + x^3)^(1/3)])/272

Rule 245

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 270

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 283

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 384

Int[1/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/c, 3]}, Simp[
ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sqrt[3]*c*q), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*c*q
), x] + Simp[Log[c + d*x^3]/(6*c*q), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 399

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

Rule 6860

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*} \text {integral}& = \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 {\arctan \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 {\arctan \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 (17-3 \sqrt {17}\right ) \int \frac {1}{\sqrt [3]{-1+x^3} \left (1-\sqrt {17}+4 x^3\right )} \, dx+\frac {1}{136} \left (-17+\sqrt {17}\right ) \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx-\frac {1}{136} \left (17+\sqrt {17}\right ) \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx+\frac {1}{34} \left (17+3 \sqrt {17}\right ) \int \frac {1}{\sqrt [3]{-1+x^3} \left (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 {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{4 \sqrt {3}}+\frac {\left (1-\sqrt {17}\right ) \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{8 \sqrt {51}}-\frac {\left (1+\sqrt {17}\right ) \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{8 \sqrt {51}}+\frac {\sqrt [3]{-43+13 \sqrt {17}} \arctan \left (\frac {1-\frac {\sqrt [3]{2 \left (-3+\sqrt {17}\right )} x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{8 \sqrt {51}}+\frac {\sqrt [3]{43+13 \sqrt {17}} \arctan \left (\frac {1+\frac {\sqrt [3]{2 \left (3+\sqrt {17}\right )} x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{8 \sqrt {51}}+\frac {\sqrt [3]{-43+13 \sqrt {17}} \log \left (1-\sqrt {17}+4 x^3\right )}{48 \sqrt {17}}+\frac {\sqrt [3]{43+13 \sqrt {17}} \log \left (1+\sqrt {17}+4 x^3\right )}{48 \sqrt {17}}-\frac {\sqrt [3]{-43+13 \sqrt {17}} \log \left (-\frac {\sqrt [3]{-3+\sqrt {17}} x}{2^{2/3}}-\sqrt [3]{-1+x^3}\right )}{16 \sqrt {17}}-\frac {\sqrt [3]{43+13 \sqrt {17}} \log \left (\frac {\sqrt [3]{3+\sqrt {17}} x}{2^{2/3}}-\sqrt [3]{-1+x^3}\right )}{16 \sqrt {17}}-\frac {1}{8} \log \left (-x+\sqrt [3]{-1+x^3}\right )+\frac {1}{272} \left (17-\sqrt {17}\right ) \log \left (-x+\sqrt [3]{-1+x^3}\right )+\frac {1}{272} \left (17+\sqrt {17}\right ) \log \left (-x+\sqrt [3]{-1+x^3}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-1+x^6\right )}{x^6 \left (-2+x^3+2 x^6\right )} \, dx=\frac {\left (-4-x^3\right ) \left (-1+x^3\right )^{2/3}}{40 x^5}+\frac {1}{12} \text {RootSum}\left [-1-3 \text {$\#$1}^3+2 \text {$\#$1}^6\&,\frac {\log (x)-\log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^3-\log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-3 \text {$\#$1}+4 \text {$\#$1}^4}\&\right ] \]

[In]

Integrate[((-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

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 1.

Time = 43.21 (sec) , antiderivative size = 6680, normalized size of antiderivative = 61.28

\[\text {output too large to display}\]

[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

Fricas [F(-2)]

Exception generated. \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-1+x^6\right )}{x^6 \left (-2+x^3+2 x^6\right )} \, dx=\text {Exception raised: TypeError} \]

[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)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-1+x^6\right )}{x^6 \left (-2+x^3+2 x^6\right )} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.28 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-1+x^6\right )}{x^6 \left (-2+x^3+2 x^6\right )} \, dx=\int { \frac {{\left (x^{6} - 1\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{6} + x^{3} - 2\right )} x^{6}} \,d x } \]

[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)

Giac [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.28 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-1+x^6\right )}{x^6 \left (-2+x^3+2 x^6\right )} \, dx=\int { \frac {{\left (x^{6} - 1\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{6} + x^{3} - 2\right )} x^{6}} \,d x } \]

[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)

Mupad [N/A]

Not integrable

Time = 5.84 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.28 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-1+x^6\right )}{x^6 \left (-2+x^3+2 x^6\right )} \, dx=\int \frac {{\left (x^3-1\right )}^{2/3}\,\left (x^6-1\right )}{x^6\,\left (2\,x^6+x^3-2\right )} \,d x \]

[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)