\(\int \frac {(-1+x^3)^{2/3} (1-x^3+x^6)}{x^6 (-2-x^3+2 x^6)} \, dx\) [1672]

Optimal result

Integrand size = 37, antiderivative size = 112 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1-x^3+x^6\right )}{x^6 \left (-2-x^3+2 x^6\right )} \, dx=\frac {\left (4-19 x^3\right ) \left (-1+x^3\right )^{2/3}}{40 x^5}+\frac {1}{12} \text {RootSum}\left [1-5 \text {$\#$1}^3+2 \text {$\#$1}^6\&,\frac {-3 \log (x)+3 \log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right )+17 \log (x) \text {$\#$1}^3-17 \log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-5 \text {$\#$1}+4 \text {$\#$1}^4}\&\right ] \]

[Out]

Unintegrable

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(505\) vs. \(2(112)=224\).

Time = 0.63 (sec) , antiderivative size = 505, normalized size of antiderivative = 4.51, number of steps used = 13, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {6860, 270, 283, 245, 399, 384} \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1-x^3+x^6\right )}{x^6 \left (-2-x^3+2 x^6\right )} \, dx=-\frac {\left (19+3 \sqrt {17}\right ) \arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{8 \sqrt {51}}+\frac {\left (19-3 \sqrt {17}\right ) \arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{8 \sqrt {51}}+\frac {1}{4} \sqrt {3} \arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )-\frac {\sqrt [3]{80295-19471 \sqrt {17}} \arctan \left (\frac {\frac {\sqrt [3]{2 \left (5-\sqrt {17}\right )} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{8 \sqrt {51}}+\frac {\sqrt [3]{80295+19471 \sqrt {17}} \arctan \left (\frac {\frac {\sqrt [3]{2 \left (5+\sqrt {17}\right )} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{8 \sqrt {51}}-\frac {\sqrt [3]{80295-19471 \sqrt {17}} \log \left (4 x^3-\sqrt {17}-1\right )}{48 \sqrt {17}}+\frac {\sqrt [3]{80295+19471 \sqrt {17}} \log \left (4 x^3+\sqrt {17}-1\right )}{48 \sqrt {17}}+\frac {\sqrt [3]{80295-19471 \sqrt {17}} \log \left (\frac {\sqrt [3]{5-\sqrt {17}} x}{2^{2/3}}-\sqrt [3]{x^3-1}\right )}{16 \sqrt {17}}-\frac {\sqrt [3]{80295+19471 \sqrt {17}} \log \left (\frac {\sqrt [3]{5+\sqrt {17}} x}{2^{2/3}}-\sqrt [3]{x^3-1}\right )}{16 \sqrt {17}}+\frac {1}{272} \left (51+19 \sqrt {17}\right ) \log \left (\sqrt [3]{x^3-1}-x\right )+\frac {1}{272} \left (51-19 \sqrt {17}\right ) \log \left (\sqrt [3]{x^3-1}-x\right )-\frac {3}{8} \log \left (\sqrt [3]{x^3-1}-x\right )-\frac {\left (x^3-1\right )^{5/3}}{10 x^5}-\frac {3 \left (x^3-1\right )^{2/3}}{8 x^2} \]

[In]

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

[Out]

(-3*(-1 + x^3)^(2/3))/(8*x^2) - (-1 + x^3)^(5/3)/(10*x^5) + (Sqrt[3]*ArcTan[(1 + (2*x)/(-1 + x^3)^(1/3))/Sqrt[
3]])/4 + ((19 - 3*Sqrt[17])*ArcTan[(1 + (2*x)/(-1 + x^3)^(1/3))/Sqrt[3]])/(8*Sqrt[51]) - ((19 + 3*Sqrt[17])*Ar
cTan[(1 + (2*x)/(-1 + x^3)^(1/3))/Sqrt[3]])/(8*Sqrt[51]) - ((80295 - 19471*Sqrt[17])^(1/3)*ArcTan[(1 + ((2*(5
- Sqrt[17]))^(1/3)*x)/(-1 + x^3)^(1/3))/Sqrt[3]])/(8*Sqrt[51]) + ((80295 + 19471*Sqrt[17])^(1/3)*ArcTan[(1 + (
(2*(5 + Sqrt[17]))^(1/3)*x)/(-1 + x^3)^(1/3))/Sqrt[3]])/(8*Sqrt[51]) - ((80295 - 19471*Sqrt[17])^(1/3)*Log[-1
- Sqrt[17] + 4*x^3])/(48*Sqrt[17]) + ((80295 + 19471*Sqrt[17])^(1/3)*Log[-1 + Sqrt[17] + 4*x^3])/(48*Sqrt[17])
+ ((80295 - 19471*Sqrt[17])^(1/3)*Log[((5 - Sqrt[17])^(1/3)*x)/2^(2/3) - (-1 + x^3)^(1/3)])/(16*Sqrt[17]) - (
(80295 + 19471*Sqrt[17])^(1/3)*Log[((5 + Sqrt[17])^(1/3)*x)/2^(2/3) - (-1 + x^3)^(1/3)])/(16*Sqrt[17]) - (3*Lo
g[-x + (-1 + x^3)^(1/3)])/8 + ((51 - 19*Sqrt[17])*Log[-x + (-1 + x^3)^(1/3)])/272 + ((51 + 19*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 {3 \left (-1+x^3\right )^{2/3}}{4 x^3}+\frac {\left (11-6 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 (11-6 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 {3}{4} \int \frac {\left (-1+x^3\right )^{2/3}}{x^3} \, dx \\ & = -\frac {3 \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 \left (\frac {\left (-6+\frac {38}{\sqrt {17}}\right ) \left (-1+x^3\right )^{2/3}}{-1-\sqrt {17}+4 x^3}+\frac {\left (-6-\frac {38}{\sqrt {17}}\right ) \left (-1+x^3\right )^{2/3}}{-1+\sqrt {17}+4 x^3}\right ) \, dx+\frac {3}{4} \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx \\ & = -\frac {3 \left (-1+x^3\right )^{2/3}}{8 x^2}-\frac {\left (-1+x^3\right )^{5/3}}{10 x^5}+\frac {1}{4} \sqrt {3} \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )-\frac {3}{8} \log \left (-x+\sqrt [3]{-1+x^3}\right )+\frac {1}{34} \left (-51+19 \sqrt {17}\right ) \int \frac {\left (-1+x^3\right )^{2/3}}{-1-\sqrt {17}+4 x^3} \, dx-\frac {1}{34} \left (51+19 \sqrt {17}\right ) \int \frac {\left (-1+x^3\right )^{2/3}}{-1+\sqrt {17}+4 x^3} \, dx \\ & = -\frac {3 \left (-1+x^3\right )^{2/3}}{8 x^2}-\frac {\left (-1+x^3\right )^{5/3}}{10 x^5}+\frac {1}{4} \sqrt {3} \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )-\frac {3}{8} \log \left (-x+\sqrt [3]{-1+x^3}\right )+\frac {1}{34} \left (119-27 \sqrt {17}\right ) \int \frac {1}{\sqrt [3]{-1+x^3} \left (-1-\sqrt {17}+4 x^3\right )} \, dx+\frac {1}{136} \left (-51+19 \sqrt {17}\right ) \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx-\frac {1}{136} \left (51+19 \sqrt {17}\right ) \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx+\frac {1}{34} \left (119+27 \sqrt {17}\right ) \int \frac {1}{\sqrt [3]{-1+x^3} \left (-1+\sqrt {17}+4 x^3\right )} \, dx \\ & = -\frac {3 \left (-1+x^3\right )^{2/3}}{8 x^2}-\frac {\left (-1+x^3\right )^{5/3}}{10 x^5}+\frac {1}{4} \sqrt {3} \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )+\frac {\left (19-3 \sqrt {17}\right ) \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{8 \sqrt {51}}-\frac {\left (19+3 \sqrt {17}\right ) \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{8 \sqrt {51}}-\frac {\sqrt [3]{642360-155768 \sqrt {17}} \arctan \left (\frac {1+\frac {\sqrt [3]{2 \left (5-\sqrt {17}\right )} x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{16 \sqrt {51}}+\frac {\sqrt [3]{80295+19471 \sqrt {17}} \arctan \left (\frac {1+\frac {\sqrt [3]{2 \left (5+\sqrt {17}\right )} x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{8 \sqrt {51}}-\frac {\sqrt [3]{642360-155768 \sqrt {17}} \log \left (-1-\sqrt {17}+4 x^3\right )}{96 \sqrt {17}}+\frac {\sqrt [3]{80295+19471 \sqrt {17}} \log \left (-1+\sqrt {17}+4 x^3\right )}{48 \sqrt {17}}+\frac {\sqrt [3]{642360-155768 \sqrt {17}} \log \left (\frac {\sqrt [3]{5-\sqrt {17}} x}{2^{2/3}}-\sqrt [3]{-1+x^3}\right )}{32 \sqrt {17}}-\frac {\sqrt [3]{80295+19471 \sqrt {17}} \log \left (\frac {\sqrt [3]{5+\sqrt {17}} x}{2^{2/3}}-\sqrt [3]{-1+x^3}\right )}{16 \sqrt {17}}-\frac {3}{8} \log \left (-x+\sqrt [3]{-1+x^3}\right )+\frac {1}{272} \left (51-19 \sqrt {17}\right ) \log \left (-x+\sqrt [3]{-1+x^3}\right )+\frac {1}{272} \left (51+19 \sqrt {17}\right ) \log \left (-x+\sqrt [3]{-1+x^3}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1-x^3+x^6\right )}{x^6 \left (-2-x^3+2 x^6\right )} \, dx=\frac {\left (4-19 x^3\right ) \left (-1+x^3\right )^{2/3}}{40 x^5}+\frac {1}{12} \text {RootSum}\left [1-5 \text {$\#$1}^3+2 \text {$\#$1}^6\&,\frac {-3 \log (x)+3 \log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right )+17 \log (x) \text {$\#$1}^3-17 \log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-5 \text {$\#$1}+4 \text {$\#$1}^4}\&\right ] \]

[In]

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

[Out]

((4 - 19*x^3)*(-1 + x^3)^(2/3))/(40*x^5) + RootSum[1 - 5*#1^3 + 2*#1^6 & , (-3*Log[x] + 3*Log[(-1 + x^3)^(1/3)
- x*#1] + 17*Log[x]*#1^3 - 17*Log[(-1 + x^3)^(1/3) - x*#1]*#1^3)/(-5*#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 = 33.41 (sec) , antiderivative size = 9294, normalized size of antiderivative = 82.98

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

[In]

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

[Out]

result too large to display

Fricas [F(-2)]

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

[In]

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

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.33 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1-x^3+x^6\right )}{x^6 \left (-2-x^3+2 x^6\right )} \, dx=\int { \frac {{\left (x^{6} - x^{3} + 1\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{6} - x^{3} - 2\right )} x^{6}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.33 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1-x^3+x^6\right )}{x^6 \left (-2-x^3+2 x^6\right )} \, dx=\int { \frac {{\left (x^{6} - x^{3} + 1\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{6} - x^{3} - 2\right )} x^{6}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 5.89 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.32 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1-x^3+x^6\right )}{x^6 \left (-2-x^3+2 x^6\right )} \, dx=\int -\frac {{\left (x^3-1\right )}^{2/3}\,\left (x^6-x^3+1\right )}{x^6\,\left (-2\,x^6+x^3+2\right )} \,d x \]

[In]

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

[Out]

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