\(\int \frac {(1+x^3)^{2/3} (1-2 x^3+2 x^6)}{x^6 (-1-x^3+2 x^6)} \, dx\) [2576]

Optimal result

Integrand size = 39, antiderivative size = 221 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (1-2 x^3+2 x^6\right )}{x^6 \left (-1-x^3+2 x^6\right )} \, dx=\frac {\left (2-13 x^3\right ) \left (1+x^3\right )^{2/3}}{10 x^5}-\frac {10 \arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{1+x^3}}\right )}{3 \sqrt {3}}-\frac {2^{2/3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{1+x^3}}\right )}{3 \sqrt {3}}-\frac {10}{9} \log \left (x+\sqrt [3]{1+x^3}\right )+\frac {1}{9} 2^{2/3} \log \left (-2 x+2^{2/3} \sqrt [3]{1+x^3}\right )+\frac {5}{9} \log \left (x^2-x \sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right )-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{1+x^3}+\sqrt [3]{2} \left (1+x^3\right )^{2/3}\right )}{9 \sqrt [3]{2}} \]

[Out]

1/10*(-13*x^3+2)*(x^3+1)^(2/3)/x^5-10/9*3^(1/2)*arctan(3^(1/2)*x/(-x+2*(x^3+1)^(1/3)))-1/9*2^(2/3)*arctan(3^(1
/2)*x/(x+2^(2/3)*(x^3+1)^(1/3)))*3^(1/2)-10/9*ln(x+(x^3+1)^(1/3))+1/9*2^(2/3)*ln(-2*x+2^(2/3)*(x^3+1)^(1/3))+5
/9*ln(x^2-x*(x^3+1)^(1/3)+(x^3+1)^(2/3))-1/18*ln(2*x^2+2^(2/3)*x*(x^3+1)^(1/3)+2^(1/3)*(x^3+1)^(2/3))*2^(2/3)

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(529\) vs. \(2(221)=442\).

Time = 0.67 (sec) , antiderivative size = 529, normalized size of antiderivative = 2.39, number of steps used = 34, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.564, Rules used = {6860, 2178, 2177, 245, 2174, 371, 270, 283, 2183, 399, 384, 495, 502, 206, 31, 648, 631, 210, 642, 455, 52, 57} \[ \int \frac {\left (1+x^3\right )^{2/3} \left (1-2 x^3+2 x^6\right )}{x^6 \left (-1-x^3+2 x^6\right )} \, dx=\frac {10 \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{x^3+1}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {2\ 2^{2/3} \arctan \left (\frac {\frac {2 \sqrt [3]{2} x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{9 \sqrt {3}}+\frac {2^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} (x+1)}{\sqrt [3]{x^3+1}}}{\sqrt {3}}\right )}{9 \sqrt {3}}-\frac {2^{2/3} \arctan \left (\frac {\frac {\sqrt [3]{2} (x+1)}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{9 \sqrt {3}}-\frac {2^{2/3} \arctan \left (\frac {2^{2/3} \sqrt [3]{x^3+1}+1}{\sqrt {3}}\right )}{9 \sqrt {3}}-\frac {1}{27} 2^{2/3} \log \left (1-x^3\right )+\frac {\log \left (1-x^3\right )}{27 \sqrt [3]{2}}+\frac {5}{9} \log \left (2 x^3+1\right )+\frac {\log \left (\frac {2^{2/3} (x+1)^2}{\left (x^3+1\right )^{2/3}}-\frac {\sqrt [3]{2} (x+1)}{\sqrt [3]{x^3+1}}+1\right )}{27 \sqrt [3]{2}}-\frac {1}{27} 2^{2/3} \log \left (\frac {\sqrt [3]{2} (x+1)}{\sqrt [3]{x^3+1}}+1\right )-\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{x^3+1}\right )}{9 \sqrt [3]{2}}-\frac {5}{3} \log \left (-\sqrt [3]{x^3+1}-x\right )+\frac {1}{9} 2^{2/3} \log \left (\sqrt [3]{2} x-\sqrt [3]{x^3+1}\right )-\frac {\log \left (-2^{2/3} \sqrt [3]{x^3+1}+x+1\right )}{18 \sqrt [3]{2}}+\frac {\log \left (2^{2/3} \sqrt [3]{x^3+1}-x-1\right )}{6 \sqrt [3]{2}}+\frac {\left (x^3+1\right )^{5/3}}{5 x^5}-\frac {3 \left (x^3+1\right )^{2/3}}{2 x^2}-\frac {\log \left (-(1-x)^2 (x+1)\right )}{18 \sqrt [3]{2}}+\frac {\log \left ((1-x)^2 (x+1)\right )}{54 \sqrt [3]{2}} \]

[In]

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

[Out]

(-3*(1 + x^3)^(2/3))/(2*x^2) + (1 + x^3)^(5/3)/(5*x^5) + (10*ArcTan[(1 - (2*x)/(1 + x^3)^(1/3))/Sqrt[3]])/(3*S
qrt[3]) - (2*2^(2/3)*ArcTan[(1 + (2*2^(1/3)*x)/(1 + x^3)^(1/3))/Sqrt[3]])/(9*Sqrt[3]) + (2^(2/3)*ArcTan[(1 - (
2*2^(1/3)*(1 + x))/(1 + x^3)^(1/3))/Sqrt[3]])/(9*Sqrt[3]) - (2^(2/3)*ArcTan[(1 + (2^(1/3)*(1 + x))/(1 + x^3)^(
1/3))/Sqrt[3]])/(9*Sqrt[3]) - (2^(2/3)*ArcTan[(1 + 2^(2/3)*(1 + x^3)^(1/3))/Sqrt[3]])/(9*Sqrt[3]) - Log[-((1 -
x)^2*(1 + x))]/(18*2^(1/3)) + Log[(1 - x)^2*(1 + x)]/(54*2^(1/3)) + Log[1 - x^3]/(27*2^(1/3)) - (2^(2/3)*Log[
1 - x^3])/27 + (5*Log[1 + 2*x^3])/9 + Log[1 + (2^(2/3)*(1 + x)^2)/(1 + x^3)^(2/3) - (2^(1/3)*(1 + x))/(1 + x^3
)^(1/3)]/(27*2^(1/3)) - (2^(2/3)*Log[1 + (2^(1/3)*(1 + x))/(1 + x^3)^(1/3)])/27 - Log[2^(1/3) - (1 + x^3)^(1/3
)]/(9*2^(1/3)) - (5*Log[-x - (1 + x^3)^(1/3)])/3 + (2^(2/3)*Log[2^(1/3)*x - (1 + x^3)^(1/3)])/9 - Log[1 + x -
2^(2/3)*(1 + x^3)^(1/3)]/(18*2^(1/3)) + Log[-1 - x + 2^(2/3)*(1 + x^3)^(1/3)]/(6*2^(1/3))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 57

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, Simp[-L
og[RemoveContent[a + b*x, x]]/(2*b*q), x] + (Dist[3/(2*b), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/
3)], x] - Dist[3/(2*b*q), Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && PosQ
[(b*c - a*d)/b]

Rule 206

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x]
/; FreeQ[{a, b}, x]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

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 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

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 455

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(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] && EqQ[m
- n + 1, 0]

Rule 495

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

Rule 502

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

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 2174

Int[1/(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^3)^(1/3)), x_Symbol] :> Simp[Sqrt[3]*(ArcTan[(1 - 2^(1/3)*Rt[b,
3]*((c - d*x)/(d*(a + b*x^3)^(1/3))))/Sqrt[3]]/(2^(4/3)*Rt[b, 3]*c)), x] + (Simp[Log[(c + d*x)^2*(c - d*x)]/(2
^(7/3)*Rt[b, 3]*c), x] - Simp[(3*Log[Rt[b, 3]*(c - d*x) + 2^(2/3)*d*(a + b*x^3)^(1/3)])/(2^(7/3)*Rt[b, 3]*c),
x]) /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^3 + a*d^3, 0]

Rule 2177

Int[((e_.) + (f_.)*(x_))/(((c_.) + (d_.)*(x_))*((a_) + (b_.)*(x_)^3)^(1/3)), x_Symbol] :> Dist[f/d, Int[1/(a +
b*x^3)^(1/3), x], x] + Dist[(d*e - c*f)/d, Int[1/((c + d*x)*(a + b*x^3)^(1/3)), x], x] /; FreeQ[{a, b, c, d,
e, f}, x]

Rule 2178

Int[((a_) + (b_.)*(x_)^3)^(2/3)/((c_) + (d_.)*(x_)), x_Symbol] :> Simp[(a + b*x^3)^(2/3)/(2*d), x] + (Dist[1/d
^2, Int[(a*d^2 + b*c^2*x)/((c + d*x)*(a + b*x^3)^(1/3)), x], x] - Dist[b*(c/d^2), Int[x/(a + b*x^3)^(1/3), x],
x]) /; FreeQ[{a, b, c, d}, x]

Rule 2183

Int[(Px_.)*((c_) + (d_.)*(x_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^3)^(p_.), x_Symbol] :> Dist[1/c^q, Int[E
xpandIntegrand[(c^3 - d^3*x^3)^q*(a + b*x^3)^p, Px/(c - d*x)^q, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] &&
PolyQ[Px, x] && EqQ[d^2 - c*e, 0] && ILtQ[q, 0] && RationalQ[p] && EqQ[Denominator[p], 3]

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}}{9 (-1+x)}-\frac {\left (1+x^3\right )^{2/3}}{x^6}+\frac {3 \left (1+x^3\right )^{2/3}}{x^3}+\frac {(-2-x) \left (1+x^3\right )^{2/3}}{9 \left (1+x+x^2\right )}-\frac {20 \left (1+x^3\right )^{2/3}}{3 \left (1+2 x^3\right )}\right ) \, dx \\ & = \frac {1}{9} \int \frac {\left (1+x^3\right )^{2/3}}{-1+x} \, dx+\frac {1}{9} \int \frac {(-2-x) \left (1+x^3\right )^{2/3}}{1+x+x^2} \, dx+3 \int \frac {\left (1+x^3\right )^{2/3}}{x^3} \, dx-\frac {20}{3} \int \frac {\left (1+x^3\right )^{2/3}}{1+2 x^3} \, dx-\int \frac {\left (1+x^3\right )^{2/3}}{x^6} \, dx \\ & = \frac {1}{18} \left (1+x^3\right )^{2/3}-\frac {3 \left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\left (1+x^3\right )^{5/3}}{5 x^5}+\frac {1}{9} \int \frac {x}{\sqrt [3]{1+x^3}} \, dx+\frac {1}{9} \int \frac {1+x}{(-1+x) \sqrt [3]{1+x^3}} \, dx+\frac {1}{9} \int \left (-\frac {2 \left (1+x^3\right )^{2/3}}{1-x^3}+\frac {x \left (1+x^3\right )^{2/3}}{1-x^3}+\frac {x^2 \left (1+x^3\right )^{2/3}}{1-x^3}\right ) \, dx+3 \int \frac {1}{\sqrt [3]{1+x^3}} \, dx-\frac {10}{3} \int \frac {1}{\sqrt [3]{1+x^3}} \, dx-\frac {10}{3} \int \frac {1}{\sqrt [3]{1+x^3} \left (1+2 x^3\right )} \, dx \\ & = \frac {1}{18} \left (1+x^3\right )^{2/3}-\frac {3 \left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\left (1+x^3\right )^{5/3}}{5 x^5}+\frac {10 \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {10 \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\sqrt {3} \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )+\frac {1}{18} x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},-x^3\right )+\frac {5}{9} \log \left (1+2 x^3\right )-\frac {5}{3} \log \left (-x-\sqrt [3]{1+x^3}\right )+\frac {1}{6} \log \left (-x+\sqrt [3]{1+x^3}\right )+\frac {1}{9} \int \frac {1}{\sqrt [3]{1+x^3}} \, dx+\frac {1}{9} \int \frac {x \left (1+x^3\right )^{2/3}}{1-x^3} \, dx+\frac {1}{9} \int \frac {x^2 \left (1+x^3\right )^{2/3}}{1-x^3} \, dx+\frac {2}{9} \int \frac {1}{(-1+x) \sqrt [3]{1+x^3}} \, dx-\frac {2}{9} \int \frac {\left (1+x^3\right )^{2/3}}{1-x^3} \, dx \\ & = \frac {1}{18} \left (1+x^3\right )^{2/3}-\frac {3 \left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\left (1+x^3\right )^{5/3}}{5 x^5}+\frac {10 \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {29 \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3}}+\sqrt {3} \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )-\frac {\arctan \left (\frac {1+\frac {\sqrt [3]{2} (1+x)}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{3 \sqrt [3]{2} \sqrt {3}}+\frac {1}{18} x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},-x^3\right )-\frac {\log \left (-(1-x)^2 (1+x)\right )}{18 \sqrt [3]{2}}+\frac {5}{9} \log \left (1+2 x^3\right )-\frac {5}{3} \log \left (-x-\sqrt [3]{1+x^3}\right )+\frac {1}{9} \log \left (-x+\sqrt [3]{1+x^3}\right )+\frac {\log \left (-1-x+2^{2/3} \sqrt [3]{1+x^3}\right )}{6 \sqrt [3]{2}}+\frac {1}{27} \text {Subst}\left (\int \frac {(1+x)^{2/3}}{1-x} \, dx,x,x^3\right )-\frac {1}{9} \int \frac {x}{\sqrt [3]{1+x^3}} \, dx+\frac {2}{9} \int \frac {1}{\sqrt [3]{1+x^3}} \, dx+\frac {2}{9} \int \frac {x}{\left (1-x^3\right ) \sqrt [3]{1+x^3}} \, dx-\frac {4}{9} \int \frac {1}{\left (1-x^3\right ) \sqrt [3]{1+x^3}} \, dx \\ & = -\frac {3 \left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\left (1+x^3\right )^{5/3}}{5 x^5}+\frac {10 \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {2\ 2^{2/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3}}-\frac {\arctan \left (\frac {1+\frac {\sqrt [3]{2} (1+x)}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{3 \sqrt [3]{2} \sqrt {3}}-\frac {\log \left (-(1-x)^2 (1+x)\right )}{18 \sqrt [3]{2}}-\frac {1}{27} 2^{2/3} \log \left (1-x^3\right )+\frac {5}{9} \log \left (1+2 x^3\right )-\frac {5}{3} \log \left (-x-\sqrt [3]{1+x^3}\right )+\frac {1}{9} 2^{2/3} \log \left (\sqrt [3]{2} x-\sqrt [3]{1+x^3}\right )+\frac {\log \left (-1-x+2^{2/3} \sqrt [3]{1+x^3}\right )}{6 \sqrt [3]{2}}+\frac {2}{27} \int \frac {1}{(1-x) \sqrt [3]{1+x^3}} \, dx+\frac {2}{27} \text {Subst}\left (\int \frac {1}{(1-x) \sqrt [3]{1+x}} \, dx,x,x^3\right )-\frac {2}{9} \text {Subst}\left (\int \frac {1}{1+2 x^3} \, dx,x,\frac {1+x}{\sqrt [3]{1+x^3}}\right ) \\ & = -\frac {3 \left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\left (1+x^3\right )^{5/3}}{5 x^5}+\frac {10 \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {2\ 2^{2/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3}}-\frac {2^{2/3} \arctan \left (\frac {1+\frac {\sqrt [3]{2} (1+x)}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3}}-\frac {\log \left (-(1-x)^2 (1+x)\right )}{18 \sqrt [3]{2}}+\frac {\log \left ((1-x)^2 (1+x)\right )}{54 \sqrt [3]{2}}+\frac {\log \left (1-x^3\right )}{27 \sqrt [3]{2}}-\frac {1}{27} 2^{2/3} \log \left (1-x^3\right )+\frac {5}{9} \log \left (1+2 x^3\right )-\frac {5}{3} \log \left (-x-\sqrt [3]{1+x^3}\right )+\frac {1}{9} 2^{2/3} \log \left (\sqrt [3]{2} x-\sqrt [3]{1+x^3}\right )-\frac {\log \left (1+x-2^{2/3} \sqrt [3]{1+x^3}\right )}{18 \sqrt [3]{2}}+\frac {\log \left (-1-x+2^{2/3} \sqrt [3]{1+x^3}\right )}{6 \sqrt [3]{2}}-\frac {2}{27} \text {Subst}\left (\int \frac {1}{1+\sqrt [3]{2} x} \, dx,x,\frac {1+x}{\sqrt [3]{1+x^3}}\right )-\frac {2}{27} \text {Subst}\left (\int \frac {2-\sqrt [3]{2} x}{1-\sqrt [3]{2} x+2^{2/3} x^2} \, dx,x,\frac {1+x}{\sqrt [3]{1+x^3}}\right )-\frac {1}{9} \text {Subst}\left (\int \frac {1}{2^{2/3}+\sqrt [3]{2} x+x^2} \, dx,x,\sqrt [3]{1+x^3}\right )+\frac {\text {Subst}\left (\int \frac {1}{\sqrt [3]{2}-x} \, dx,x,\sqrt [3]{1+x^3}\right )}{9 \sqrt [3]{2}} \\ & = -\frac {3 \left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\left (1+x^3\right )^{5/3}}{5 x^5}+\frac {10 \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {2\ 2^{2/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3}}-\frac {2^{2/3} \arctan \left (\frac {1+\frac {\sqrt [3]{2} (1+x)}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3}}-\frac {\log \left (-(1-x)^2 (1+x)\right )}{18 \sqrt [3]{2}}+\frac {\log \left ((1-x)^2 (1+x)\right )}{54 \sqrt [3]{2}}+\frac {\log \left (1-x^3\right )}{27 \sqrt [3]{2}}-\frac {1}{27} 2^{2/3} \log \left (1-x^3\right )+\frac {5}{9} \log \left (1+2 x^3\right )-\frac {1}{27} 2^{2/3} \log \left (1+\frac {\sqrt [3]{2} (1+x)}{\sqrt [3]{1+x^3}}\right )-\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1+x^3}\right )}{9 \sqrt [3]{2}}-\frac {5}{3} \log \left (-x-\sqrt [3]{1+x^3}\right )+\frac {1}{9} 2^{2/3} \log \left (\sqrt [3]{2} x-\sqrt [3]{1+x^3}\right )-\frac {\log \left (1+x-2^{2/3} \sqrt [3]{1+x^3}\right )}{18 \sqrt [3]{2}}+\frac {\log \left (-1-x+2^{2/3} \sqrt [3]{1+x^3}\right )}{6 \sqrt [3]{2}}-\frac {1}{9} \text {Subst}\left (\int \frac {1}{1-\sqrt [3]{2} x+2^{2/3} x^2} \, dx,x,\frac {1+x}{\sqrt [3]{1+x^3}}\right )+\frac {\text {Subst}\left (\int \frac {-\sqrt [3]{2}+2\ 2^{2/3} x}{1-\sqrt [3]{2} x+2^{2/3} x^2} \, dx,x,\frac {1+x}{\sqrt [3]{1+x^3}}\right )}{27 \sqrt [3]{2}}+\frac {1}{9} 2^{2/3} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2^{2/3} \sqrt [3]{1+x^3}\right ) \\ & = -\frac {3 \left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\left (1+x^3\right )^{5/3}}{5 x^5}+\frac {10 \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {2\ 2^{2/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3}}-\frac {2^{2/3} \arctan \left (\frac {1+\frac {\sqrt [3]{2} (1+x)}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3}}-\frac {2^{2/3} \arctan \left (\frac {1+2^{2/3} \sqrt [3]{1+x^3}}{\sqrt {3}}\right )}{9 \sqrt {3}}-\frac {\log \left (-(1-x)^2 (1+x)\right )}{18 \sqrt [3]{2}}+\frac {\log \left ((1-x)^2 (1+x)\right )}{54 \sqrt [3]{2}}+\frac {\log \left (1-x^3\right )}{27 \sqrt [3]{2}}-\frac {1}{27} 2^{2/3} \log \left (1-x^3\right )+\frac {5}{9} \log \left (1+2 x^3\right )+\frac {\log \left (1+\frac {2^{2/3} (1+x)^2}{\left (1+x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1+x)}{\sqrt [3]{1+x^3}}\right )}{27 \sqrt [3]{2}}-\frac {1}{27} 2^{2/3} \log \left (1+\frac {\sqrt [3]{2} (1+x)}{\sqrt [3]{1+x^3}}\right )-\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1+x^3}\right )}{9 \sqrt [3]{2}}-\frac {5}{3} \log \left (-x-\sqrt [3]{1+x^3}\right )+\frac {1}{9} 2^{2/3} \log \left (\sqrt [3]{2} x-\sqrt [3]{1+x^3}\right )-\frac {\log \left (1+x-2^{2/3} \sqrt [3]{1+x^3}\right )}{18 \sqrt [3]{2}}+\frac {\log \left (-1-x+2^{2/3} \sqrt [3]{1+x^3}\right )}{6 \sqrt [3]{2}}-\frac {1}{9} 2^{2/3} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{2} (1+x)}{\sqrt [3]{1+x^3}}\right ) \\ & = -\frac {3 \left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\left (1+x^3\right )^{5/3}}{5 x^5}+\frac {10 \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {2\ 2^{2/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3}}+\frac {2^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} (1+x)}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3}}-\frac {2^{2/3} \arctan \left (\frac {1+\frac {\sqrt [3]{2} (1+x)}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3}}-\frac {2^{2/3} \arctan \left (\frac {1+2^{2/3} \sqrt [3]{1+x^3}}{\sqrt {3}}\right )}{9 \sqrt {3}}-\frac {\log \left (-(1-x)^2 (1+x)\right )}{18 \sqrt [3]{2}}+\frac {\log \left ((1-x)^2 (1+x)\right )}{54 \sqrt [3]{2}}+\frac {\log \left (1-x^3\right )}{27 \sqrt [3]{2}}-\frac {1}{27} 2^{2/3} \log \left (1-x^3\right )+\frac {5}{9} \log \left (1+2 x^3\right )+\frac {\log \left (1+\frac {2^{2/3} (1+x)^2}{\left (1+x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1+x)}{\sqrt [3]{1+x^3}}\right )}{27 \sqrt [3]{2}}-\frac {1}{27} 2^{2/3} \log \left (1+\frac {\sqrt [3]{2} (1+x)}{\sqrt [3]{1+x^3}}\right )-\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1+x^3}\right )}{9 \sqrt [3]{2}}-\frac {5}{3} \log \left (-x-\sqrt [3]{1+x^3}\right )+\frac {1}{9} 2^{2/3} \log \left (\sqrt [3]{2} x-\sqrt [3]{1+x^3}\right )-\frac {\log \left (1+x-2^{2/3} \sqrt [3]{1+x^3}\right )}{18 \sqrt [3]{2}}+\frac {\log \left (-1-x+2^{2/3} \sqrt [3]{1+x^3}\right )}{6 \sqrt [3]{2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.49 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.95 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (1-2 x^3+2 x^6\right )}{x^6 \left (-1-x^3+2 x^6\right )} \, dx=\frac {1}{90} \left (\frac {9 \left (2-13 x^3\right ) \left (1+x^3\right )^{2/3}}{x^5}+100 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x-2 \sqrt [3]{1+x^3}}\right )-10\ 2^{2/3} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{1+x^3}}\right )-100 \log \left (x+\sqrt [3]{1+x^3}\right )+10\ 2^{2/3} \log \left (-2 x+2^{2/3} \sqrt [3]{1+x^3}\right )+50 \log \left (x^2-x \sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right )-5\ 2^{2/3} \log \left (2 x^2+2^{2/3} x \sqrt [3]{1+x^3}+\sqrt [3]{2} \left (1+x^3\right )^{2/3}\right )\right ) \]

[In]

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

[Out]

((9*(2 - 13*x^3)*(1 + x^3)^(2/3))/x^5 + 100*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x - 2*(1 + x^3)^(1/3))] - 10*2^(2/3)*S
qrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(1 + x^3)^(1/3))] - 100*Log[x + (1 + x^3)^(1/3)] + 10*2^(2/3)*Log[-2*x
+ 2^(2/3)*(1 + x^3)^(1/3)] + 50*Log[x^2 - x*(1 + x^3)^(1/3) + (1 + x^3)^(2/3)] - 5*2^(2/3)*Log[2*x^2 + 2^(2/3)
*x*(1 + x^3)^(1/3) + 2^(1/3)*(1 + x^3)^(2/3)])/90

Maple [A] (verified)

Time = 5.69 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.86

method	result	size
pseudoelliptic	\(\frac {10 \,2^{\frac {2}{3}} \ln \left (\frac {-2^{\frac {1}{3}} x +\left (x^{3}+1\right )^{\frac {1}{3}}}{x}\right ) x^{5}-100 \ln \left (\frac {x +\left (x^{3}+1\right )^{\frac {1}{3}}}{x}\right ) x^{5}+\left (-117 x^{3}+18\right ) \left (x^{3}+1\right )^{\frac {2}{3}}-5 x^{5} \left (\left (-2 \arctan \left (\frac {\sqrt {3}\, \left (x +2^{\frac {2}{3}} \left (x^{3}+1\right )^{\frac {1}{3}}\right )}{3 x}\right ) \sqrt {3}+\ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} x \left (x^{3}+1\right )^{\frac {1}{3}}+\left (x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right )\right ) 2^{\frac {2}{3}}+20 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x -2 \left (x^{3}+1\right )^{\frac {1}{3}}\right )}{3 x}\right )-10 \ln \left (\frac {x^{2}-x \left (x^{3}+1\right )^{\frac {1}{3}}+\left (x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right )\right )}{90 x^{5}}\)	\(191\)
risch	\(\text {Expression too large to display}\)	\(1357\)

[In]

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

[Out]

1/90*(10*2^(2/3)*ln((-2^(1/3)*x+(x^3+1)^(1/3))/x)*x^5-100*ln((x+(x^3+1)^(1/3))/x)*x^5+(-117*x^3+18)*(x^3+1)^(2
/3)-5*x^5*((-2*arctan(1/3*3^(1/2)/x*(x+2^(2/3)*(x^3+1)^(1/3)))*3^(1/2)+ln((2^(2/3)*x^2+2^(1/3)*x*(x^3+1)^(1/3)
+(x^3+1)^(2/3))/x^2))*2^(2/3)+20*3^(1/2)*arctan(1/3*3^(1/2)*(x-2*(x^3+1)^(1/3))/x)-10*ln((x^2-x*(x^3+1)^(1/3)+
(x^3+1)^(2/3))/x^2)))/x^5

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 367 vs. \(2 (166) = 332\).

Time = 2.03 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.66 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (1-2 x^3+2 x^6\right )}{x^6 \left (-1-x^3+2 x^6\right )} \, dx=-\frac {10 \cdot 4^{\frac {1}{3}} \sqrt {3} x^{5} \arctan \left (\frac {3 \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (5 \, x^{7} - 4 \, x^{4} - x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} - 6 \cdot 4^{\frac {1}{3}} \sqrt {3} {\left (19 \, x^{8} + 16 \, x^{5} + x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}} - \sqrt {3} {\left (71 \, x^{9} + 111 \, x^{6} + 33 \, x^{3} + 1\right )}}{3 \, {\left (109 \, x^{9} + 105 \, x^{6} + 3 \, x^{3} - 1\right )}}\right ) - 300 \, \sqrt {3} x^{5} \arctan \left (\frac {4 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 2 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (x^{3} + 1\right )}}{7 \, x^{3} - 1}\right ) - 10 \cdot 4^{\frac {1}{3}} x^{5} \log \left (\frac {3 \cdot 4^{\frac {2}{3}} {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 6 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} x - 4^{\frac {1}{3}} {\left (x^{3} - 1\right )}}{x^{3} - 1}\right ) + 5 \cdot 4^{\frac {1}{3}} x^{5} \log \left (\frac {6 \cdot 4^{\frac {1}{3}} {\left (5 \, x^{4} + x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} + 4^{\frac {2}{3}} {\left (19 \, x^{6} + 16 \, x^{3} + 1\right )} + 24 \, {\left (2 \, x^{5} + x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x^{6} - 2 \, x^{3} + 1}\right ) + 150 \, x^{5} \log \left (\frac {2 \, x^{3} + 3 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} x + 1}{2 \, x^{3} + 1}\right ) + 27 \, {\left (13 \, x^{3} - 2\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{270 \, x^{5}} \]

[In]

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

[Out]

-1/270*(10*4^(1/3)*sqrt(3)*x^5*arctan(1/3*(3*4^(2/3)*sqrt(3)*(5*x^7 - 4*x^4 - x)*(x^3 + 1)^(2/3) - 6*4^(1/3)*s
qrt(3)*(19*x^8 + 16*x^5 + x^2)*(x^3 + 1)^(1/3) - sqrt(3)*(71*x^9 + 111*x^6 + 33*x^3 + 1))/(109*x^9 + 105*x^6 +
3*x^3 - 1)) - 300*sqrt(3)*x^5*arctan((4*sqrt(3)*(x^3 + 1)^(1/3)*x^2 + 2*sqrt(3)*(x^3 + 1)^(2/3)*x + sqrt(3)*(
x^3 + 1))/(7*x^3 - 1)) - 10*4^(1/3)*x^5*log((3*4^(2/3)*(x^3 + 1)^(1/3)*x^2 - 6*(x^3 + 1)^(2/3)*x - 4^(1/3)*(x^
3 - 1))/(x^3 - 1)) + 5*4^(1/3)*x^5*log((6*4^(1/3)*(5*x^4 + x)*(x^3 + 1)^(2/3) + 4^(2/3)*(19*x^6 + 16*x^3 + 1)
+ 24*(2*x^5 + x^2)*(x^3 + 1)^(1/3))/(x^6 - 2*x^3 + 1)) + 150*x^5*log((2*x^3 + 3*(x^3 + 1)^(1/3)*x^2 + 3*(x^3 +
1)^(2/3)*x + 1)/(2*x^3 + 1)) + 27*(13*x^3 - 2)*(x^3 + 1)^(2/3))/x^5

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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