3.23.77 \(\int \frac {1}{\sqrt [3]{-x+x^3} (1+x^6)} \, dx\)

Optimal. Leaf size=173 \[ -\frac {1}{6} \text {RootSum}\left [\text {$\#$1}^6-\text {$\#$1}^3+1\& ,\frac {\log \left (\sqrt [3]{x^3-x}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ]-\frac {\log \left (2^{2/3} \sqrt [3]{x^3-x}-2 x\right )}{6 \sqrt [3]{2}}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^3-x}+x}\right )}{2 \sqrt [3]{2} \sqrt {3}}+\frac {\log \left (2^{2/3} \sqrt [3]{x^3-x} x+\sqrt [3]{2} \left (x^3-x\right )^{2/3}+2 x^2\right )}{12 \sqrt [3]{2}} \]

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Rubi [C]  time = 1.71, antiderivative size = 1319, normalized size of antiderivative = 7.62, number of steps used = 25, number of rules used = 12, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.632, Rules used = {2056, 6715, 2074, 2148, 6728, 377, 200, 31, 634, 617, 204, 628} \begin {gather*} \frac {\sqrt [3]{x} \sqrt [3]{x^2-1} \tan ^{-1}\left (\frac {1-\frac {\sqrt [3]{2} \left (1-x^{2/3}\right )}{\sqrt [3]{x^2-1}}}{\sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{x^3-x}}+\frac {\sqrt [3]{x} \sqrt [3]{x^2-1} \tan ^{-1}\left (\frac {\frac {\sqrt [3]{2} \left (2 x^{2/3}-i \sqrt {3}+1\right )}{\sqrt [3]{x^2-1}}+2}{2 \sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{x^3-x}}+\frac {\sqrt [3]{x} \sqrt [3]{x^2-1} \tan ^{-1}\left (\frac {\frac {\sqrt [3]{2} \left (2 x^{2/3}+i \sqrt {3}+1\right )}{\sqrt [3]{x^2-1}}+2}{2 \sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{x^3-x}}+\frac {\left (3 i-\sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{x^2-1} \tan ^{-1}\left (\frac {1-\frac {2 x^{2/3}}{\sqrt [3]{\frac {i+\sqrt {3}}{i-\sqrt {3}}} \sqrt [3]{x^2-1}}}{\sqrt {3}}\right )}{6\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}} \sqrt [3]{x^3-x}}-\frac {\left (1+i \sqrt {3}\right )^{2/3} \sqrt [3]{x} \sqrt [3]{x^2-1} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{\frac {i+\sqrt {3}}{i-\sqrt {3}}} x^{2/3}}{\sqrt [3]{x^2-1}}}{\sqrt {3}}\right )}{2\ 2^{2/3} \sqrt {3} \sqrt [3]{x^3-x}}+\frac {\sqrt [3]{x} \sqrt [3]{x^2-1} \log \left (-\left (-2 i x^{2/3}-\sqrt {3}+i\right )^2 \left (2 i x^{2/3}-\sqrt {3}+i\right )\right )}{24 \sqrt [3]{2} \sqrt [3]{x^3-x}}+\frac {\sqrt [3]{x} \sqrt [3]{x^2-1} \log \left (\left (-2 i x^{2/3}+\sqrt {3}+i\right )^2 \left (2 i x^{2/3}+\sqrt {3}+i\right )\right )}{24 \sqrt [3]{2} \sqrt [3]{x^3-x}}+\frac {\sqrt [3]{x} \sqrt [3]{x^2-1} \log \left (-\left (\left (1-x^{2/3}\right ) \left (x^{2/3}+1\right )^2\right )\right )}{24 \sqrt [3]{2} \sqrt [3]{x^3-x}}-\frac {\sqrt [3]{x} \sqrt [3]{x^2-1} \log \left (\frac {\left (1-i \sqrt {3}\right )^{2/3} x^{4/3}}{\left (x^2-1\right )^{2/3}}-\frac {2^{2/3} x^{2/3}}{\sqrt [3]{x^2-1}}+\left (1+i \sqrt {3}\right )^{2/3}\right )}{12 \sqrt [3]{\frac {i+\sqrt {3}}{i-\sqrt {3}}} \sqrt [3]{x^3-x}}-\frac {\sqrt [3]{\frac {i+\sqrt {3}}{i-\sqrt {3}}} \sqrt [3]{x} \sqrt [3]{x^2-1} \log \left (\frac {\left (1+i \sqrt {3}\right )^{2/3} x^{4/3}}{\left (x^2-1\right )^{2/3}}-\frac {2^{2/3} x^{2/3}}{\sqrt [3]{x^2-1}}+\left (1-i \sqrt {3}\right )^{2/3}\right )}{12 \sqrt [3]{x^3-x}}+\frac {\sqrt [3]{x} \sqrt [3]{x^2-1} \log \left (\frac {\sqrt [3]{1-i \sqrt {3}} x^{2/3}}{\sqrt [3]{x^2-1}}+\sqrt [3]{1+i \sqrt {3}}\right )}{6 \sqrt [3]{\frac {i+\sqrt {3}}{i-\sqrt {3}}} \sqrt [3]{x^3-x}}+\frac {\sqrt [3]{\frac {i+\sqrt {3}}{i-\sqrt {3}}} \sqrt [3]{x} \sqrt [3]{x^2-1} \log \left (\frac {\sqrt [3]{1+i \sqrt {3}} x^{2/3}}{\sqrt [3]{x^2-1}}+\sqrt [3]{1-i \sqrt {3}}\right )}{6 \sqrt [3]{x^3-x}}-\frac {\sqrt [3]{x} \sqrt [3]{x^2-1} \log \left (2 x^{2/3}-2\ 2^{2/3} \sqrt [3]{x^2-1}-i \sqrt {3}+1\right )}{8 \sqrt [3]{2} \sqrt [3]{x^3-x}}-\frac {\sqrt [3]{x} \sqrt [3]{x^2-1} \log \left (2 x^{2/3}-2\ 2^{2/3} \sqrt [3]{x^2-1}+i \sqrt {3}+1\right )}{8 \sqrt [3]{2} \sqrt [3]{x^3-x}}-\frac {\sqrt [3]{x} \sqrt [3]{x^2-1} \log \left (-x^{2/3}+2^{2/3} \sqrt [3]{x^2-1}+1\right )}{8 \sqrt [3]{2} \sqrt [3]{x^3-x}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[1/((-x + x^3)^(1/3)*(1 + x^6)),x]

[Out]

(x^(1/3)*(-1 + x^2)^(1/3)*ArcTan[(1 - (2^(1/3)*(1 - x^(2/3)))/(-1 + x^2)^(1/3))/Sqrt[3]])/(4*2^(1/3)*Sqrt[3]*(
-x + x^3)^(1/3)) + (x^(1/3)*(-1 + x^2)^(1/3)*ArcTan[(2 + (2^(1/3)*(1 - I*Sqrt[3] + 2*x^(2/3)))/(-1 + x^2)^(1/3
))/(2*Sqrt[3])])/(4*2^(1/3)*Sqrt[3]*(-x + x^3)^(1/3)) + (x^(1/3)*(-1 + x^2)^(1/3)*ArcTan[(2 + (2^(1/3)*(1 + I*
Sqrt[3] + 2*x^(2/3)))/(-1 + x^2)^(1/3))/(2*Sqrt[3])])/(4*2^(1/3)*Sqrt[3]*(-x + x^3)^(1/3)) + ((3*I - Sqrt[3])*
x^(1/3)*(-1 + x^2)^(1/3)*ArcTan[(1 - (2*x^(2/3))/(((I + Sqrt[3])/(I - Sqrt[3]))^(1/3)*(-1 + x^2)^(1/3)))/Sqrt[
3]])/(6*2^(2/3)*(1 - I*Sqrt[3])^(1/3)*(-x + x^3)^(1/3)) - ((1 + I*Sqrt[3])^(2/3)*x^(1/3)*(-1 + x^2)^(1/3)*ArcT
an[(1 - (2*((I + Sqrt[3])/(I - Sqrt[3]))^(1/3)*x^(2/3))/(-1 + x^2)^(1/3))/Sqrt[3]])/(2*2^(2/3)*Sqrt[3]*(-x + x
^3)^(1/3)) + (x^(1/3)*(-1 + x^2)^(1/3)*Log[-((I - Sqrt[3] - (2*I)*x^(2/3))^2*(I - Sqrt[3] + (2*I)*x^(2/3)))])/
(24*2^(1/3)*(-x + x^3)^(1/3)) + (x^(1/3)*(-1 + x^2)^(1/3)*Log[(I + Sqrt[3] - (2*I)*x^(2/3))^2*(I + Sqrt[3] + (
2*I)*x^(2/3))])/(24*2^(1/3)*(-x + x^3)^(1/3)) + (x^(1/3)*(-1 + x^2)^(1/3)*Log[-((1 - x^(2/3))*(1 + x^(2/3))^2)
])/(24*2^(1/3)*(-x + x^3)^(1/3)) - (x^(1/3)*(-1 + x^2)^(1/3)*Log[(1 + I*Sqrt[3])^(2/3) + ((1 - I*Sqrt[3])^(2/3
)*x^(4/3))/(-1 + x^2)^(2/3) - (2^(2/3)*x^(2/3))/(-1 + x^2)^(1/3)])/(12*((I + Sqrt[3])/(I - Sqrt[3]))^(1/3)*(-x
+ x^3)^(1/3)) - (((I + Sqrt[3])/(I - Sqrt[3]))^(1/3)*x^(1/3)*(-1 + x^2)^(1/3)*Log[(1 - I*Sqrt[3])^(2/3) + ((1
+ I*Sqrt[3])^(2/3)*x^(4/3))/(-1 + x^2)^(2/3) - (2^(2/3)*x^(2/3))/(-1 + x^2)^(1/3)])/(12*(-x + x^3)^(1/3)) + (
x^(1/3)*(-1 + x^2)^(1/3)*Log[(1 + I*Sqrt[3])^(1/3) + ((1 - I*Sqrt[3])^(1/3)*x^(2/3))/(-1 + x^2)^(1/3)])/(6*((I
+ Sqrt[3])/(I - Sqrt[3]))^(1/3)*(-x + x^3)^(1/3)) + (((I + Sqrt[3])/(I - Sqrt[3]))^(1/3)*x^(1/3)*(-1 + x^2)^(
1/3)*Log[(1 - I*Sqrt[3])^(1/3) + ((1 + I*Sqrt[3])^(1/3)*x^(2/3))/(-1 + x^2)^(1/3)])/(6*(-x + x^3)^(1/3)) - (x^
(1/3)*(-1 + x^2)^(1/3)*Log[1 - I*Sqrt[3] + 2*x^(2/3) - 2*2^(2/3)*(-1 + x^2)^(1/3)])/(8*2^(1/3)*(-x + x^3)^(1/3
)) - (x^(1/3)*(-1 + x^2)^(1/3)*Log[1 + I*Sqrt[3] + 2*x^(2/3) - 2*2^(2/3)*(-1 + x^2)^(1/3)])/(8*2^(1/3)*(-x + x
^3)^(1/3)) - (x^(1/3)*(-1 + x^2)^(1/3)*Log[1 - x^(2/3) + 2^(2/3)*(-1 + x^2)^(1/3)])/(8*2^(1/3)*(-x + x^3)^(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 200

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 204

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

Rule 377

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

Rule 617

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 628

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 634

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 2056

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rule 2148

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 6715

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

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 {1}{\sqrt [3]{-x+x^3} \left (1+x^6\right )} \, dx &=\frac {\left (\sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \int \frac {1}{\sqrt [3]{x} \sqrt [3]{-1+x^2} \left (1+x^6\right )} \, dx}{\sqrt [3]{-x+x^3}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^3} \left (1+x^9\right )} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x+x^3}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{9 (1+x) \sqrt [3]{-1+x^3}}+\frac {2-x}{9 \left (1-x+x^2\right ) \sqrt [3]{-1+x^3}}+\frac {2-x^3}{3 \sqrt [3]{-1+x^3} \left (1-x^3+x^6\right )}\right ) \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x+x^3}}\\ &=\frac {\left (\sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{(1+x) \sqrt [3]{-1+x^3}} \, dx,x,x^{2/3}\right )}{6 \sqrt [3]{-x+x^3}}+\frac {\left (\sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {2-x}{\left (1-x+x^2\right ) \sqrt [3]{-1+x^3}} \, dx,x,x^{2/3}\right )}{6 \sqrt [3]{-x+x^3}}+\frac {\left (\sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {2-x^3}{\sqrt [3]{-1+x^3} \left (1-x^3+x^6\right )} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x+x^3}}\\ &=\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \tan ^{-1}\left (\frac {1-\frac {\sqrt [3]{2} \left (1-x^{2/3}\right )}{\sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (-\left (\left (1-x^{2/3}\right ) \left (1+x^{2/3}\right )^2\right )\right )}{24 \sqrt [3]{2} \sqrt [3]{-x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (1-x^{2/3}+2^{2/3} \sqrt [3]{-1+x^2}\right )}{8 \sqrt [3]{2} \sqrt [3]{-x+x^3}}+\frac {\left (\sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {-1-i \sqrt {3}}{\left (-1-i \sqrt {3}+2 x\right ) \sqrt [3]{-1+x^3}}+\frac {-1+i \sqrt {3}}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt [3]{-1+x^3}}\right ) \, dx,x,x^{2/3}\right )}{6 \sqrt [3]{-x+x^3}}+\frac {\left (\sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {-1-i \sqrt {3}}{\sqrt [3]{-1+x^3} \left (-1-i \sqrt {3}+2 x^3\right )}+\frac {-1+i \sqrt {3}}{\sqrt [3]{-1+x^3} \left (-1+i \sqrt {3}+2 x^3\right )}\right ) \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x+x^3}}\\ &=\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \tan ^{-1}\left (\frac {1-\frac {\sqrt [3]{2} \left (1-x^{2/3}\right )}{\sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (-\left (\left (1-x^{2/3}\right ) \left (1+x^{2/3}\right )^2\right )\right )}{24 \sqrt [3]{2} \sqrt [3]{-x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (1-x^{2/3}+2^{2/3} \sqrt [3]{-1+x^2}\right )}{8 \sqrt [3]{2} \sqrt [3]{-x+x^3}}+\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1-i \sqrt {3}+2 x\right ) \sqrt [3]{-1+x^3}} \, dx,x,x^{2/3}\right )}{6 \sqrt [3]{-x+x^3}}+\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^3} \left (-1-i \sqrt {3}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x+x^3}}+\frac {\left (\left (-1+i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt [3]{-1+x^3}} \, dx,x,x^{2/3}\right )}{6 \sqrt [3]{-x+x^3}}+\frac {\left (\left (-1+i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^3} \left (-1+i \sqrt {3}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x+x^3}}\\ &=\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \tan ^{-1}\left (\frac {1-\frac {\sqrt [3]{2} \left (1-x^{2/3}\right )}{\sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \tan ^{-1}\left (\frac {2+\frac {\sqrt [3]{2} \left (1-i \sqrt {3}+2 x^{2/3}\right )}{\sqrt [3]{-1+x^2}}}{2 \sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \tan ^{-1}\left (\frac {2+\frac {\sqrt [3]{2} \left (1+i \sqrt {3}+2 x^{2/3}\right )}{\sqrt [3]{-1+x^2}}}{2 \sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (-\left (i-\sqrt {3}-2 i x^{2/3}\right )^2 \left (i-\sqrt {3}+2 i x^{2/3}\right )\right )}{24 \sqrt [3]{2} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (\left (i+\sqrt {3}-2 i x^{2/3}\right )^2 \left (i+\sqrt {3}+2 i x^{2/3}\right )\right )}{24 \sqrt [3]{2} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (-\left (\left (1-x^{2/3}\right ) \left (1+x^{2/3}\right )^2\right )\right )}{24 \sqrt [3]{2} \sqrt [3]{-x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (1-i \sqrt {3}+2 x^{2/3}-2\ 2^{2/3} \sqrt [3]{-1+x^2}\right )}{8 \sqrt [3]{2} \sqrt [3]{-x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (1+i \sqrt {3}+2 x^{2/3}-2\ 2^{2/3} \sqrt [3]{-1+x^2}\right )}{8 \sqrt [3]{2} \sqrt [3]{-x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (1-x^{2/3}+2^{2/3} \sqrt [3]{-1+x^2}\right )}{8 \sqrt [3]{2} \sqrt [3]{-x+x^3}}+\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-i \sqrt {3}-\left (1-i \sqrt {3}\right ) x^3} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{2 \sqrt [3]{-x+x^3}}+\frac {\left (\left (-1+i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{-1+i \sqrt {3}-\left (1+i \sqrt {3}\right ) x^3} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{2 \sqrt [3]{-x+x^3}}\\ &=\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \tan ^{-1}\left (\frac {1-\frac {\sqrt [3]{2} \left (1-x^{2/3}\right )}{\sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \tan ^{-1}\left (\frac {2+\frac {\sqrt [3]{2} \left (1-i \sqrt {3}+2 x^{2/3}\right )}{\sqrt [3]{-1+x^2}}}{2 \sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \tan ^{-1}\left (\frac {2+\frac {\sqrt [3]{2} \left (1+i \sqrt {3}+2 x^{2/3}\right )}{\sqrt [3]{-1+x^2}}}{2 \sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (-\left (i-\sqrt {3}-2 i x^{2/3}\right )^2 \left (i-\sqrt {3}+2 i x^{2/3}\right )\right )}{24 \sqrt [3]{2} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (\left (i+\sqrt {3}-2 i x^{2/3}\right )^2 \left (i+\sqrt {3}+2 i x^{2/3}\right )\right )}{24 \sqrt [3]{2} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (-\left (\left (1-x^{2/3}\right ) \left (1+x^{2/3}\right )^2\right )\right )}{24 \sqrt [3]{2} \sqrt [3]{-x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (1-i \sqrt {3}+2 x^{2/3}-2\ 2^{2/3} \sqrt [3]{-1+x^2}\right )}{8 \sqrt [3]{2} \sqrt [3]{-x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (1+i \sqrt {3}+2 x^{2/3}-2\ 2^{2/3} \sqrt [3]{-1+x^2}\right )}{8 \sqrt [3]{2} \sqrt [3]{-x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (1-x^{2/3}+2^{2/3} \sqrt [3]{-1+x^2}\right )}{8 \sqrt [3]{2} \sqrt [3]{-x+x^3}}+\frac {\left (\left (-1+i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{-\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{1+i \sqrt {3}} x} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{6 \left (1-i \sqrt {3}\right )^{2/3} \sqrt [3]{-x+x^3}}+\frac {\left (\left (-1+i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {-2 \sqrt [3]{1-i \sqrt {3}}+\sqrt [3]{1+i \sqrt {3}} x}{\left (1-i \sqrt {3}\right )^{2/3}-\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{1+i \sqrt {3}} x+\left (1+i \sqrt {3}\right )^{2/3} x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{6 \left (1-i \sqrt {3}\right )^{2/3} \sqrt [3]{-x+x^3}}+\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{-\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{1-i \sqrt {3}} x} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{6 \left (1+i \sqrt {3}\right )^{2/3} \sqrt [3]{-x+x^3}}+\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {-2 \sqrt [3]{1+i \sqrt {3}}+\sqrt [3]{1-i \sqrt {3}} x}{\left (1+i \sqrt {3}\right )^{2/3}-\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{1+i \sqrt {3}} x+\left (1-i \sqrt {3}\right )^{2/3} x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{6 \left (1+i \sqrt {3}\right )^{2/3} \sqrt [3]{-x+x^3}}\\ &=\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \tan ^{-1}\left (\frac {1-\frac {\sqrt [3]{2} \left (1-x^{2/3}\right )}{\sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \tan ^{-1}\left (\frac {2+\frac {\sqrt [3]{2} \left (1-i \sqrt {3}+2 x^{2/3}\right )}{\sqrt [3]{-1+x^2}}}{2 \sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \tan ^{-1}\left (\frac {2+\frac {\sqrt [3]{2} \left (1+i \sqrt {3}+2 x^{2/3}\right )}{\sqrt [3]{-1+x^2}}}{2 \sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (-\left (i-\sqrt {3}-2 i x^{2/3}\right )^2 \left (i-\sqrt {3}+2 i x^{2/3}\right )\right )}{24 \sqrt [3]{2} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (\left (i+\sqrt {3}-2 i x^{2/3}\right )^2 \left (i+\sqrt {3}+2 i x^{2/3}\right )\right )}{24 \sqrt [3]{2} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (-\left (\left (1-x^{2/3}\right ) \left (1+x^{2/3}\right )^2\right )\right )}{24 \sqrt [3]{2} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (\sqrt [3]{1+i \sqrt {3}}+\frac {\sqrt [3]{1-i \sqrt {3}} x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{6 \sqrt [3]{\frac {i+\sqrt {3}}{i-\sqrt {3}}} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (\sqrt [3]{1-i \sqrt {3}}+\frac {\sqrt [3]{1+i \sqrt {3}} x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{6 \sqrt [3]{1+i \sqrt {3}} \sqrt [3]{-x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (1-i \sqrt {3}+2 x^{2/3}-2\ 2^{2/3} \sqrt [3]{-1+x^2}\right )}{8 \sqrt [3]{2} \sqrt [3]{-x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (1+i \sqrt {3}+2 x^{2/3}-2\ 2^{2/3} \sqrt [3]{-1+x^2}\right )}{8 \sqrt [3]{2} \sqrt [3]{-x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (1-x^{2/3}+2^{2/3} \sqrt [3]{-1+x^2}\right )}{8 \sqrt [3]{2} \sqrt [3]{-x+x^3}}-\frac {\left (\left (-1+i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-i \sqrt {3}\right )^{2/3}-\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{1+i \sqrt {3}} x+\left (1+i \sqrt {3}\right )^{2/3} x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{4 \sqrt [3]{1-i \sqrt {3}} \sqrt [3]{-x+x^3}}+\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {-\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{1+i \sqrt {3}}+2 \left (1-i \sqrt {3}\right )^{2/3} x}{\left (1+i \sqrt {3}\right )^{2/3}-\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{1+i \sqrt {3}} x+\left (1-i \sqrt {3}\right )^{2/3} x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{12 \sqrt [3]{1-i \sqrt {3}} \left (1+i \sqrt {3}\right )^{2/3} \sqrt [3]{-x+x^3}}-\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+i \sqrt {3}\right )^{2/3}-\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{1+i \sqrt {3}} x+\left (1-i \sqrt {3}\right )^{2/3} x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{4 \sqrt [3]{1+i \sqrt {3}} \sqrt [3]{-x+x^3}}+\frac {\left (\left (-1+i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {-\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{1+i \sqrt {3}}+2 \left (1+i \sqrt {3}\right )^{2/3} x}{\left (1-i \sqrt {3}\right )^{2/3}-\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{1+i \sqrt {3}} x+\left (1+i \sqrt {3}\right )^{2/3} x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{12 \left (1-i \sqrt {3}\right )^{2/3} \sqrt [3]{1+i \sqrt {3}} \sqrt [3]{-x+x^3}}\\ &=\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \tan ^{-1}\left (\frac {1-\frac {\sqrt [3]{2} \left (1-x^{2/3}\right )}{\sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \tan ^{-1}\left (\frac {2+\frac {\sqrt [3]{2} \left (1-i \sqrt {3}+2 x^{2/3}\right )}{\sqrt [3]{-1+x^2}}}{2 \sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \tan ^{-1}\left (\frac {2+\frac {\sqrt [3]{2} \left (1+i \sqrt {3}+2 x^{2/3}\right )}{\sqrt [3]{-1+x^2}}}{2 \sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (-\left (i-\sqrt {3}-2 i x^{2/3}\right )^2 \left (i-\sqrt {3}+2 i x^{2/3}\right )\right )}{24 \sqrt [3]{2} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (\left (i+\sqrt {3}-2 i x^{2/3}\right )^2 \left (i+\sqrt {3}+2 i x^{2/3}\right )\right )}{24 \sqrt [3]{2} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (-\left (\left (1-x^{2/3}\right ) \left (1+x^{2/3}\right )^2\right )\right )}{24 \sqrt [3]{2} \sqrt [3]{-x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (\left (1+i \sqrt {3}\right )^{2/3}+\frac {\left (1-i \sqrt {3}\right )^{2/3} x^{4/3}}{\left (-1+x^2\right )^{2/3}}-\frac {2^{2/3} x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{12 \sqrt [3]{\frac {i+\sqrt {3}}{i-\sqrt {3}}} \sqrt [3]{-x+x^3}}-\frac {\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (\left (1-i \sqrt {3}\right )^{2/3}+\frac {\left (1+i \sqrt {3}\right )^{2/3} x^{4/3}}{\left (-1+x^2\right )^{2/3}}-\frac {2^{2/3} x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{12 \sqrt [3]{1+i \sqrt {3}} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (\sqrt [3]{1+i \sqrt {3}}+\frac {\sqrt [3]{1-i \sqrt {3}} x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{6 \sqrt [3]{\frac {i+\sqrt {3}}{i-\sqrt {3}}} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (\sqrt [3]{1-i \sqrt {3}}+\frac {\sqrt [3]{1+i \sqrt {3}} x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{6 \sqrt [3]{1+i \sqrt {3}} \sqrt [3]{-x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (1-i \sqrt {3}+2 x^{2/3}-2\ 2^{2/3} \sqrt [3]{-1+x^2}\right )}{8 \sqrt [3]{2} \sqrt [3]{-x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (1+i \sqrt {3}+2 x^{2/3}-2\ 2^{2/3} \sqrt [3]{-1+x^2}\right )}{8 \sqrt [3]{2} \sqrt [3]{-x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (1-x^{2/3}+2^{2/3} \sqrt [3]{-1+x^2}\right )}{8 \sqrt [3]{2} \sqrt [3]{-x+x^3}}-\frac {\left (\left (-1+i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{1+i \sqrt {3}} x^{2/3}}{\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{-1+x^2}}\right )}{2\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}} \sqrt [3]{-x+x^3}}-\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{1-i \sqrt {3}} x^{2/3}}{\sqrt [3]{1+i \sqrt {3}} \sqrt [3]{-1+x^2}}\right )}{2\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}} \sqrt [3]{-x+x^3}}\\ &=\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \tan ^{-1}\left (\frac {1-\frac {\sqrt [3]{2} \left (1-x^{2/3}\right )}{\sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \tan ^{-1}\left (\frac {2+\frac {\sqrt [3]{2} \left (1-i \sqrt {3}+2 x^{2/3}\right )}{\sqrt [3]{-1+x^2}}}{2 \sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \tan ^{-1}\left (\frac {2+\frac {\sqrt [3]{2} \left (1+i \sqrt {3}+2 x^{2/3}\right )}{\sqrt [3]{-1+x^2}}}{2 \sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{-x+x^3}}+\frac {\left (3 i-\sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^2} \tan ^{-1}\left (\frac {1-\frac {2 x^{2/3}}{\sqrt [3]{\frac {i+\sqrt {3}}{i-\sqrt {3}}} \sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{6\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}} \sqrt [3]{-x+x^3}}-\frac {\left (3 i+\sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^2} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{\frac {i+\sqrt {3}}{i-\sqrt {3}}} x^{2/3}}{\sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{6\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (-\left (i-\sqrt {3}-2 i x^{2/3}\right )^2 \left (i-\sqrt {3}+2 i x^{2/3}\right )\right )}{24 \sqrt [3]{2} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (\left (i+\sqrt {3}-2 i x^{2/3}\right )^2 \left (i+\sqrt {3}+2 i x^{2/3}\right )\right )}{24 \sqrt [3]{2} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (-\left (\left (1-x^{2/3}\right ) \left (1+x^{2/3}\right )^2\right )\right )}{24 \sqrt [3]{2} \sqrt [3]{-x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (\left (1+i \sqrt {3}\right )^{2/3}+\frac {\left (1-i \sqrt {3}\right )^{2/3} x^{4/3}}{\left (-1+x^2\right )^{2/3}}-\frac {2^{2/3} x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{12 \sqrt [3]{\frac {i+\sqrt {3}}{i-\sqrt {3}}} \sqrt [3]{-x+x^3}}-\frac {\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (\left (1-i \sqrt {3}\right )^{2/3}+\frac {\left (1+i \sqrt {3}\right )^{2/3} x^{4/3}}{\left (-1+x^2\right )^{2/3}}-\frac {2^{2/3} x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{12 \sqrt [3]{1+i \sqrt {3}} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (\sqrt [3]{1+i \sqrt {3}}+\frac {\sqrt [3]{1-i \sqrt {3}} x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{6 \sqrt [3]{\frac {i+\sqrt {3}}{i-\sqrt {3}}} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (\sqrt [3]{1-i \sqrt {3}}+\frac {\sqrt [3]{1+i \sqrt {3}} x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{6 \sqrt [3]{1+i \sqrt {3}} \sqrt [3]{-x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (1-i \sqrt {3}+2 x^{2/3}-2\ 2^{2/3} \sqrt [3]{-1+x^2}\right )}{8 \sqrt [3]{2} \sqrt [3]{-x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (1+i \sqrt {3}+2 x^{2/3}-2\ 2^{2/3} \sqrt [3]{-1+x^2}\right )}{8 \sqrt [3]{2} \sqrt [3]{-x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (1-x^{2/3}+2^{2/3} \sqrt [3]{-1+x^2}\right )}{8 \sqrt [3]{2} \sqrt [3]{-x+x^3}}\\ \end {align*}

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Mathematica [A]  time = 0.29, size = 153, normalized size = 0.88 \begin {gather*} -\frac {\sqrt [3]{\frac {1}{x^2}-1} x \left (4 \text {RootSum}\left [\text {$\#$1}^6+\text {$\#$1}^3+1\&,\frac {\log \left (\sqrt [3]{\frac {1}{x^2}-1}-\text {$\#$1}\right )}{\text {$\#$1}}\&\right ]+2^{2/3} \left (-2 \log \left (2^{2/3} \sqrt [3]{\frac {1}{x^2}-1}+2\right )+\log \left (\sqrt [3]{2} \left (\frac {1}{x^2}-1\right )^{2/3}-2^{2/3} \sqrt [3]{\frac {1}{x^2}-1}+2\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {2^{2/3} \sqrt [3]{\frac {1}{x^2}-1}-1}{\sqrt {3}}\right )\right )\right )}{24 \sqrt [3]{x \left (x^2-1\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/24*((-1 + x^(-2))^(1/3)*x*(2^(2/3)*(2*Sqrt[3]*ArcTan[(-1 + 2^(2/3)*(-1 + x^(-2))^(1/3))/Sqrt[3]] - 2*Log[2
+ 2^(2/3)*(-1 + x^(-2))^(1/3)] + Log[2 - 2^(2/3)*(-1 + x^(-2))^(1/3) + 2^(1/3)*(-1 + x^(-2))^(2/3)]) + 4*RootS
um[1 + #1^3 + #1^6 & , Log[(-1 + x^(-2))^(1/3) - #1]/#1 & ]))/(x*(-1 + x^2))^(1/3)

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IntegrateAlgebraic [A]  time = 0.00, size = 173, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-x+x^3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}-\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{-x+x^3}\right )}{6 \sqrt [3]{2}}+\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{-x+x^3}+\sqrt [3]{2} \left (-x+x^3\right )^{2/3}\right )}{12 \sqrt [3]{2}}-\frac {1}{6} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{-x+x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

[Out]

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

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giac [B]  time = 0.43, size = 967, normalized size = 5.59

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*sqrt(3)*2^(2/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3) + 2*(-1/x^2 + 1)^(1/3))) + 1/6*(sqrt(3)*cos(4/9*pi)^
5 - 10*sqrt(3)*cos(4/9*pi)^3*sin(4/9*pi)^2 + 5*sqrt(3)*cos(4/9*pi)*sin(4/9*pi)^4 - 5*cos(4/9*pi)^4*sin(4/9*pi)
+ 10*cos(4/9*pi)^2*sin(4/9*pi)^3 - sin(4/9*pi)^5 + sqrt(3)*cos(4/9*pi)^2 - sqrt(3)*sin(4/9*pi)^2 - 2*cos(4/9*
pi)*sin(4/9*pi))*arctan(1/2*((-I*sqrt(3) - 1)*cos(4/9*pi) + 2*(-1/x^2 + 1)^(1/3))/((1/2*I*sqrt(3) + 1/2)*sin(4
/9*pi))) + 1/6*(sqrt(3)*cos(2/9*pi)^5 - 10*sqrt(3)*cos(2/9*pi)^3*sin(2/9*pi)^2 + 5*sqrt(3)*cos(2/9*pi)*sin(2/9
*pi)^4 - 5*cos(2/9*pi)^4*sin(2/9*pi) + 10*cos(2/9*pi)^2*sin(2/9*pi)^3 - sin(2/9*pi)^5 + sqrt(3)*cos(2/9*pi)^2
- sqrt(3)*sin(2/9*pi)^2 - 2*cos(2/9*pi)*sin(2/9*pi))*arctan(1/2*((-I*sqrt(3) - 1)*cos(2/9*pi) + 2*(-1/x^2 + 1)
^(1/3))/((1/2*I*sqrt(3) + 1/2)*sin(2/9*pi))) - 1/6*(sqrt(3)*cos(1/9*pi)^5 - 10*sqrt(3)*cos(1/9*pi)^3*sin(1/9*p
i)^2 + 5*sqrt(3)*cos(1/9*pi)*sin(1/9*pi)^4 + 5*cos(1/9*pi)^4*sin(1/9*pi) - 10*cos(1/9*pi)^2*sin(1/9*pi)^3 + si
n(1/9*pi)^5 - sqrt(3)*cos(1/9*pi)^2 + sqrt(3)*sin(1/9*pi)^2 - 2*cos(1/9*pi)*sin(1/9*pi))*arctan(-1/2*((-I*sqrt
(3) - 1)*cos(1/9*pi) - 2*(-1/x^2 + 1)^(1/3))/((1/2*I*sqrt(3) + 1/2)*sin(1/9*pi))) + 1/12*(5*sqrt(3)*cos(4/9*pi
)^4*sin(4/9*pi) - 10*sqrt(3)*cos(4/9*pi)^2*sin(4/9*pi)^3 + sqrt(3)*sin(4/9*pi)^5 + cos(4/9*pi)^5 - 10*cos(4/9*
pi)^3*sin(4/9*pi)^2 + 5*cos(4/9*pi)*sin(4/9*pi)^4 + 2*sqrt(3)*cos(4/9*pi)*sin(4/9*pi) + cos(4/9*pi)^2 - sin(4/
9*pi)^2)*log((-I*sqrt(3)*cos(4/9*pi) - cos(4/9*pi))*(-1/x^2 + 1)^(1/3) + (-1/x^2 + 1)^(2/3) + 1) + 1/12*(5*sqr
t(3)*cos(2/9*pi)^4*sin(2/9*pi) - 10*sqrt(3)*cos(2/9*pi)^2*sin(2/9*pi)^3 + sqrt(3)*sin(2/9*pi)^5 + cos(2/9*pi)^
5 - 10*cos(2/9*pi)^3*sin(2/9*pi)^2 + 5*cos(2/9*pi)*sin(2/9*pi)^4 + 2*sqrt(3)*cos(2/9*pi)*sin(2/9*pi) + cos(2/9
*pi)^2 - sin(2/9*pi)^2)*log((-I*sqrt(3)*cos(2/9*pi) - cos(2/9*pi))*(-1/x^2 + 1)^(1/3) + (-1/x^2 + 1)^(2/3) + 1
) + 1/12*(5*sqrt(3)*cos(1/9*pi)^4*sin(1/9*pi) - 10*sqrt(3)*cos(1/9*pi)^2*sin(1/9*pi)^3 + sqrt(3)*sin(1/9*pi)^5
- cos(1/9*pi)^5 + 10*cos(1/9*pi)^3*sin(1/9*pi)^2 - 5*cos(1/9*pi)*sin(1/9*pi)^4 - 2*sqrt(3)*cos(1/9*pi)*sin(1/
9*pi) + cos(1/9*pi)^2 - sin(1/9*pi)^2)*log((I*sqrt(3)*cos(1/9*pi) + cos(1/9*pi))*(-1/x^2 + 1)^(1/3) + (-1/x^2
+ 1)^(2/3) + 1) + 1/24*2^(2/3)*log(2^(2/3) + 2^(1/3)*(-1/x^2 + 1)^(1/3) + (-1/x^2 + 1)^(2/3)) - 1/12*2^(2/3)*l
og(abs(-2^(1/3) + (-1/x^2 + 1)^(1/3)))

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maple [B]  time = 70.15, size = 3898, normalized size = 22.53

method	result	size
trager	\(\text {Expression too large to display}\)	\(3898\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*RootOf(_Z^3+4)*ln((2190*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)*RootOf(_Z^3+4)^3*x^2+26532
*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)^2*RootOf(_Z^3+4)^2*x^2+10656*(x^3-x)^(2/3)*RootOf(_Z^3
+4)^2*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)+1776*(x^3-x)^(1/3)*RootOf(_Z^3+4)^2*x+83430*(x^3-
x)^(1/3)*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)*RootOf(_Z^3+4)*x-8760*RootOf(_Z^3+4)^3*RootOf(
RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)-106128*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)^
2*RootOf(_Z^3+4)^2+9125*RootOf(_Z^3+4)*x^2+110550*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)*x^2+1
0353*(x^3-x)^(2/3)-1095*RootOf(_Z^3+4)-13266*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2))/(x^2+1))+
RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)*ln(-(4422*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+
144*_Z^2)*RootOf(_Z^3+4)^3*x^2+52560*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)^2*RootOf(_Z^3+4)^2
*x^2+21312*(x^3-x)^(2/3)*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)+3552*(x^3-x)^
(1/3)*RootOf(_Z^3+4)^2*x-124236*(x^3-x)^(1/3)*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)*RootOf(_Z
^3+4)*x-17688*RootOf(_Z^3+4)^3*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)-210240*RootOf(RootOf(_Z^
3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)^2*RootOf(_Z^3+4)^2-19899*RootOf(_Z^3+4)*x^2-236520*RootOf(RootOf(_Z^3+4)
^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)*x^2-27810*(x^3-x)^(2/3)+8107*RootOf(_Z^3+4)+96360*RootOf(RootOf(_Z^3+4)^2+12
*_Z*RootOf(_Z^3+4)+144*_Z^2))/(x^2+1))-1/2*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_
Z^2)*RootOf(_Z^3+3*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)*RootOf(_Z^3+4)^2)*ln(-(54*RootOf(_Z^
3+3*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)*RootOf(_Z^3+4)^2)*RootOf(RootOf(_Z^3+4)^2+12*_Z*Roo
tOf(_Z^3+4)+144*_Z^2)^2*RootOf(_Z^3+4)^4*x^2-156*RootOf(_Z^3+3*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+14
4*_Z^2)*RootOf(_Z^3+4)^2)^2*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)*RootOf(_Z^3+4)^2*(x^3-x)^(1
/3)*x-216*RootOf(_Z^3+3*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)*RootOf(_Z^3+4)^2)*RootOf(RootOf
(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)^2*RootOf(_Z^3+4)^4+21*RootOf(_Z^3+3*RootOf(RootOf(_Z^3+4)^2+12*_Z*Ro
otOf(_Z^3+4)+144*_Z^2)*RootOf(_Z^3+4)^2)*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)*RootOf(_Z^3+4)
^2*x^2+39*(x^3-x)^(2/3)*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)+65*RootOf(_Z^3
+3*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)*RootOf(_Z^3+4)^2)^2*(x^3-x)^(1/3)*x+150*RootOf(_Z^3+
4)^2*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)*RootOf(_Z^3+3*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf
(_Z^3+4)+144*_Z^2)*RootOf(_Z^3+4)^2)-33*RootOf(_Z^3+3*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)*R
ootOf(_Z^3+4)^2)*x^2+52*(x^3-x)^(2/3)-11*RootOf(_Z^3+3*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)*
RootOf(_Z^3+4)^2))/(3*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)*x^2-12*RootOf(Ro
otOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)*RootOf(_Z^3+4)^2-4*x^2+3))+1/2*ln((204*RootOf(_Z^3+3*RootOf(Root
Of(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)*RootOf(_Z^3+4)^2)^2*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+1
44*_Z^2)*RootOf(_Z^3+4)^2*(x^3-x)^(1/3)*x-222*RootOf(_Z^3+3*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_
Z^2)*RootOf(_Z^3+4)^2)*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)*RootOf(_Z^3+4)^2*x^2+129*(x^3-x)
^(2/3)*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)-25*RootOf(_Z^3+3*RootOf(RootOf(
_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)*RootOf(_Z^3+4)^2)^2*(x^3-x)^(1/3)*x+111*RootOf(_Z^3+4)^2*RootOf(RootO
f(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)*RootOf(_Z^3+3*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2
)*RootOf(_Z^3+4)^2)+62*RootOf(_Z^3+3*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)*RootOf(_Z^3+4)^2)*
x^2-68*(x^3-x)^(2/3)-31*RootOf(_Z^3+3*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)*RootOf(_Z^3+4)^2)
)/(3*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)*x^2-12*RootOf(RootOf(_Z^3+4)^2+12
*_Z*RootOf(_Z^3+4)+144*_Z^2)*RootOf(_Z^3+4)^2-4*x^2+3))*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(
_Z^3+4)+144*_Z^2)*RootOf(_Z^3+3*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)*RootOf(_Z^3+4)^2)-1/6*l
n((204*RootOf(_Z^3+3*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)*RootOf(_Z^3+4)^2)^2*RootOf(RootOf(
_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)*RootOf(_Z^3+4)^2*(x^3-x)^(1/3)*x-222*RootOf(_Z^3+3*RootOf(RootOf(_Z^3
+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)*RootOf(_Z^3+4)^2)*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)*
RootOf(_Z^3+4)^2*x^2+129*(x^3-x)^(2/3)*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)
-25*RootOf(_Z^3+3*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)*RootOf(_Z^3+4)^2)^2*(x^3-x)^(1/3)*x+1
11*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)*RootOf(_Z^3+3*RootOf(RootOf(_Z^3+4)
^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)*RootOf(_Z^3+4)^2)+62*RootOf(_Z^3+3*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3
+4)+144*_Z^2)*RootOf(_Z^3+4)^2)*x^2-68*(x^3-x)^(2/3)-31*RootOf(_Z^3+3*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^
3+4)+144*_Z^2)*RootOf(_Z^3+4)^2))/(3*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)*x
^2-12*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)*RootOf(_Z^3+4)^2-4*x^2+3))*RootOf(_Z^3+3*RootOf(R
ootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)*RootOf(_Z^3+4)^2)+1/6*RootOf(_Z^3+3*RootOf(RootOf(_Z^3+4)^2+12*
_Z*RootOf(_Z^3+4)+144*_Z^2)*RootOf(_Z^3+4)^2)^2*ln(-(36*RootOf(_Z^3+3*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^
3+4)+144*_Z^2)*RootOf(_Z^3+4)^2)^2*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)*RootOf(_Z^3+4)^2*x^2
-129*RootOf(_Z^3+3*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)*RootOf(_Z^3+4)^2)*RootOf(RootOf(_Z^3
+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)*RootOf(_Z^3+4)^2*(x^3-x)^(1/3)*x-18*RootOf(_Z^3+3*RootOf(RootOf(_Z^3+4)^2
+12*_Z*RootOf(_Z^3+4)+144*_Z^2)*RootOf(_Z^3+4)^2)^2*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)*Roo
tOf(_Z^3+4)^2+129*(x^3-x)^(2/3)*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)+62*Roo
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Of(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)*x^2-12*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^
2)*RootOf(_Z^3+4)^2+3*x^2+1))-1/2*RootOf(_Z^3+3*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)*RootOf(
_Z^3+4)^2)^2*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)*RootOf(_Z^3+4)^2*ln((-45*RootOf(_Z^3+3*Roo
tOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)*RootOf(_Z^3+4)^2)^2*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_
Z^3+4)+144*_Z^2)^2*RootOf(_Z^3+4)^4*x^2+180*RootOf(_Z^3+3*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^
2)*RootOf(_Z^3+4)^2)^2*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)^2*RootOf(_Z^3+4)^4+93*RootOf(_Z^
3+3*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)*RootOf(_Z^3+4)^2)^2*RootOf(RootOf(_Z^3+4)^2+12*_Z*R
ootOf(_Z^3+4)+144*_Z^2)*RootOf(_Z^3+4)^2*x^2-39*RootOf(_Z^3+3*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144
*_Z^2)*RootOf(_Z^3+4)^2)*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)*RootOf(_Z^3+4)^2*(x^3-x)^(1/3)
*x+39*(x^3-x)^(2/3)*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)-177*RootOf(_Z^3+3*
RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)*RootOf(_Z^3+4)^2)^2*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootO
f(_Z^3+4)+144*_Z^2)*RootOf(_Z^3+4)^2-44*RootOf(_Z^3+3*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)*R
ootOf(_Z^3+4)^2)^2*x^2+65*RootOf(_Z^3+3*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)*RootOf(_Z^3+4)^
2)*(x^3-x)^(1/3)*x-65*(x^3-x)^(2/3)+33*RootOf(_Z^3+3*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)*Ro
otOf(_Z^3+4)^2)^2)/(3*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)*x^2-12*RootOf(Ro
otOf(_Z^3+4)^2+12*_Z*RootOf(_Z^3+4)+144*_Z^2)*RootOf(_Z^3+4)^2+3*x^2+1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((x^3 - x)^(1/3)*(x^6 + 1)),x)

[Out]

int(1/((x^3 - x)^(1/3)*(x^6 + 1)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [3]{x \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + 1\right ) \left (x^{4} - x^{2} + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/((x*(x - 1)*(x + 1))**(1/3)*(x**2 + 1)*(x**4 - x**2 + 1)), x)

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