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

Optimal. Leaf size=163 \[ \frac {1}{6} \text {RootSum}\left [\text {$\#$1}^6-\text {$\#$1}^3+1\& ,\frac {\log \left (\sqrt [3]{x^3+x}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ]+\frac {\log \left (2^{2/3} \sqrt [3]{x^3+x}-2 x\right )}{6 \sqrt [3]{2}}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^3+x}+x}\right )}{2 \sqrt [3]{2} \sqrt {3}}-\frac {\log \left (2^{2/3} \sqrt [3]{x^3+x} x+\sqrt [3]{2} \left (x^3+x\right )^{2/3}+2 x^2\right )}{12 \sqrt [3]{2}} \]

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Rubi [C]  time = 1.62, antiderivative size = 1281, normalized size of antiderivative = 7.86, number of steps used = 25, number of rules used = 12, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.706, 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 {2-\frac {\sqrt [3]{2} \left (-2 x^{2/3}-i \sqrt {3}+1\right )}{\sqrt [3]{x^2+1}}}{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 {2-\frac {\sqrt [3]{2} \left (-2 x^{2/3}+i \sqrt {3}+1\right )}{\sqrt [3]{x^2+1}}}{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 (x^{2/3}+1\right )}{\sqrt [3]{x^2+1}}+1}{\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 (1-x^{2/3}\right )^2 \left (x^{2/3}+1\right )\right )}{24 \sqrt [3]{2} \sqrt [3]{x^3+x}}-\frac {\sqrt [3]{x} \sqrt [3]{x^2+1} \log \left (\left (-2 x^{2/3}-i \sqrt {3}+1\right ) \left (2 x^{2/3}-i \sqrt {3}+1\right )^2\right )}{24 \sqrt [3]{2} \sqrt [3]{x^3+x}}-\frac {\sqrt [3]{x} \sqrt [3]{x^2+1} \log \left (\left (-2 x^{2/3}+i \sqrt {3}+1\right ) \left (2 x^{2/3}+i \sqrt {3}+1\right )^2\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 (x^{2/3}-2^{2/3} \sqrt [3]{x^2+1}+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 (-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}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[1/((x + x^3)^(1/3)*(-1 + x^6)),x]

[Out]

-1/4*(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])])
/(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[(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)) - ((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)*ArcTan[(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[(1 - x^(2/3))^2*(1 + x^(2/3))])/(24*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))*(1 - I*Sqrt[3] + 2*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*x^(2/3))*(1 + I*Sqrt[3] + 2*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*Sqr
t[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 + x^(2/3) - 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 + 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))

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 (1-x^{2/3}\right )^2 \left (1+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 (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 (1-x^{2/3}\right )^2 \left (1+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 (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 {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} \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 (1-x^{2/3}\right )^2 \left (1+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 (1-i \sqrt {3}-2 x^{2/3}\right ) \left (1-i \sqrt {3}+2 x^{2/3}\right )^2\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}-2 x^{2/3}\right ) \left (1+i \sqrt {3}+2 x^{2/3}\right )^2\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 {\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 {\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 {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} \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 (1-x^{2/3}\right )^2 \left (1+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 (1-i \sqrt {3}-2 x^{2/3}\right ) \left (1-i \sqrt {3}+2 x^{2/3}\right )^2\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}-2 x^{2/3}\right ) \left (1+i \sqrt {3}+2 x^{2/3}\right )^2\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 {\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 {\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 {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} \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 (1-x^{2/3}\right )^2 \left (1+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 (1-i \sqrt {3}-2 x^{2/3}\right ) \left (1-i \sqrt {3}+2 x^{2/3}\right )^2\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}-2 x^{2/3}\right ) \left (1+i \sqrt {3}+2 x^{2/3}\right )^2\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+x^{2/3}-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+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 {\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 {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} \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 (1-x^{2/3}\right )^2 \left (1+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 (1-i \sqrt {3}-2 x^{2/3}\right ) \left (1-i \sqrt {3}+2 x^{2/3}\right )^2\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}-2 x^{2/3}\right ) \left (1+i \sqrt {3}+2 x^{2/3}\right )^2\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+x^{2/3}-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+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 {\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 {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} \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 {\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 (1-x^{2/3}\right )^2 \left (1+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 (1-i \sqrt {3}-2 x^{2/3}\right ) \left (1-i \sqrt {3}+2 x^{2/3}\right )^2\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}-2 x^{2/3}\right ) \left (1+i \sqrt {3}+2 x^{2/3}\right )^2\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+x^{2/3}-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+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}}\\ \end {align*}

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Mathematica [A]  time = 0.25, size = 155, normalized size = 0.95 \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^{2/3} \sqrt [3]{\frac {1}{x^2}+1}\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^3+x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((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*RootSum[1 - #1^3
+ #1^6 & , Log[(1 + x^(-2))^(1/3) - #1]/#1 & ]))/(24*(x + x^3)^(1/3))

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IntegrateAlgebraic [A]  time = 0.00, size = 163, 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]

-1/2*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(x + x^3)^(1/3))]/(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 = 0.32, size = 941, normalized size = 5.77

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 - s
qrt(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*pi)^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 + sin(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*s
in(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*sqrt(3)*co
s(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) + c
os(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)*log(abs(-2^(1/
3) + (1/x^2 + 1)^(1/3)))

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maple [F]  time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (x^{3}+x \right )^{\frac {1}{3}} \left (x^{6}-1\right )}\, dx\]

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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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\frac {3 \, {\left (9 \, x^{7} + 3 \, x^{5} - x^{3} + 5 \, x\right )}}{80 \, {\left (x^{\frac {19}{3}} - x^{\frac {1}{3}}\right )} {\left (x^{2} + 1\right )}^{\frac {1}{3}}} - \int \frac {9 \, {\left (9 \, x^{6} + 3 \, x^{4} - x^{2} + 5\right )}}{40 \, {\left (x^{\frac {37}{3}} - 2 \, x^{\frac {19}{3}} + x^{\frac {1}{3}}\right )} {\left (x^{2} + 1\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]

-3/80*(9*x^7 + 3*x^5 - x^3 + 5*x)/((x^(19/3) - x^(1/3))*(x^2 + 1)^(1/3)) - integrate(9/40*(9*x^6 + 3*x^4 - x^2
+ 5)/((x^(37/3) - 2*x^(19/3) + x^(1/3))*(x^2 + 1)^(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^6-1\right )\,{\left (x^3+x\right )}^{1/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((x^6 - 1)*(x + x^3)^(1/3)),x)

[Out]

int(1/((x^6 - 1)*(x + x^3)^(1/3)), 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^{2} + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 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**2 + 1))**(1/3)*(x - 1)*(x + 1)*(x**2 - x + 1)*(x**2 + x + 1)), x)

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