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

Optimal. Leaf size=237 \[ \frac {1}{2} \text {RootSum}\left [\text {$\#$1}^6-\text {$\#$1}^3+1\& ,\frac {\log \left (\sqrt [3]{x^3+x}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ]-\log \left (\sqrt [3]{x^3+x}-x\right )+\frac {\log \left (2^{2/3} \sqrt [3]{x^3+x}-2 x\right )}{2 \sqrt [3]{2}}+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3+x}+x}\right )-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^3+x}+x}\right )}{2 \sqrt [3]{2}}+\frac {1}{2} \log \left (\sqrt [3]{x^3+x} x+\left (x^3+x\right )^{2/3}+x^2\right )-\frac {\log \left (2^{2/3} \sqrt [3]{x^3+x} x+\sqrt [3]{2} \left (x^3+x\right )^{2/3}+2 x^2\right )}{4 \sqrt [3]{2}} \]

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Rubi [C]  time = 2.05, antiderivative size = 1380, normalized size of antiderivative = 5.82, number of steps used = 28, number of rules used = 14, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {2056, 6715, 6725, 239, 2074, 2148, 6728, 377, 200, 31, 634, 617, 204, 628} \begin {gather*} -\frac {\sqrt {3} \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]{x^3+x}}-\frac {\sqrt {3} \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]{x^3+x}}-\frac {\sqrt {3} \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]{x^3+x}}+\frac {\sqrt {3} \sqrt [3]{x} \sqrt [3]{x^2+1} \tan ^{-1}\left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2+1}}+1}{\sqrt {3}}\right )}{\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 )}{2\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}} \sqrt [3]{x^3+x}}+\frac {\sqrt {3} \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]{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 )}{8 \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 )}{8 \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 )}{8 \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 )}{4 \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 )}{4 \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 )}{2 \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 )}{2 \sqrt [3]{x^3+x}}-\frac {3 \sqrt [3]{x} \sqrt [3]{x^2+1} \log \left (x^{2/3}-\sqrt [3]{x^2+1}\right )}{2 \sqrt [3]{x^3+x}}+\frac {3 \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 {3 \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 {3 \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 + 2*x^6)/((x + x^3)^(1/3)*(-1 + x^6)),x]

[Out]

-1/4*(Sqrt[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*Sq
rt[3])])/(2^(1/3)*(x + x^3)^(1/3)) - (Sqrt[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)*(x + x^3)^(1/3)) - (Sqrt[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)*(x + x^3)^(1/3)) + (Sqrt[3]*x^(1/3)*(1 + x
^2)^(1/3)*ArcTan[(1 + (2*x^(2/3))/(1 + x^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]])/(2*2^(2/3)*
(1 - I*Sqrt[3])^(1/3)*(x + x^3)^(1/3)) + (Sqrt[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)*(x + x^3)^(1/3)) - (x^(1/3
)*(1 + x^2)^(1/3)*Log[(1 - x^(2/3))^2*(1 + x^(2/3))])/(8*2^(1/3)*(x + x^3)^(1/3)) - (x^(1/3)*(1 + x^2)^(1/3)*L
og[(1 - I*Sqrt[3] - 2*x^(2/3))*(1 - I*Sqrt[3] + 2*x^(2/3))^2])/(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))*(1 + I*Sqrt[3] + 2*x^(2/3))^2])/(8*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)])/(4*((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)])/(4*(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)])/(2*((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)])/(2*(x + x^3)^(1/3)) - (3*x^(1/3)*(1 + x^2)^(1/3)*Log[x^(2/3) - (1 + x^2)^(1/3)])/(2*
(x + x^3)^(1/3)) + (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)) + (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)) + (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 239

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 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 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

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+2 x^6}{\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+2 x^6}{\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+2 x^9}{\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 {2}{\sqrt [3]{1+x^3}}+\frac {3}{\sqrt [3]{1+x^3} \left (-1+x^9\right )}\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 \frac {1}{\sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x+x^3}}+\frac {\left (9 \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 {\sqrt {3} \sqrt [3]{x} \sqrt [3]{1+x^2} \tan ^{-1}\left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{\sqrt [3]{x+x^3}}-\frac {3 \sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (x^{2/3}-\sqrt [3]{1+x^2}\right )}{2 \sqrt [3]{x+x^3}}+\frac {\left (9 \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 {\sqrt {3} \sqrt [3]{x} \sqrt [3]{1+x^2} \tan ^{-1}\left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{\sqrt [3]{x+x^3}}-\frac {3 \sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (x^{2/3}-\sqrt [3]{1+x^2}\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 )}{2 \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 )}{2 \sqrt [3]{x+x^3}}+\frac {\left (3 \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} \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]{x+x^3}}+\frac {\sqrt {3} \sqrt [3]{x} \sqrt [3]{1+x^2} \tan ^{-1}\left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{\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 )}{8 \sqrt [3]{2} \sqrt [3]{x+x^3}}-\frac {3 \sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (x^{2/3}-\sqrt [3]{1+x^2}\right )}{2 \sqrt [3]{x+x^3}}+\frac {3 \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 )}{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+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} \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]{x+x^3}}+\frac {\sqrt {3} \sqrt [3]{x} \sqrt [3]{1+x^2} \tan ^{-1}\left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{\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 )}{8 \sqrt [3]{2} \sqrt [3]{x+x^3}}-\frac {3 \sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (x^{2/3}-\sqrt [3]{1+x^2}\right )}{2 \sqrt [3]{x+x^3}}+\frac {3 \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 )}{2 \sqrt [3]{x+x^3}}+\frac {\left (3 \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 )}{2 \sqrt [3]{x+x^3}}+\frac {\left (3 \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} \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]{x+x^3}}-\frac {\sqrt {3} \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]{x+x^3}}-\frac {\sqrt {3} \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]{x+x^3}}+\frac {\sqrt {3} \sqrt [3]{x} \sqrt [3]{1+x^2} \tan ^{-1}\left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{\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 )}{8 \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 )}{8 \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 )}{8 \sqrt [3]{2} \sqrt [3]{x+x^3}}-\frac {3 \sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (x^{2/3}-\sqrt [3]{1+x^2}\right )}{2 \sqrt [3]{x+x^3}}+\frac {3 \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 {3 \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 {3 \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 (3 \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 (3 \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} \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]{x+x^3}}-\frac {\sqrt {3} \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]{x+x^3}}-\frac {\sqrt {3} \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]{x+x^3}}+\frac {\sqrt {3} \sqrt [3]{x} \sqrt [3]{1+x^2} \tan ^{-1}\left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{\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 )}{8 \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 )}{8 \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 )}{8 \sqrt [3]{2} \sqrt [3]{x+x^3}}-\frac {3 \sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (x^{2/3}-\sqrt [3]{1+x^2}\right )}{2 \sqrt [3]{x+x^3}}+\frac {3 \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 {3 \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 {3 \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 )}{2 \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 )}{2 \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 )}{2 \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 )}{2 \left (1+i \sqrt {3}\right )^{2/3} \sqrt [3]{x+x^3}}\\ &=-\frac {\sqrt {3} \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]{x+x^3}}-\frac {\sqrt {3} \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]{x+x^3}}-\frac {\sqrt {3} \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]{x+x^3}}+\frac {\sqrt {3} \sqrt [3]{x} \sqrt [3]{1+x^2} \tan ^{-1}\left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{\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 )}{8 \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 )}{8 \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 )}{8 \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 )}{2 \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 )}{2 \sqrt [3]{1+i \sqrt {3}} \sqrt [3]{x+x^3}}-\frac {3 \sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (x^{2/3}-\sqrt [3]{1+x^2}\right )}{2 \sqrt [3]{x+x^3}}+\frac {3 \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 {3 \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 {3 \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 (3 \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 )}{4 \sqrt [3]{1-i \sqrt {3}} \left (1+i \sqrt {3}\right )^{2/3} \sqrt [3]{x+x^3}}+\frac {\left (3 \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 )}{4 \left (1-i \sqrt {3}\right )^{2/3} \sqrt [3]{1+i \sqrt {3}} \sqrt [3]{x+x^3}}\\ &=-\frac {\sqrt {3} \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]{x+x^3}}-\frac {\sqrt {3} \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]{x+x^3}}-\frac {\sqrt {3} \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]{x+x^3}}+\frac {\sqrt {3} \sqrt [3]{x} \sqrt [3]{1+x^2} \tan ^{-1}\left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{\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 )}{8 \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 )}{8 \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 )}{8 \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 )}{4 \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 )}{4 \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 )}{2 \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 )}{2 \sqrt [3]{1+i \sqrt {3}} \sqrt [3]{x+x^3}}-\frac {3 \sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (x^{2/3}-\sqrt [3]{1+x^2}\right )}{2 \sqrt [3]{x+x^3}}+\frac {3 \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 {3 \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 {3 \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 (3 \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 (3 \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} \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]{x+x^3}}-\frac {\sqrt {3} \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]{x+x^3}}-\frac {\sqrt {3} \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]{x+x^3}}+\frac {\sqrt {3} \sqrt [3]{x} \sqrt [3]{1+x^2} \tan ^{-1}\left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{\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 )}{2\ 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 )}{2\ 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 )}{8 \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 )}{8 \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 )}{8 \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 )}{4 \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 )}{4 \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 )}{2 \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 )}{2 \sqrt [3]{1+i \sqrt {3}} \sqrt [3]{x+x^3}}-\frac {3 \sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (x^{2/3}-\sqrt [3]{1+x^2}\right )}{2 \sqrt [3]{x+x^3}}+\frac {3 \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 {3 \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 {3 \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 [F]  time = 0.26, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+2 x^6}{\sqrt [3]{x+x^3} \left (-1+x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.00, size = 237, normalized size = 1.00 \begin {gather*} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x+x^3}}\right )-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{x+x^3}}\right )}{2 \sqrt [3]{2}}-\log \left (-x+\sqrt [3]{x+x^3}\right )+\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{x+x^3}\right )}{2 \sqrt [3]{2}}+\frac {1}{2} \log \left (x^2+x \sqrt [3]{x+x^3}+\left (x+x^3\right )^{2/3}\right )-\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 )}{4 \sqrt [3]{2}}+\frac {1}{2} \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 + 2*x^6)/((x + x^3)^(1/3)*(-1 + x^6)),x]

[Out]

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

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (re
sidue poly has multiple non-linear factors)

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giac [B]  time = 0.55, size = 995, normalized size = 4.20

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sqrt(3)*2^(2/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3) + 2*(1/x^2 + 1)^(1/3))) - 1/2*(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*p
i))) - 1/2*(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 - sq
rt(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))) + 1/2*(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))) - 1/4*(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*s
in(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) - 1/4*(5*sqrt(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 - s
in(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) - 1/4*(5*s
qrt(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)
- 1/8*2^(2/3)*log(2^(2/3) + 2^(1/3)*(1/x^2 + 1)^(1/3) + (1/x^2 + 1)^(2/3)) + 1/4*2^(2/3)*log(abs(-2^(1/3) + (
1/x^2 + 1)^(1/3))) - sqrt(3)*arctan(1/3*sqrt(3)*(2*(1/x^2 + 1)^(1/3) + 1)) + 1/2*log((1/x^2 + 1)^(2/3) + (1/x^
2 + 1)^(1/3) + 1) - log(abs((1/x^2 + 1)^(1/3) - 1))

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maple [F]  time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {2 x^{6}+1}{\left (x^{3}+x \right )^{\frac {1}{3}} \left (x^{6}-1\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((2*x^6+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*} \int \frac {2 \, x^{6} + 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((2*x^6+1)/(x^3+x)^(1/3)/(x^6-1),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {2\,x^6+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((2*x^6 + 1)/((x^6 - 1)*(x + x^3)^(1/3)),x)

[Out]

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

[Out]

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

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