\(\int \frac {1}{\sqrt [3]{x+x^3} (-1+x^6)} \, dx\) [2201]

Optimal result

Integrand size = 17, antiderivative size = 163 \[ \int \frac {1}{\sqrt [3]{x+x^3} \left (-1+x^6\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{x+x^3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}+\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{x+x^3}\right )}{6 \sqrt [3]{2}}-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{x+x^3}+\sqrt [3]{2} \left (x+x^3\right )^{2/3}\right )}{12 \sqrt [3]{2}}+\frac {1}{6} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{x+x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]

[Out]

Unintegrable

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.87 (sec) , antiderivative size = 1232, normalized size of antiderivative = 7.56, number of steps used = 25, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.941, Rules used = {2081, 6847, 2099, 2174, 2183, 384, 502, 206, 31, 648, 631, 210, 642, 455, 57, 6860} \[ \int \frac {1}{\sqrt [3]{x+x^3} \left (-1+x^6\right )} \, dx=\frac {\sqrt [3]{x} \sqrt [3]{x^2+1} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \left (x^{2/3}+1\right )}{\sqrt [3]{x^2+1}}}{\sqrt {3}}\right )}{6 \sqrt [3]{2} \sqrt {3} \sqrt [3]{x^3+x}}-\frac {\sqrt [3]{x} \sqrt [3]{x^2+1} \arctan \left (\frac {\frac {\sqrt [3]{2} \left (x^{2/3}+1\right )}{\sqrt [3]{x^2+1}}+1}{\sqrt {3}}\right )}{6 \sqrt [3]{2} \sqrt {3} \sqrt [3]{x^3+x}}-\frac {\sqrt [3]{x} \sqrt [3]{x^2+1} \arctan \left (\frac {\frac {2 \sqrt [3]{2} x^{2/3}}{\sqrt [3]{x^2+1}}+1}{\sqrt {3}}\right )}{3 \sqrt [3]{2} \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} \arctan \left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{x^2+1}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} \sqrt [3]{x^3+x}}-\frac {\sqrt [3]{x} \sqrt [3]{x^2+1} \arctan \left (\frac {\frac {2 \sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} x^{2/3}}{\sqrt [3]{x^2+1}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} \sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{x^3+x}}-\frac {\sqrt [3]{x} \sqrt [3]{x^2+1} \arctan \left (\frac {2^{2/3} \sqrt [3]{x^2+1}+1}{\sqrt {3}}\right )}{6 \sqrt [3]{2} \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 )}{36 \sqrt [3]{2} \sqrt [3]{x^3+x}}-\frac {\sqrt [3]{x} \sqrt [3]{x^2+1} \log \left (1-x^2\right )}{36 \sqrt [3]{2} \sqrt [3]{x^3+x}}-\frac {\sqrt [3]{x} \sqrt [3]{x^2+1} \log \left (2 x^2-i \sqrt {3}+1\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 (2 x^2+i \sqrt {3}+1\right )}{12 \sqrt [3]{x^3+x}}+\frac {\sqrt [3]{x} \sqrt [3]{x^2+1} \log \left (\frac {2^{2/3} \left (x^{2/3}+1\right )^2}{\left (x^2+1\right )^{2/3}}-\frac {\sqrt [3]{2} \left (x^{2/3}+1\right )}{\sqrt [3]{x^2+1}}+1\right )}{36 \sqrt [3]{2} \sqrt [3]{x^3+x}}-\frac {\sqrt [3]{x} \sqrt [3]{x^2+1} \log \left (\frac {\sqrt [3]{2} \left (x^{2/3}+1\right )}{\sqrt [3]{x^2+1}}+1\right )}{18 \sqrt [3]{2} \sqrt [3]{x^3+x}}-\frac {\sqrt [3]{x} \sqrt [3]{x^2+1} \log \left (\sqrt [3]{2}-\sqrt [3]{x^2+1}\right )}{12 \sqrt [3]{2} \sqrt [3]{x^3+x}}+\frac {\sqrt [3]{x} \sqrt [3]{x^2+1} \log \left (\sqrt [3]{2} x^{2/3}-\sqrt [3]{x^2+1}\right )}{6 \sqrt [3]{2} \sqrt [3]{x^3+x}}+\frac {\sqrt [3]{x} \sqrt [3]{x^2+1} \log \left (\sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} x^{2/3}-\sqrt [3]{x^2+1}\right )}{4 \sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \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 )}{12 \sqrt [3]{2} \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 (x^{2/3}-\sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{x^2+1}\right )}{4 \sqrt [3]{x^3+x}} \]

[In]

Int[1/((x + x^3)^(1/3)*(-1 + x^6)),x]

[Out]

(x^(1/3)*(1 + x^2)^(1/3)*ArcTan[(1 - (2*2^(1/3)*(1 + x^(2/3)))/(1 + x^2)^(1/3))/Sqrt[3]])/(6*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]])/(6*2
^(1/3)*Sqrt[3]*(x + x^3)^(1/3)) - (x^(1/3)*(1 + x^2)^(1/3)*ArcTan[(1 + (2*2^(1/3)*x^(2/3))/(1 + x^2)^(1/3))/Sq
rt[3]])/(3*2^(1/3)*Sqrt[3]*(x + x^3)^(1/3)) - ((-((I - Sqrt[3])/(I + Sqrt[3])))^(1/3)*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*Sqrt[3]*(x + x^
3)^(1/3)) - (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*Sqrt[3]*(-((I - Sqrt[3])/(I + Sqrt[3])))^(1/3)*(x + x^3)^(1/3)) - (x^(1/3)*(1 + x^2)^(1/3)
*ArcTan[(1 + 2^(2/3)*(1 + x^2)^(1/3))/Sqrt[3]])/(6*2^(1/3)*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))])/(36*2^(1/3)*(x + x^3)^(1/3)) - (x^(1/3)*(1 + x^2)^(1/3)*Log[1 - x^2])/(36
*2^(1/3)*(x + x^3)^(1/3)) - (x^(1/3)*(1 + x^2)^(1/3)*Log[1 - I*Sqrt[3] + 2*x^2])/(12*(-((I - Sqrt[3])/(I + Sqr
t[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*Sqr
t[3] + 2*x^2])/(12*(x + x^3)^(1/3)) + (x^(1/3)*(1 + x^2)^(1/3)*Log[1 + (2^(2/3)*(1 + x^(2/3))^2)/(1 + x^2)^(2/
3) - (2^(1/3)*(1 + x^(2/3)))/(1 + x^2)^(1/3)])/(36*2^(1/3)*(x + x^3)^(1/3)) - (x^(1/3)*(1 + x^2)^(1/3)*Log[1 +
(2^(1/3)*(1 + x^(2/3)))/(1 + x^2)^(1/3)])/(18*2^(1/3)*(x + x^3)^(1/3)) - (x^(1/3)*(1 + x^2)^(1/3)*Log[2^(1/3)
- (1 + x^2)^(1/3)])/(12*2^(1/3)*(x + x^3)^(1/3)) + (x^(1/3)*(1 + x^2)^(1/3)*Log[2^(1/3)*x^(2/3) - (1 + x^2)^(
1/3)])/(6*2^(1/3)*(x + x^3)^(1/3)) + (x^(1/3)*(1 + x^2)^(1/3)*Log[(-((I - Sqrt[3])/(I + Sqrt[3])))^(1/3)*x^(2/
3) - (1 + x^2)^(1/3)])/(4*(-((I - Sqrt[3])/(I + Sqrt[3])))^(1/3)*(x + x^3)^(1/3)) + (x^(1/3)*(1 + x^2)^(1/3)*L
og[1 + x^(2/3) - 2^(2/3)*(1 + x^2)^(1/3)])/(12*2^(1/3)*(x + x^3)^(1/3)) + ((-((I - Sqrt[3])/(I + Sqrt[3])))^(1
/3)*x^(1/3)*(1 + x^2)^(1/3)*Log[x^(2/3) - (-((I - Sqrt[3])/(I + Sqrt[3])))^(1/3)*(1 + x^2)^(1/3)])/(4*(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 57

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

Rule 206

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

Rule 210

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

Rule 384

Int[1/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/c, 3]}, Simp[
ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sqrt[3]*c*q), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*c*q
), x] + Simp[Log[c + d*x^3]/(6*c*q), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 455

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

Rule 502

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

Rule 631

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

Rule 642

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

Rule 648

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

Rule 2081

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 2099

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 2174

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

Rule 2183

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

Rule 6847

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 6860

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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} \arctan \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 ) \text {Subst}\left (\int \left (-\frac {2}{\left (1-x^3\right ) \sqrt [3]{1+x^3}}+\frac {x}{\left (1-x^3\right ) \sqrt [3]{1+x^3}}+\frac {x^2}{\left (1-x^3\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 ) \text {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} \arctan \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 ) \text {Subst}\left (\int \frac {x}{\left (1-x^3\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 ) \text {Subst}\left (\int \frac {x^2}{\left (1-x^3\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 ) \text {Subst}\left (\int \frac {1}{\left (1-x^3\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{3 \sqrt [3]{x+x^3}}+\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {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 ) \text {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} \arctan \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} \arctan \left (\frac {1+\frac {2 \sqrt [3]{2} x^{2/3}}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{3 \sqrt [3]{2} \sqrt {3} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{x} \sqrt [3]{1+x^2} \arctan \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 \sqrt {3} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \arctan \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 \sqrt {3} \sqrt [3]{-\frac {i-\sqrt {3}}{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 (1-x^2\right )}{18 \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\right )}{12 \sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (1+i \sqrt {3}+2 x^2\right )}{12 \sqrt [3]{x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (\sqrt [3]{2} x^{2/3}-\sqrt [3]{1+x^2}\right )}{6 \sqrt [3]{2} \sqrt [3]{x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (\sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {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]{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]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (x^{2/3}-\sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{1+x^2}\right )}{4 \sqrt [3]{x+x^3}}+\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{(1-x) \sqrt [3]{1+x}} \, dx,x,x^2\right )}{18 \sqrt [3]{x+x^3}}+\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{(1-x) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{18 \sqrt [3]{x+x^3}}-\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{1+2 x^3} \, dx,x,\frac {1+x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{6 \sqrt [3]{x+x^3}} \\ & = -\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \arctan \left (\frac {1+\frac {\sqrt [3]{2} \left (1+x^{2/3}\right )}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{6 \sqrt [3]{2} \sqrt {3} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \arctan \left (\frac {1+\frac {2 \sqrt [3]{2} x^{2/3}}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{3 \sqrt [3]{2} \sqrt {3} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{x} \sqrt [3]{1+x^2} \arctan \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 \sqrt {3} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \arctan \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 \sqrt {3} \sqrt [3]{-\frac {i-\sqrt {3}}{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 )}{36 \sqrt [3]{2} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (1-x^2\right )}{36 \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\right )}{12 \sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (1+i \sqrt {3}+2 x^2\right )}{12 \sqrt [3]{x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (\sqrt [3]{2} x^{2/3}-\sqrt [3]{1+x^2}\right )}{6 \sqrt [3]{2} \sqrt [3]{x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (\sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {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]{x} \sqrt [3]{1+x^2} \log \left (1+x^{2/3}-2^{2/3} \sqrt [3]{1+x^2}\right )}{12 \sqrt [3]{2} \sqrt [3]{x+x^3}}+\frac {\sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (x^{2/3}-\sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{1+x^2}\right )}{4 \sqrt [3]{x+x^3}}-\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt [3]{2} x} \, dx,x,\frac {1+x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{18 \sqrt [3]{x+x^3}}-\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {2-\sqrt [3]{2} x}{1-\sqrt [3]{2} x+2^{2/3} x^2} \, dx,x,\frac {1+x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{18 \sqrt [3]{x+x^3}}-\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{2^{2/3}+\sqrt [3]{2} x+x^2} \, dx,x,\sqrt [3]{1+x^2}\right )}{12 \sqrt [3]{x+x^3}}+\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{2}-x} \, dx,x,\sqrt [3]{1+x^2}\right )}{12 \sqrt [3]{2} \sqrt [3]{x+x^3}} \\ & = -\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \arctan \left (\frac {1+\frac {\sqrt [3]{2} \left (1+x^{2/3}\right )}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{6 \sqrt [3]{2} \sqrt {3} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \arctan \left (\frac {1+\frac {2 \sqrt [3]{2} x^{2/3}}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{3 \sqrt [3]{2} \sqrt {3} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{x} \sqrt [3]{1+x^2} \arctan \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 \sqrt {3} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \arctan \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 \sqrt {3} \sqrt [3]{-\frac {i-\sqrt {3}}{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 )}{36 \sqrt [3]{2} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (1-x^2\right )}{36 \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\right )}{12 \sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (1+i \sqrt {3}+2 x^2\right )}{12 \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (1+\frac {\sqrt [3]{2} \left (1+x^{2/3}\right )}{\sqrt [3]{1+x^2}}\right )}{18 \sqrt [3]{2} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (\sqrt [3]{2}-\sqrt [3]{1+x^2}\right )}{12 \sqrt [3]{2} \sqrt [3]{x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (\sqrt [3]{2} x^{2/3}-\sqrt [3]{1+x^2}\right )}{6 \sqrt [3]{2} \sqrt [3]{x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (\sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {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]{x} \sqrt [3]{1+x^2} \log \left (1+x^{2/3}-2^{2/3} \sqrt [3]{1+x^2}\right )}{12 \sqrt [3]{2} \sqrt [3]{x+x^3}}+\frac {\sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (x^{2/3}-\sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{1+x^2}\right )}{4 \sqrt [3]{x+x^3}}-\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt [3]{2} x+2^{2/3} x^2} \, dx,x,\frac {1+x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{12 \sqrt [3]{x+x^3}}+\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {-\sqrt [3]{2}+2\ 2^{2/3} x}{1-\sqrt [3]{2} x+2^{2/3} x^2} \, dx,x,\frac {1+x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{36 \sqrt [3]{2} \sqrt [3]{x+x^3}}+\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2^{2/3} \sqrt [3]{1+x^2}\right )}{6 \sqrt [3]{2} \sqrt [3]{x+x^3}} \\ & = -\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \arctan \left (\frac {1+\frac {\sqrt [3]{2} \left (1+x^{2/3}\right )}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{6 \sqrt [3]{2} \sqrt {3} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \arctan \left (\frac {1+\frac {2 \sqrt [3]{2} x^{2/3}}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{3 \sqrt [3]{2} \sqrt {3} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{x} \sqrt [3]{1+x^2} \arctan \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 \sqrt {3} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \arctan \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 \sqrt {3} \sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \arctan \left (\frac {1+2^{2/3} \sqrt [3]{1+x^2}}{\sqrt {3}}\right )}{6 \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 )}{36 \sqrt [3]{2} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (1-x^2\right )}{36 \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\right )}{12 \sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (1+i \sqrt {3}+2 x^2\right )}{12 \sqrt [3]{x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (1+\frac {2^{2/3} \left (1+x^{2/3}\right )^2}{\left (1+x^2\right )^{2/3}}-\frac {\sqrt [3]{2} \left (1+x^{2/3}\right )}{\sqrt [3]{1+x^2}}\right )}{36 \sqrt [3]{2} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (1+\frac {\sqrt [3]{2} \left (1+x^{2/3}\right )}{\sqrt [3]{1+x^2}}\right )}{18 \sqrt [3]{2} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (\sqrt [3]{2}-\sqrt [3]{1+x^2}\right )}{12 \sqrt [3]{2} \sqrt [3]{x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (\sqrt [3]{2} x^{2/3}-\sqrt [3]{1+x^2}\right )}{6 \sqrt [3]{2} \sqrt [3]{x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (\sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {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]{x} \sqrt [3]{1+x^2} \log \left (1+x^{2/3}-2^{2/3} \sqrt [3]{1+x^2}\right )}{12 \sqrt [3]{2} \sqrt [3]{x+x^3}}+\frac {\sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (x^{2/3}-\sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{1+x^2}\right )}{4 \sqrt [3]{x+x^3}}-\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{2} \left (1+x^{2/3}\right )}{\sqrt [3]{1+x^2}}\right )}{6 \sqrt [3]{2} \sqrt [3]{x+x^3}} \\ & = \frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \left (1+x^{2/3}\right )}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{6 \sqrt [3]{2} \sqrt {3} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \arctan \left (\frac {1+\frac {\sqrt [3]{2} \left (1+x^{2/3}\right )}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{6 \sqrt [3]{2} \sqrt {3} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \arctan \left (\frac {1+\frac {2 \sqrt [3]{2} x^{2/3}}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{3 \sqrt [3]{2} \sqrt {3} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{x} \sqrt [3]{1+x^2} \arctan \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 \sqrt {3} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \arctan \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 \sqrt {3} \sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \arctan \left (\frac {1+2^{2/3} \sqrt [3]{1+x^2}}{\sqrt {3}}\right )}{6 \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 )}{36 \sqrt [3]{2} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (1-x^2\right )}{36 \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\right )}{12 \sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (1+i \sqrt {3}+2 x^2\right )}{12 \sqrt [3]{x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (1+\frac {2^{2/3} \left (1+x^{2/3}\right )^2}{\left (1+x^2\right )^{2/3}}-\frac {\sqrt [3]{2} \left (1+x^{2/3}\right )}{\sqrt [3]{1+x^2}}\right )}{36 \sqrt [3]{2} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (1+\frac {\sqrt [3]{2} \left (1+x^{2/3}\right )}{\sqrt [3]{1+x^2}}\right )}{18 \sqrt [3]{2} \sqrt [3]{x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (\sqrt [3]{2}-\sqrt [3]{1+x^2}\right )}{12 \sqrt [3]{2} \sqrt [3]{x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (\sqrt [3]{2} x^{2/3}-\sqrt [3]{1+x^2}\right )}{6 \sqrt [3]{2} \sqrt [3]{x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (\sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {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]{x} \sqrt [3]{1+x^2} \log \left (1+x^{2/3}-2^{2/3} \sqrt [3]{1+x^2}\right )}{12 \sqrt [3]{2} \sqrt [3]{x+x^3}}+\frac {\sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (x^{2/3}-\sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{1+x^2}\right )}{4 \sqrt [3]{x+x^3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.73 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.22 \[ \int \frac {1}{\sqrt [3]{x+x^3} \left (-1+x^6\right )} \, dx=\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \left (-2^{2/3} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2^{2/3} \sqrt [3]{1+x^2}}\right )-2 \log \left (-2 x^{2/3}+2^{2/3} \sqrt [3]{1+x^2}\right )+\log \left (2 x^{4/3}+2^{2/3} x^{2/3} \sqrt [3]{1+x^2}+\sqrt [3]{2} \left (1+x^2\right )^{2/3}\right )\right )+4 \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-2 \log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{1+x^2}-x^{2/3} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]\right )}{24 \sqrt [3]{x+x^3}} \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Time = 58.23 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.86

method	result	size
pseudoelliptic	\(\frac {2^{\frac {2}{3}} \ln \left (\frac {-2^{\frac {1}{3}} x +{\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}}{x}\right )}{12}-\frac {2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} {\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}} x +{\left (x \left (x^{2}+1\right )\right )}^{\frac {2}{3}}}{x^{2}}\right )}{24}+\frac {\sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (2^{\frac {2}{3}} {\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}+x \right )}{3 x}\right )}{12}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +{\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}\right )}{6}\)	\(140\)
trager	\(\text {Expression too large to display}\)	\(4739\)

[In]

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

[Out]

1/12*2^(2/3)*ln((-2^(1/3)*x+(x*(x^2+1))^(1/3))/x)-1/24*2^(2/3)*ln((2^(2/3)*x^2+2^(1/3)*(x*(x^2+1))^(1/3)*x+(x*
(x^2+1))^(2/3))/x^2)+1/12*3^(1/2)*2^(2/3)*arctan(1/3*3^(1/2)*(2^(2/3)*(x*(x^2+1))^(1/3)+x)/x)+1/6*sum(ln((-_R*
x+(x*(x^2+1))^(1/3))/x)/_R,_R=RootOf(_Z^6-_Z^3+1))

Fricas [F(-2)]

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

[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)

Sympy [N/A]

Not integrable

Time = 0.78 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.20 \[ \int \frac {1}{\sqrt [3]{x+x^3} \left (-1+x^6\right )} \, dx=\int \frac {1}{\sqrt [3]{x \left (x^{2} + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )}\, dx \]

[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)

Maxima [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.52 \[ \int \frac {1}{\sqrt [3]{x+x^3} \left (-1+x^6\right )} \, dx=\int { \frac {1}{{\left (x^{6} - 1\right )} {\left (x^{3} + x\right )}^{\frac {1}{3}}} \,d x } \]

[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)

Giac [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.37 (sec) , antiderivative size = 941, normalized size of antiderivative = 5.77 \[ \int \frac {1}{\sqrt [3]{x+x^3} \left (-1+x^6\right )} \, dx=\text {Too large to display} \]

[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)))

Mupad [N/A]

Not integrable

Time = 5.67 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.10 \[ \int \frac {1}{\sqrt [3]{x+x^3} \left (-1+x^6\right )} \, dx=\int \frac {1}{\left (x^6-1\right )\,{\left (x^3+x\right )}^{1/3}} \,d x \]

[In]

int(1/((x^6 - 1)*(x + x^3)^(1/3)),x)

[Out]

int(1/((x^6 - 1)*(x + x^3)^(1/3)), x)