3.1.58 \(\int \frac {\sqrt [3]{1-x^3}}{1+x} \, dx\) [58]

Optimal. Leaf size=482 \[ \sqrt [3]{1-x^3}+\frac {\sqrt [3]{2} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\tan ^{-1}\left (\frac {1+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}-\frac {\tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\sqrt [3]{2} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\sqrt [3]{2} \tan ^{-1}\left (\frac {1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{3} \sqrt [3]{2} \log \left (1+x^3\right )+\frac {\log \left (2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}-\frac {\log \left (1+\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}+\frac {1}{3} \sqrt [3]{2} \log \left (1+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )-\frac {\log \left (2 \sqrt [3]{2}+\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}\right )}{6\ 2^{2/3}}+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2^{2/3}}-\frac {1}{2} \log \left (-x-\sqrt [3]{1-x^3}\right )+\frac {\log \left (-\sqrt [3]{2} x-\sqrt [3]{1-x^3}\right )}{2^{2/3}} \]

[Out]

(-x^3+1)^(1/3)-1/3*2^(1/3)*ln(x^3+1)+1/6*ln(2^(2/3)+(-1+x)/(-x^3+1)^(1/3))*2^(1/3)-1/6*ln(1+2^(2/3)*(1-x)^2/(-
x^3+1)^(2/3)-2^(1/3)*(1-x)/(-x^3+1)^(1/3))*2^(1/3)+1/3*2^(1/3)*ln(1+2^(1/3)*(1-x)/(-x^3+1)^(1/3))-1/12*ln(2*2^
(1/3)+(1-x)^2/(-x^3+1)^(2/3)+2^(2/3)*(1-x)/(-x^3+1)^(1/3))*2^(1/3)+1/2*ln(2^(1/3)-(-x^3+1)^(1/3))*2^(1/3)-1/2*
ln(-x-(-x^3+1)^(1/3))+1/2*ln(-2^(1/3)*x-(-x^3+1)^(1/3))*2^(1/3)+1/3*2^(1/3)*arctan(1/3*(1-2*2^(1/3)*(1-x)/(-x^
3+1)^(1/3))*3^(1/2))*3^(1/2)+1/6*arctan(1/3*(1+2^(1/3)*(1-x)/(-x^3+1)^(1/3))*3^(1/2))*2^(1/3)*3^(1/2)-1/3*arct
an(1/3*(1-2*x/(-x^3+1)^(1/3))*3^(1/2))*3^(1/2)+1/3*2^(1/3)*arctan(1/3*(1-2*2^(1/3)*x/(-x^3+1)^(1/3))*3^(1/2))*
3^(1/2)-1/3*2^(1/3)*arctan(1/3*(1+2^(2/3)*(-x^3+1)^(1/3))*3^(1/2))*3^(1/2)

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Rubi [A]
time = 0.20, antiderivative size = 482, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 15, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.882, Rules used = {2181, 420, 493, 298, 31, 648, 631, 210, 642, 495, 337, 503, 455, 52, 59} \begin {gather*} \frac {\sqrt [3]{2} \text {ArcTan}\left (\frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\text {ArcTan}\left (\frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}-\frac {\text {ArcTan}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\sqrt [3]{2} \text {ArcTan}\left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\sqrt [3]{2} \text {ArcTan}\left (\frac {2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\sqrt [3]{1-x^3}-\frac {1}{3} \sqrt [3]{2} \log \left (x^3+1\right )+\frac {\log \left (2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}-\frac {\log \left (\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3\ 2^{2/3}}+\frac {1}{3} \sqrt [3]{2} \log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )-\frac {\log \left (\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}\right )}{6\ 2^{2/3}}+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2^{2/3}}-\frac {1}{2} \log \left (-\sqrt [3]{1-x^3}-x\right )+\frac {\log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )}{2^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - x^3)^(1/3)/(1 + x),x]

[Out]

(1 - x^3)^(1/3) + (2^(1/3)*ArcTan[(1 - (2*2^(1/3)*(1 - x))/(1 - x^3)^(1/3))/Sqrt[3]])/Sqrt[3] + ArcTan[(1 + (2
^(1/3)*(1 - x))/(1 - x^3)^(1/3))/Sqrt[3]]/(2^(2/3)*Sqrt[3]) - ArcTan[(1 - (2*x)/(1 - x^3)^(1/3))/Sqrt[3]]/Sqrt
[3] + (2^(1/3)*ArcTan[(1 - (2*2^(1/3)*x)/(1 - x^3)^(1/3))/Sqrt[3]])/Sqrt[3] - (2^(1/3)*ArcTan[(1 + 2^(2/3)*(1
- x^3)^(1/3))/Sqrt[3]])/Sqrt[3] - (2^(1/3)*Log[1 + x^3])/3 + Log[2^(2/3) - (1 - x)/(1 - x^3)^(1/3)]/(3*2^(2/3)
) - Log[1 + (2^(2/3)*(1 - x)^2)/(1 - x^3)^(2/3) - (2^(1/3)*(1 - x))/(1 - x^3)^(1/3)]/(3*2^(2/3)) + (2^(1/3)*Lo
g[1 + (2^(1/3)*(1 - x))/(1 - x^3)^(1/3)])/3 - Log[2*2^(1/3) + (1 - x)^2/(1 - x^3)^(2/3) + (2^(2/3)*(1 - x))/(1
- x^3)^(1/3)]/(6*2^(2/3)) + Log[2^(1/3) - (1 - x^3)^(1/3)]/2^(2/3) - Log[-x - (1 - x^3)^(1/3)]/2 + Log[-(2^(1
/3)*x) - (1 - x^3)^(1/3)]/2^(2/3)

Rule 31

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

Rule 52

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

Rule 59

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, Simp[-L
og[RemoveContent[a + b*x, x]]/(2*b*q^2), x] + (-Dist[3/(2*b*q), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x
)^(1/3)], x] - Dist[3/(2*b*q^2), 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 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 298

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Dist[-(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(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 337

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

Rule 420

Int[((a_) + (b_.)*(x_)^3)^(1/3)/((c_) + (d_.)*(x_)^3), x_Symbol] :> With[{q = Rt[b/a, 3]}, Dist[9*(a/(c*q)), S
ubst[Int[x/((4 - a*x^3)*(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 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 493

Int[((e_.)*(x_))^(m_.)/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[b/(b*c - a*d), I
nt[(e*x)^m/(a + b*x^n), x], x] - Dist[d/(b*c - a*d), Int[(e*x)^m/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e,
m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0]

Rule 495

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

Rule 503

Int[(x_)/(((a_) + (b_.)*(x_)^3)^(2/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/c, 3]}, Si
mp[-ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sqrt[3]*c*q^2), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/
(2*c*q^2), x] + Simp[Log[c + d*x^3]/(6*c*q^2), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[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 2181

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

Rubi steps

\begin {align*} \int \frac {\sqrt [3]{1-x^3}}{1+x} \, dx &=\int \frac {\sqrt [3]{1-x^3}}{1+x} \, dx\\ \end {align*}

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Mathematica [F]
time = 45.62, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{1-x^3}}{1+x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(1 - x^3)^(1/3)/(1 + x),x]

[Out]

Integrate[(1 - x^3)^(1/3)/(1 + x), x]

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 12.52, size = 2990, normalized size = 6.20

method	result	size
risch	\(\text {Expression too large to display}\)	\(2990\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x^3-1)/(-x^3+1)^(2/3)+(1/2*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*ln(-(2*RootOf(RootOf(_Z^3-2)^2+_Z
*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^3*x^3+4*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^
2*x^3+2*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^3*x^2+4*RootOf(RootOf(_Z^3-2)^2+_Z*Root
Of(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^2*x^2+2*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^3*x+4
*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^2*x+5*(x^6-2*x^3+1)^(2/3)*RootOf(_Z^3-2)^2*R
ootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)+8*RootOf(_Z^3-2)^2*(x^6-2*x^3+1)^(1/3)*x^2+10*RootOf(RootOf(_Z^
3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)*(x^6-2*x^3+1)^(1/3)*x^2+7*RootOf(_Z^3-2)*x^4+14*RootOf(RootOf(_Z
^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*x^4+8*RootOf(_Z^3-2)^2*(x^6-2*x^3+1)^(1/3)*x+10*RootOf(RootOf(_Z^3-2)^2+_Z*Roo
tOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)*(x^6-2*x^3+1)^(1/3)*x+13*RootOf(_Z^3-2)*x^3+26*RootOf(RootOf(_Z^3-2)^2+_Z*Roo
tOf(_Z^3-2)+_Z^2)*x^3+8*RootOf(_Z^3-2)^2*(x^6-2*x^3+1)^(1/3)+10*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2
)*RootOf(_Z^3-2)*(x^6-2*x^3+1)^(1/3)+20*RootOf(_Z^3-2)*x^2+40*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*
x^2+13*RootOf(_Z^3-2)*x+26*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*x+8*(x^6-2*x^3+1)^(2/3)+7*RootOf(_Z
^3-2)+14*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2))/(x^2+x+1)/(1+x)^2)-1/2*ln(-(2*RootOf(RootOf(_Z^3-2)^
2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^3*x^3+4*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3
-2)^2*x^3+2*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^3*x^2+4*RootOf(RootOf(_Z^3-2)^2+_Z*
RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^2*x^2+2*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^3
*x+4*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^2*x-5*(x^6-2*x^3+1)^(2/3)*RootOf(_Z^3-2)
^2*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)-2*RootOf(_Z^3-2)^2*(x^6-2*x^3+1)^(1/3)*x^2-10*RootOf(RootOf
(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)*(x^6-2*x^3+1)^(1/3)*x^2-7*RootOf(_Z^3-2)*x^4-14*RootOf(RootO
f(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*x^4-2*RootOf(_Z^3-2)^2*(x^6-2*x^3+1)^(1/3)*x-10*RootOf(RootOf(_Z^3-2)^2+_Z
*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)*(x^6-2*x^3+1)^(1/3)*x-9*RootOf(_Z^3-2)*x^3-18*RootOf(RootOf(_Z^3-2)^2+_Z*
RootOf(_Z^3-2)+_Z^2)*x^3-2*RootOf(_Z^3-2)^2*(x^6-2*x^3+1)^(1/3)-10*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_
Z^2)*RootOf(_Z^3-2)*(x^6-2*x^3+1)^(1/3)-16*RootOf(_Z^3-2)*x^2-32*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^
2)*x^2-9*RootOf(_Z^3-2)*x-18*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*x-2*(x^6-2*x^3+1)^(2/3)-7*RootOf(
_Z^3-2)-14*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2))/(x^2+x+1)/(1+x)^2)*RootOf(_Z^3-2)-1/2*ln(-(2*RootO
f(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^3*x^3+4*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z
^2)^2*RootOf(_Z^3-2)^2*x^3+2*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^3*x^2+4*RootOf(Roo
tOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^2*x^2+2*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)
*RootOf(_Z^3-2)^3*x+4*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^2*x-5*(x^6-2*x^3+1)^(2/
3)*RootOf(_Z^3-2)^2*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)-2*RootOf(_Z^3-2)^2*(x^6-2*x^3+1)^(1/3)*x^2
-10*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)*(x^6-2*x^3+1)^(1/3)*x^2-7*RootOf(_Z^3-2)*x^
4-14*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*x^4-2*RootOf(_Z^3-2)^2*(x^6-2*x^3+1)^(1/3)*x-10*RootOf(Ro
otOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)*(x^6-2*x^3+1)^(1/3)*x-9*RootOf(_Z^3-2)*x^3-18*RootOf(Roo
tOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*x^3-2*RootOf(_Z^3-2)^2*(x^6-2*x^3+1)^(1/3)-10*RootOf(RootOf(_Z^3-2)^2+_Z
*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)*(x^6-2*x^3+1)^(1/3)-16*RootOf(_Z^3-2)*x^2-32*RootOf(RootOf(_Z^3-2)^2+_Z*R
ootOf(_Z^3-2)+_Z^2)*x^2-9*RootOf(_Z^3-2)*x-18*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*x-2*(x^6-2*x^3+1
)^(2/3)-7*RootOf(_Z^3-2)-14*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2))/(x^2+x+1)/(1+x)^2)*RootOf(RootOf(
_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)-1/3*ln((RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^4*x
^6-RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^4*x^3+8*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(
_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^2*x^6-6*(x^6-2*x^3+1)^(2/3)*RootOf(_Z^3-2)^2*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z
^3-2)+_Z^2)*x^2-10*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*RootOf(_Z^3-2)^2*x^3+16*x^6-12*(x^6-2*x^3+1
)^(1/3)*x^4+2*RootOf(_Z^3-2)^2*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)-24*x^3+12*(x^6-2*x^3+1)^(1/3)*x
+8)/(-1+x)/(x^2+x+1))+1/6*ln((RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^4*x^6-RootOf(Ro
otOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)^2*RootOf(_Z^3-2)^4*x^3+2*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2
)*RootOf(_Z^3-2)^2*x^6-6*RootOf(_Z^3-2)^2*RootOf(RootOf(_Z^3-2)^2+_Z*RootOf(_Z^3-2)+_Z^2)*(x^6-2*x^3+1)^(1/3)*
x^4+6*(x^6-2*x^3+1)^(2/3)*RootOf(_Z^3-2)^2*Root...

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - x^3)^(1/3)/(x + 1),x)

[Out]

int((1 - x^3)^(1/3)/(x + 1), x)

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