\(\int \frac {1-x}{(1+x+x^2) \sqrt [3]{1+x^3}} \, dx\) [102]

Optimal result

Integrand size = 23, antiderivative size = 119 \[ \int \frac {1-x}{\left (1+x+x^2\right ) \sqrt [3]{1+x^3}} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} (1+x)}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2}}-\frac {\log \left (1+\frac {2^{2/3} (1+x)^2}{\left (1+x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1+x)}{\sqrt [3]{1+x^3}}\right )}{2 \sqrt [3]{2}}+\frac {\log \left (1+\frac {\sqrt [3]{2} (1+x)}{\sqrt [3]{1+x^3}}\right )}{\sqrt [3]{2}} \]

[Out]

-1/4*ln(1+2^(2/3)*(1+x)^2/(x^3+1)^(2/3)-2^(1/3)*(1+x)/(x^3+1)^(1/3))*2^(2/3)+1/2*ln(1+2^(1/3)*(1+x)/(x^3+1)^(1
/3))*2^(2/3)-1/2*arctan(1/3*(1-2*2^(1/3)*(1+x)/(x^3+1)^(1/3))*3^(1/2))*3^(1/2)*2^(2/3)

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(357\) vs. \(2(119)=238\).

Time = 0.20 (sec) , antiderivative size = 357, normalized size of antiderivative = 3.00, number of steps used = 16, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {2183, 384, 502, 2174, 206, 31, 648, 631, 210, 642, 455, 57} \[ \int \frac {1-x}{\left (1+x+x^2\right ) \sqrt [3]{1+x^3}} \, dx=\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{2} x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {2^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} (x+1)}{\sqrt [3]{x^3+1}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\arctan \left (\frac {\frac {\sqrt [3]{2} (x+1)}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {\arctan \left (\frac {2^{2/3} \sqrt [3]{x^3+1}+1}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {\log \left (1-x^3\right )}{3 \sqrt [3]{2}}-\frac {\log \left (\frac {2^{2/3} (x+1)^2}{\left (x^3+1\right )^{2/3}}-\frac {\sqrt [3]{2} (x+1)}{\sqrt [3]{x^3+1}}+1\right )}{3 \sqrt [3]{2}}+\frac {1}{3} 2^{2/3} \log \left (\frac {\sqrt [3]{2} (x+1)}{\sqrt [3]{x^3+1}}+1\right )-\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{x^3+1}\right )}{2 \sqrt [3]{2}}-\frac {\log \left (\sqrt [3]{2} x-\sqrt [3]{x^3+1}\right )}{2 \sqrt [3]{2}}+\frac {\log \left (-2^{2/3} \sqrt [3]{x^3+1}+x+1\right )}{2 \sqrt [3]{2}}-\frac {\log \left ((1-x)^2 (x+1)\right )}{6 \sqrt [3]{2}} \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 1.09 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.17 \[ \int \frac {1-x}{\left (1+x+x^2\right ) \sqrt [3]{1+x^3}} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{1+x^3}}{-2 \sqrt [3]{2}-2 \sqrt [3]{2} x+\sqrt [3]{1+x^3}}\right )+2 \log \left (\sqrt [3]{2}+\sqrt [3]{2} x+\sqrt [3]{1+x^3}\right )-\log \left (2^{2/3}+2\ 2^{2/3} x+2^{2/3} x^2-\sqrt [3]{2} (1+x) \sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right )}{2 \sqrt [3]{2}} \]

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 6.40 (sec) , antiderivative size = 714, normalized size of antiderivative = 6.00

method	result	size
trager	\(\text {Expression too large to display}\)	\(714\)

[In]

int((1-x)/(x^2+x+1)/(x^3+1)^(1/3),x,method=_RETURNVERBOSE)

[Out]

-1/2*ln(-((x^3+1)^(2/3)*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)^2+2*RootOf(RootOf(_
Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)^2*RootOf(_Z^3-4)^2*x-(x^3+1)^(1/3)*RootOf(_Z^3-4)^2*x-(x^3+1)^(1/3)*RootO
f(_Z^3-4)^2-2*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*x^2-2*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z
^3-4)+4*_Z^2)*x-2*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2))/(x^2+x+1))*RootOf(_Z^3-4)-ln(-((x^3+1)^
(2/3)*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)^2+2*RootOf(RootOf(_Z^3-4)^2+2*_Z*Root
Of(_Z^3-4)+4*_Z^2)^2*RootOf(_Z^3-4)^2*x-(x^3+1)^(1/3)*RootOf(_Z^3-4)^2*x-(x^3+1)^(1/3)*RootOf(_Z^3-4)^2-2*Root
Of(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*x^2-2*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*x-2*
RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2))/(x^2+x+1))*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*
_Z^2)+1/2*RootOf(_Z^3-4)*ln(((x^3+1)^(2/3)*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)^
2+2*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)^2*RootOf(_Z^3-4)^2*x+2*(x^3+1)^(1/3)*RootOf(_Z^3-4)*Ro
otOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*x+2*(x^3+1)^(1/3)*RootOf(_Z^3-4)*RootOf(RootOf(_Z^3-4)^2+2*_
Z*RootOf(_Z^3-4)+4*_Z^2)+2*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*x^2+2*(x^3+1)^(2/3)+6*RootOf(Ro
otOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*x+2*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2))/(x^2+x+1))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 268 vs. \(2 (93) = 186\).

Time = 5.27 (sec) , antiderivative size = 268, normalized size of antiderivative = 2.25 \[ \int \frac {1-x}{\left (1+x+x^2\right ) \sqrt [3]{1+x^3}} \, dx=\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {1}{6}} {\left (2^{\frac {5}{6}} {\left (x^{6} + 7 \, x^{5} + 10 \, x^{4} + 7 \, x^{3} + 10 \, x^{2} + 7 \, x + 1\right )} - 4 \, \sqrt {2} {\left (x^{5} + x^{4} - 3 \, x^{3} - 3 \, x^{2} + x + 1\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 4 \cdot 2^{\frac {1}{6}} {\left (x^{4} + 4 \, x^{3} + 5 \, x^{2} + 4 \, x + 1\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}\right )}}{6 \, {\left (3 \, x^{6} + 9 \, x^{5} + 6 \, x^{4} + x^{3} + 6 \, x^{2} + 9 \, x + 3\right )}}\right ) - \frac {1}{12} \cdot 2^{\frac {2}{3}} \log \left (\frac {2^{\frac {2}{3}} {\left (x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{2} + 3 \, x + 1\right )} - 2^{\frac {1}{3}} {\left (x^{4} - 3 \, x^{2} + 1\right )} - 4 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} {\left (x^{2} + x\right )}}{x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1}\right ) + \frac {1}{6} \cdot 2^{\frac {2}{3}} \log \left (\frac {2^{\frac {2}{3}} {\left (x^{2} + x + 1\right )} + 2 \cdot 2^{\frac {1}{3}} {\left (x^{3} + 1\right )}^{\frac {1}{3}} {\left (x + 1\right )} + 2 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2} + x + 1}\right ) \]

[In]

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

[Out]

1/6*sqrt(3)*2^(2/3)*arctan(1/6*sqrt(3)*2^(1/6)*(2^(5/6)*(x^6 + 7*x^5 + 10*x^4 + 7*x^3 + 10*x^2 + 7*x + 1) - 4*
sqrt(2)*(x^5 + x^4 - 3*x^3 - 3*x^2 + x + 1)*(x^3 + 1)^(1/3) + 4*2^(1/6)*(x^4 + 4*x^3 + 5*x^2 + 4*x + 1)*(x^3 +
1)^(2/3))/(3*x^6 + 9*x^5 + 6*x^4 + x^3 + 6*x^2 + 9*x + 3)) - 1/12*2^(2/3)*log((2^(2/3)*(x^3 + 1)^(2/3)*(x^2 +
3*x + 1) - 2^(1/3)*(x^4 - 3*x^2 + 1) - 4*(x^3 + 1)^(1/3)*(x^2 + x))/(x^4 + 2*x^3 + 3*x^2 + 2*x + 1)) + 1/6*2^
(2/3)*log((2^(2/3)*(x^2 + x + 1) + 2*2^(1/3)*(x^3 + 1)^(1/3)*(x + 1) + 2*(x^3 + 1)^(2/3))/(x^2 + x + 1))

Sympy [F]

\[ \int \frac {1-x}{\left (1+x+x^2\right ) \sqrt [3]{1+x^3}} \, dx=- \int \frac {x}{x^{2} \sqrt [3]{x^{3} + 1} + x \sqrt [3]{x^{3} + 1} + \sqrt [3]{x^{3} + 1}}\, dx - \int \left (- \frac {1}{x^{2} \sqrt [3]{x^{3} + 1} + x \sqrt [3]{x^{3} + 1} + \sqrt [3]{x^{3} + 1}}\right )\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {1-x}{\left (1+x+x^2\right ) \sqrt [3]{1+x^3}} \, dx=\int { -\frac {x - 1}{{\left (x^{3} + 1\right )}^{\frac {1}{3}} {\left (x^{2} + x + 1\right )}} \,d x } \]

[In]

integrate((1-x)/(x^2+x+1)/(x^3+1)^(1/3),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {1-x}{\left (1+x+x^2\right ) \sqrt [3]{1+x^3}} \, dx=\int { -\frac {x - 1}{{\left (x^{3} + 1\right )}^{\frac {1}{3}} {\left (x^{2} + x + 1\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1-x}{\left (1+x+x^2\right ) \sqrt [3]{1+x^3}} \, dx=-\int \frac {x-1}{{\left (x^3+1\right )}^{1/3}\,\left (x^2+x+1\right )} \,d x \]

[In]

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

[Out]

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