3.1.28 \(\int \frac {x (1-x^3)}{1-x^3+x^6} \, dx\) [28]

Optimal. Leaf size=411 \[ \frac {\left (i-\sqrt {3}\right ) \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{3\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}}}-\frac {\left (i+\sqrt {3}\right ) \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{3\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}}}-\frac {\left (3-i \sqrt {3}\right ) \log \left (\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} x\right )}{9\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}}}-\frac {\left (3+i \sqrt {3}\right ) \log \left (\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} x\right )}{9\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}}}+\frac {\left (3-i \sqrt {3}\right ) \log \left (\left (1-i \sqrt {3}\right )^{2/3}+\sqrt [3]{2 \left (1-i \sqrt {3}\right )} x+2^{2/3} x^2\right )}{18\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}}}+\frac {\left (3+i \sqrt {3}\right ) \log \left (\left (1+i \sqrt {3}\right )^{2/3}+\sqrt [3]{2 \left (1+i \sqrt {3}\right )} x+2^{2/3} x^2\right )}{18\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}}} \]

[Out]

1/6*arctan(1/3*(1+2*2^(1/3)*x/(1-I*3^(1/2))^(1/3))*3^(1/2))*(I-3^(1/2))*2^(1/3)/(1-I*3^(1/2))^(1/3)-1/18*ln(-2
^(1/3)*x+(1-I*3^(1/2))^(1/3))*(3-I*3^(1/2))*2^(1/3)/(1-I*3^(1/2))^(1/3)+1/36*ln(2^(2/3)*x^2+2^(1/3)*x*(1-I*3^(
1/2))^(1/3)+(1-I*3^(1/2))^(2/3))*(3-I*3^(1/2))*2^(1/3)/(1-I*3^(1/2))^(1/3)-1/18*ln(-2^(1/3)*x+(1+I*3^(1/2))^(1
/3))*(3+I*3^(1/2))*2^(1/3)/(1+I*3^(1/2))^(1/3)+1/36*ln(2^(2/3)*x^2+2^(1/3)*x*(1+I*3^(1/2))^(1/3)+(1+I*3^(1/2))
^(2/3))*(3+I*3^(1/2))*2^(1/3)/(1+I*3^(1/2))^(1/3)-1/6*arctan(1/3*(1+2*2^(1/3)*x/(1+I*3^(1/2))^(1/3))*3^(1/2))*
(3^(1/2)+I)*2^(1/3)/(1+I*3^(1/2))^(1/3)

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Rubi [A]
time = 0.19, antiderivative size = 411, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1524, 298, 31, 648, 631, 210, 642} \begin {gather*} \frac {\left (-\sqrt {3}+i\right ) \text {ArcTan}\left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{3\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}}}-\frac {\left (\sqrt {3}+i\right ) \text {ArcTan}\left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{3\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}}}+\frac {\left (3-i \sqrt {3}\right ) \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1-i \sqrt {3}\right )} x+\left (1-i \sqrt {3}\right )^{2/3}\right )}{18\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}}}+\frac {\left (3+i \sqrt {3}\right ) \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1+i \sqrt {3}\right )} x+\left (1+i \sqrt {3}\right )^{2/3}\right )}{18\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}}}-\frac {\left (3-i \sqrt {3}\right ) \log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt {3}}\right )}{9\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}}}-\frac {\left (3+i \sqrt {3}\right ) \log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt {3}}\right )}{9\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x*(1 - x^3))/(1 - x^3 + x^6),x]

[Out]

((I - Sqrt[3])*ArcTan[(1 + (2*x)/((1 - I*Sqrt[3])/2)^(1/3))/Sqrt[3]])/(3*2^(2/3)*(1 - I*Sqrt[3])^(1/3)) - ((I
+ Sqrt[3])*ArcTan[(1 + (2*x)/((1 + I*Sqrt[3])/2)^(1/3))/Sqrt[3]])/(3*2^(2/3)*(1 + I*Sqrt[3])^(1/3)) - ((3 - I*
Sqrt[3])*Log[(1 - I*Sqrt[3])^(1/3) - 2^(1/3)*x])/(9*2^(2/3)*(1 - I*Sqrt[3])^(1/3)) - ((3 + I*Sqrt[3])*Log[(1 +
I*Sqrt[3])^(1/3) - 2^(1/3)*x])/(9*2^(2/3)*(1 + I*Sqrt[3])^(1/3)) + ((3 - I*Sqrt[3])*Log[(1 - I*Sqrt[3])^(2/3)
+ (2*(1 - I*Sqrt[3]))^(1/3)*x + 2^(2/3)*x^2])/(18*2^(2/3)*(1 - I*Sqrt[3])^(1/3)) + ((3 + I*Sqrt[3])*Log[(1 +
I*Sqrt[3])^(2/3) + (2*(1 + I*Sqrt[3]))^(1/3)*x + 2^(2/3)*x^2])/(18*2^(2/3)*(1 + I*Sqrt[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 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 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 1524

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_)))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> Wi
th[{q = Rt[b^2 - 4*a*c, 2]}, Dist[e/2 + (2*c*d - b*e)/(2*q), Int[(f*x)^m/(b/2 - q/2 + c*x^n), x], x] + Dist[e/
2 - (2*c*d - b*e)/(2*q), Int[(f*x)^m/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[n2
, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {x \left (1-x^3\right )}{1-x^3+x^6} \, dx &=\frac {1}{6} \left (-3+i \sqrt {3}\right ) \int \frac {x}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}+x^3} \, dx-\frac {1}{6} \left (3+i \sqrt {3}\right ) \int \frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}+x^3} \, dx\\ &=-\frac {\left (3-i \sqrt {3}\right ) \int \frac {1}{-\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}+x} \, dx}{9\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}}}+\frac {\left (3-i \sqrt {3}\right ) \int \frac {-\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}+x}{\left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}+\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )} x+x^2} \, dx}{9\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}}}-\frac {\left (3+i \sqrt {3}\right ) \int \frac {1}{-\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}+x} \, dx}{9\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}}}+\frac {\left (3+i \sqrt {3}\right ) \int \frac {-\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}+x}{\left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}+\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )} x+x^2} \, dx}{9\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}}}\\ &=-\frac {\left (3-i \sqrt {3}\right ) \log \left (\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} x\right )}{9\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}}}-\frac {\left (3+i \sqrt {3}\right ) \log \left (\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} x\right )}{9\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}}}+\frac {\left (3-i \sqrt {3}\right ) \int \frac {\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}+2 x}{\left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}+\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )} x+x^2} \, dx}{18\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}}}+\frac {1}{12} \left (-3+i \sqrt {3}\right ) \int \frac {1}{\left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}+\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )} x+x^2} \, dx-\frac {1}{12} \left (3+i \sqrt {3}\right ) \int \frac {1}{\left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}+\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )} x+x^2} \, dx+\frac {\left (3+i \sqrt {3}\right ) \int \frac {\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}+2 x}{\left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{2/3}+\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )} x+x^2} \, dx}{18\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}}}\\ &=-\frac {\left (3-i \sqrt {3}\right ) \log \left (\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} x\right )}{9\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}}}-\frac {\left (3+i \sqrt {3}\right ) \log \left (\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} x\right )}{9\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}}}+\frac {\left (3-i \sqrt {3}\right ) \log \left (\left (1-i \sqrt {3}\right )^{2/3}+\sqrt [3]{2 \left (1-i \sqrt {3}\right )} x+2^{2/3} x^2\right )}{18\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}}}+\frac {\left (3+i \sqrt {3}\right ) \log \left (\left (1+i \sqrt {3}\right )^{2/3}+\sqrt [3]{2 \left (1+i \sqrt {3}\right )} x+2^{2/3} x^2\right )}{18\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}}}+\frac {\left (3-i \sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}\right )}{3\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}}}+\frac {\left (3+i \sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}\right )}{3\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}}}\\ &=\frac {\left (i-\sqrt {3}\right ) \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{3\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}}}-\frac {\left (i+\sqrt {3}\right ) \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{3\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}}}-\frac {\left (3-i \sqrt {3}\right ) \log \left (\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} x\right )}{9\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}}}-\frac {\left (3+i \sqrt {3}\right ) \log \left (\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} x\right )}{9\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}}}+\frac {\left (3-i \sqrt {3}\right ) \log \left (\left (1-i \sqrt {3}\right )^{2/3}+\sqrt [3]{2 \left (1-i \sqrt {3}\right )} x+2^{2/3} x^2\right )}{18\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}}}+\frac {\left (3+i \sqrt {3}\right ) \log \left (\left (1+i \sqrt {3}\right )^{2/3}+\sqrt [3]{2 \left (1+i \sqrt {3}\right )} x+2^{2/3} x^2\right )}{18\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.01, size = 55, normalized size = 0.13 \begin {gather*} -\frac {1}{3} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x-\text {$\#$1})+\log (x-\text {$\#$1}) \text {$\#$1}^3}{-\text {$\#$1}+2 \text {$\#$1}^4}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.02, size = 44, normalized size = 0.11

method	result	size
default	\(-\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )}{\sum }\frac {\left (\textit {\_R}^{4}-\textit {\_R} \right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5}-\textit {\_R}^{2}}\right )}{3}\)	\(44\)
risch	\(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )}{\sum }\frac {\left (-\textit {\_R}^{4}+\textit {\_R} \right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5}-\textit {\_R}^{2}}\right )}{3}\)	\(44\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*sum((_R^4-_R)/(2*_R^5-_R^2)*ln(x-_R),_R=RootOf(_Z^6-_Z^3+1))

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 992 vs. \(2 (267) = 534\).
time = 0.47, size = 992, normalized size = 2.41 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/54*18^(2/3)*12^(1/6)*cos(2/3*arctan(sqrt(3) + 2))*log(-24*18^(1/3)*12^(1/3)*x*cos(2/3*arctan(sqrt(3) + 2))^2
+ 36*x^2 + 12*18^(1/3)*12^(1/3)*x + 18^(2/3)*12^(2/3)) + 2/27*18^(2/3)*12^(1/6)*arctan(-1/432*(6*18^(2/3)*12^
(2/3)*x - 432*cos(2/3*arctan(sqrt(3) + 2))^2 - 18^(2/3)*12^(2/3)*sqrt(-24*18^(1/3)*12^(1/3)*x*cos(2/3*arctan(s
qrt(3) + 2))^2 + 36*x^2 + 12*18^(1/3)*12^(1/3)*x + 18^(2/3)*12^(2/3)) + 216)/(cos(2/3*arctan(sqrt(3) + 2))*sin
(2/3*arctan(sqrt(3) + 2))))*sin(2/3*arctan(sqrt(3) + 2)) + 1/27*(18^(2/3)*12^(1/6)*sqrt(3)*cos(2/3*arctan(sqrt
(3) + 2)) - 18^(2/3)*12^(1/6)*sin(2/3*arctan(sqrt(3) + 2)))*arctan(1/216*(24*18^(2/3)*12^(2/3)*sqrt(3)*x*cos(2
/3*arctan(sqrt(3) + 2))^2 - 12*18^(2/3)*12^(2/3)*sqrt(3)*x - 24*(144*cos(2/3*arctan(sqrt(3) + 2))^3 + (18^(2/3
)*12^(2/3)*x - 72)*cos(2/3*arctan(sqrt(3) + 2)))*sin(2/3*arctan(sqrt(3) + 2)) - sqrt(-48*18^(1/3)*12^(1/3)*sqr
t(3)*x*cos(2/3*arctan(sqrt(3) + 2))*sin(2/3*arctan(sqrt(3) + 2)) + 48*18^(1/3)*12^(1/3)*x*cos(2/3*arctan(sqrt(
3) + 2))^2 + 144*x^2 - 24*18^(1/3)*12^(1/3)*x + 4*18^(2/3)*12^(2/3))*(2*18^(2/3)*12^(2/3)*sqrt(3)*cos(2/3*arct
an(sqrt(3) + 2))^2 - 2*18^(2/3)*12^(2/3)*cos(2/3*arctan(sqrt(3) + 2))*sin(2/3*arctan(sqrt(3) + 2)) - 18^(2/3)*
12^(2/3)*sqrt(3)) + 216*sqrt(3))/(16*cos(2/3*arctan(sqrt(3) + 2))^4 - 16*cos(2/3*arctan(sqrt(3) + 2))^2 + 3))
- 1/27*(18^(2/3)*12^(1/6)*sqrt(3)*cos(2/3*arctan(sqrt(3) + 2)) + 18^(2/3)*12^(1/6)*sin(2/3*arctan(sqrt(3) + 2)
))*arctan(-1/108*(12*18^(2/3)*12^(2/3)*sqrt(3)*x*cos(2/3*arctan(sqrt(3) + 2))^2 - 6*18^(2/3)*12^(2/3)*sqrt(3)*
x + 12*(144*cos(2/3*arctan(sqrt(3) + 2))^3 + (18^(2/3)*12^(2/3)*x - 72)*cos(2/3*arctan(sqrt(3) + 2)))*sin(2/3*
arctan(sqrt(3) + 2)) - sqrt(12*18^(1/3)*12^(1/3)*sqrt(3)*x*cos(2/3*arctan(sqrt(3) + 2))*sin(2/3*arctan(sqrt(3)
+ 2)) + 12*18^(1/3)*12^(1/3)*x*cos(2/3*arctan(sqrt(3) + 2))^2 + 36*x^2 - 6*18^(1/3)*12^(1/3)*x + 18^(2/3)*12^
(2/3))*(2*18^(2/3)*12^(2/3)*sqrt(3)*cos(2/3*arctan(sqrt(3) + 2))^2 + 2*18^(2/3)*12^(2/3)*cos(2/3*arctan(sqrt(3
) + 2))*sin(2/3*arctan(sqrt(3) + 2)) - 18^(2/3)*12^(2/3)*sqrt(3)) + 108*sqrt(3))/(16*cos(2/3*arctan(sqrt(3) +
2))^4 - 16*cos(2/3*arctan(sqrt(3) + 2))^2 + 3)) + 1/108*(18^(2/3)*12^(1/6)*sqrt(3)*sin(2/3*arctan(sqrt(3) + 2)
) - 18^(2/3)*12^(1/6)*cos(2/3*arctan(sqrt(3) + 2)))*log(48*18^(1/3)*12^(1/3)*sqrt(3)*x*cos(2/3*arctan(sqrt(3)
+ 2))*sin(2/3*arctan(sqrt(3) + 2)) + 48*18^(1/3)*12^(1/3)*x*cos(2/3*arctan(sqrt(3) + 2))^2 + 144*x^2 - 24*18^(
1/3)*12^(1/3)*x + 4*18^(2/3)*12^(2/3)) - 1/108*(18^(2/3)*12^(1/6)*sqrt(3)*sin(2/3*arctan(sqrt(3) + 2)) + 18^(2
/3)*12^(1/6)*cos(2/3*arctan(sqrt(3) + 2)))*log(-48*18^(1/3)*12^(1/3)*sqrt(3)*x*cos(2/3*arctan(sqrt(3) + 2))*si
n(2/3*arctan(sqrt(3) + 2)) + 48*18^(1/3)*12^(1/3)*x*cos(2/3*arctan(sqrt(3) + 2))^2 + 144*x^2 - 24*18^(1/3)*12^
(1/3)*x + 4*18^(2/3)*12^(2/3))

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Sympy [A]
time = 0.07, size = 22, normalized size = 0.05 \begin {gather*} - \operatorname {RootSum} {\left (19683 t^{6} - 243 t^{3} + 1, \left ( t \mapsto t \log {\left (- 27 t^{2} + x \right )} \right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-RootSum(19683*_t**6 - 243*_t**3 + 1, Lambda(_t, _t*log(-27*_t**2 + x)))

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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 824 vs. \(2 (267) = 534\).
time = 5.62, size = 824, normalized size = 2.00 \begin {gather*} \frac {1}{9} \, {\left (\sqrt {3} \cos \left (\frac {4}{9} \, \pi \right )^{5} - 10 \, \sqrt {3} \cos \left (\frac {4}{9} \, \pi \right )^{3} \sin \left (\frac {4}{9} \, \pi \right )^{2} + 5 \, \sqrt {3} \cos \left (\frac {4}{9} \, \pi \right ) \sin \left (\frac {4}{9} \, \pi \right )^{4} - 5 \, \cos \left (\frac {4}{9} \, \pi \right )^{4} \sin \left (\frac {4}{9} \, \pi \right ) + 10 \, \cos \left (\frac {4}{9} \, \pi \right )^{2} \sin \left (\frac {4}{9} \, \pi \right )^{3} - \sin \left (\frac {4}{9} \, \pi \right )^{5} + 2 \, \sqrt {3} \cos \left (\frac {4}{9} \, \pi \right )^{2} - 2 \, \sqrt {3} \sin \left (\frac {4}{9} \, \pi \right )^{2} - 4 \, \cos \left (\frac {4}{9} \, \pi \right ) \sin \left (\frac {4}{9} \, \pi \right )\right )} \arctan \left (\frac {{\left (-i \, \sqrt {3} - 1\right )} \cos \left (\frac {4}{9} \, \pi \right ) + 2 \, x}{-{\left (-i \, \sqrt {3} - 1\right )} \sin \left (\frac {4}{9} \, \pi \right )}\right ) + \frac {1}{9} \, {\left (\sqrt {3} \cos \left (\frac {2}{9} \, \pi \right )^{5} - 10 \, \sqrt {3} \cos \left (\frac {2}{9} \, \pi \right )^{3} \sin \left (\frac {2}{9} \, \pi \right )^{2} + 5 \, \sqrt {3} \cos \left (\frac {2}{9} \, \pi \right ) \sin \left (\frac {2}{9} \, \pi \right )^{4} - 5 \, \cos \left (\frac {2}{9} \, \pi \right )^{4} \sin \left (\frac {2}{9} \, \pi \right ) + 10 \, \cos \left (\frac {2}{9} \, \pi \right )^{2} \sin \left (\frac {2}{9} \, \pi \right )^{3} - \sin \left (\frac {2}{9} \, \pi \right )^{5} + 2 \, \sqrt {3} \cos \left (\frac {2}{9} \, \pi \right )^{2} - 2 \, \sqrt {3} \sin \left (\frac {2}{9} \, \pi \right )^{2} - 4 \, \cos \left (\frac {2}{9} \, \pi \right ) \sin \left (\frac {2}{9} \, \pi \right )\right )} \arctan \left (\frac {{\left (-i \, \sqrt {3} - 1\right )} \cos \left (\frac {2}{9} \, \pi \right ) + 2 \, x}{-{\left (-i \, \sqrt {3} - 1\right )} \sin \left (\frac {2}{9} \, \pi \right )}\right ) - \frac {1}{9} \, {\left (\sqrt {3} \cos \left (\frac {1}{9} \, \pi \right )^{5} - 10 \, \sqrt {3} \cos \left (\frac {1}{9} \, \pi \right )^{3} \sin \left (\frac {1}{9} \, \pi \right )^{2} + 5 \, \sqrt {3} \cos \left (\frac {1}{9} \, \pi \right ) \sin \left (\frac {1}{9} \, \pi \right )^{4} + 5 \, \cos \left (\frac {1}{9} \, \pi \right )^{4} \sin \left (\frac {1}{9} \, \pi \right ) - 10 \, \cos \left (\frac {1}{9} \, \pi \right )^{2} \sin \left (\frac {1}{9} \, \pi \right )^{3} + \sin \left (\frac {1}{9} \, \pi \right )^{5} - 2 \, \sqrt {3} \cos \left (\frac {1}{9} \, \pi \right )^{2} + 2 \, \sqrt {3} \sin \left (\frac {1}{9} \, \pi \right )^{2} - 4 \, \cos \left (\frac {1}{9} \, \pi \right ) \sin \left (\frac {1}{9} \, \pi \right )\right )} \arctan \left (-\frac {{\left (-i \, \sqrt {3} - 1\right )} \cos \left (\frac {1}{9} \, \pi \right ) - 2 \, x}{-{\left (-i \, \sqrt {3} - 1\right )} \sin \left (\frac {1}{9} \, \pi \right )}\right ) + \frac {1}{18} \, {\left (5 \, \sqrt {3} \cos \left (\frac {4}{9} \, \pi \right )^{4} \sin \left (\frac {4}{9} \, \pi \right ) - 10 \, \sqrt {3} \cos \left (\frac {4}{9} \, \pi \right )^{2} \sin \left (\frac {4}{9} \, \pi \right )^{3} + \sqrt {3} \sin \left (\frac {4}{9} \, \pi \right )^{5} + \cos \left (\frac {4}{9} \, \pi \right )^{5} - 10 \, \cos \left (\frac {4}{9} \, \pi \right )^{3} \sin \left (\frac {4}{9} \, \pi \right )^{2} + 5 \, \cos \left (\frac {4}{9} \, \pi \right ) \sin \left (\frac {4}{9} \, \pi \right )^{4} + 4 \, \sqrt {3} \cos \left (\frac {4}{9} \, \pi \right ) \sin \left (\frac {4}{9} \, \pi \right ) + 2 \, \cos \left (\frac {4}{9} \, \pi \right )^{2} - 2 \, \sin \left (\frac {4}{9} \, \pi \right )^{2}\right )} \log \left ({\left (-i \, \sqrt {3} \cos \left (\frac {4}{9} \, \pi \right ) - \cos \left (\frac {4}{9} \, \pi \right )\right )} x + x^{2} + 1\right ) + \frac {1}{18} \, {\left (5 \, \sqrt {3} \cos \left (\frac {2}{9} \, \pi \right )^{4} \sin \left (\frac {2}{9} \, \pi \right ) - 10 \, \sqrt {3} \cos \left (\frac {2}{9} \, \pi \right )^{2} \sin \left (\frac {2}{9} \, \pi \right )^{3} + \sqrt {3} \sin \left (\frac {2}{9} \, \pi \right )^{5} + \cos \left (\frac {2}{9} \, \pi \right )^{5} - 10 \, \cos \left (\frac {2}{9} \, \pi \right )^{3} \sin \left (\frac {2}{9} \, \pi \right )^{2} + 5 \, \cos \left (\frac {2}{9} \, \pi \right ) \sin \left (\frac {2}{9} \, \pi \right )^{4} + 4 \, \sqrt {3} \cos \left (\frac {2}{9} \, \pi \right ) \sin \left (\frac {2}{9} \, \pi \right ) + 2 \, \cos \left (\frac {2}{9} \, \pi \right )^{2} - 2 \, \sin \left (\frac {2}{9} \, \pi \right )^{2}\right )} \log \left ({\left (-i \, \sqrt {3} \cos \left (\frac {2}{9} \, \pi \right ) - \cos \left (\frac {2}{9} \, \pi \right )\right )} x + x^{2} + 1\right ) + \frac {1}{18} \, {\left (5 \, \sqrt {3} \cos \left (\frac {1}{9} \, \pi \right )^{4} \sin \left (\frac {1}{9} \, \pi \right ) - 10 \, \sqrt {3} \cos \left (\frac {1}{9} \, \pi \right )^{2} \sin \left (\frac {1}{9} \, \pi \right )^{3} + \sqrt {3} \sin \left (\frac {1}{9} \, \pi \right )^{5} - \cos \left (\frac {1}{9} \, \pi \right )^{5} + 10 \, \cos \left (\frac {1}{9} \, \pi \right )^{3} \sin \left (\frac {1}{9} \, \pi \right )^{2} - 5 \, \cos \left (\frac {1}{9} \, \pi \right ) \sin \left (\frac {1}{9} \, \pi \right )^{4} - 4 \, \sqrt {3} \cos \left (\frac {1}{9} \, \pi \right ) \sin \left (\frac {1}{9} \, \pi \right ) + 2 \, \cos \left (\frac {1}{9} \, \pi \right )^{2} - 2 \, \sin \left (\frac {1}{9} \, \pi \right )^{2}\right )} \log \left ({\left (i \, \sqrt {3} \cos \left (\frac {1}{9} \, \pi \right ) + \cos \left (\frac {1}{9} \, \pi \right )\right )} x + x^{2} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*(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 + 2*sqrt(3)*cos(4/9*pi)^2 - 2*sqrt(
3)*sin(4/9*pi)^2 - 4*cos(4/9*pi)*sin(4/9*pi))*arctan(1/2*((-I*sqrt(3) - 1)*cos(4/9*pi) + 2*x)/((1/2*I*sqrt(3)
+ 1/2)*sin(4/9*pi))) + 1/9*(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 + 2*sqrt(3)*
cos(2/9*pi)^2 - 2*sqrt(3)*sin(2/9*pi)^2 - 4*cos(2/9*pi)*sin(2/9*pi))*arctan(1/2*((-I*sqrt(3) - 1)*cos(2/9*pi)
+ 2*x)/((1/2*I*sqrt(3) + 1/2)*sin(2/9*pi))) - 1/9*(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 - 2*sqrt(3)*cos(1/9*pi)^2 + 2*sqrt(3)*sin(1/9*pi)^2 - 4*cos(1/9*pi)*sin(1/9*pi))*arctan(-1/2*((-I*s
qrt(3) - 1)*cos(1/9*pi) - 2*x)/((1/2*I*sqrt(3) + 1/2)*sin(1/9*pi))) + 1/18*(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*sin(4/9*
pi)^2 + 5*cos(4/9*pi)*sin(4/9*pi)^4 + 4*sqrt(3)*cos(4/9*pi)*sin(4/9*pi) + 2*cos(4/9*pi)^2 - 2*sin(4/9*pi)^2)*l
og((-I*sqrt(3)*cos(4/9*pi) - cos(4/9*pi))*x + x^2 + 1) + 1/18*(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 + 4*sqrt(3)*cos(2/9*pi)*sin(2/9*pi) + 2*cos(2/9*pi)^2 - 2*sin(2/9*pi)^2)*log((-I*sqrt(3
)*cos(2/9*pi) - cos(2/9*pi))*x + x^2 + 1) + 1/18*(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 - 4*sqrt(3)*cos(1/9*pi)*sin(1/9*pi) + 2*cos(1/9*pi)^2 - 2*sin(1/9*pi)^2)*log((I*sqrt(3)*cos(1/9*pi)
+ cos(1/9*pi))*x + x^2 + 1)

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Mupad [B]
time = 2.26, size = 281, normalized size = 0.68 \begin {gather*} \frac {\ln \left (x-\frac {2^{1/3}\,3^{2/3}\,{\left (-3+\sqrt {3}\,1{}\mathrm {i}\right )}^{2/3}}{6}\right )\,{\left (-36+\sqrt {3}\,12{}\mathrm {i}\right )}^{1/3}}{18}+\frac {\ln \left (x-\frac {{\left (-36-\sqrt {3}\,12{}\mathrm {i}\right )}^{2/3}}{12}\right )\,{\left (-36-\sqrt {3}\,12{}\mathrm {i}\right )}^{1/3}}{18}-\frac {2^{2/3}\,\ln \left (x-\frac {2^{1/3}\,{\left (-3-\sqrt {3}\,1{}\mathrm {i}\right )}^{2/3}\,{\left (3^{1/3}-3^{5/6}\,1{}\mathrm {i}\right )}^2}{24}\right )\,{\left (-3-\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,\left (3^{1/3}-3^{5/6}\,1{}\mathrm {i}\right )}{36}-\frac {2^{2/3}\,\ln \left (x-\frac {2^{1/3}\,{\left (-3-\sqrt {3}\,1{}\mathrm {i}\right )}^{2/3}\,{\left (3^{1/3}+3^{5/6}\,1{}\mathrm {i}\right )}^2}{24}\right )\,{\left (-3-\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,\left (3^{1/3}+3^{5/6}\,1{}\mathrm {i}\right )}{36}-\frac {2^{2/3}\,\ln \left (x-\frac {2^{1/3}\,{\left (-3+\sqrt {3}\,1{}\mathrm {i}\right )}^{2/3}\,{\left (3^{1/3}-3^{5/6}\,1{}\mathrm {i}\right )}^2}{24}\right )\,{\left (-3+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,\left (3^{1/3}-3^{5/6}\,1{}\mathrm {i}\right )}{36}-\frac {2^{2/3}\,\ln \left (x-\frac {2^{1/3}\,{\left (-3+\sqrt {3}\,1{}\mathrm {i}\right )}^{2/3}\,{\left (3^{1/3}+3^{5/6}\,1{}\mathrm {i}\right )}^2}{24}\right )\,{\left (-3+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,\left (3^{1/3}+3^{5/6}\,1{}\mathrm {i}\right )}{36} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x*(x^3 - 1))/(x^6 - x^3 + 1),x)

[Out]

(log(x - (2^(1/3)*3^(2/3)*(3^(1/2)*1i - 3)^(2/3))/6)*(3^(1/2)*12i - 36)^(1/3))/18 + (log(x - (- 3^(1/2)*12i -
36)^(2/3)/12)*(- 3^(1/2)*12i - 36)^(1/3))/18 - (2^(2/3)*log(x - (2^(1/3)*(- 3^(1/2)*1i - 3)^(2/3)*(3^(1/3) - 3
^(5/6)*1i)^2)/24)*(- 3^(1/2)*1i - 3)^(1/3)*(3^(1/3) - 3^(5/6)*1i))/36 - (2^(2/3)*log(x - (2^(1/3)*(- 3^(1/2)*1
i - 3)^(2/3)*(3^(1/3) + 3^(5/6)*1i)^2)/24)*(- 3^(1/2)*1i - 3)^(1/3)*(3^(1/3) + 3^(5/6)*1i))/36 - (2^(2/3)*log(
x - (2^(1/3)*(3^(1/2)*1i - 3)^(2/3)*(3^(1/3) - 3^(5/6)*1i)^2)/24)*(3^(1/2)*1i - 3)^(1/3)*(3^(1/3) - 3^(5/6)*1i
))/36 - (2^(2/3)*log(x - (2^(1/3)*(3^(1/2)*1i - 3)^(2/3)*(3^(1/3) + 3^(5/6)*1i)^2)/24)*(3^(1/2)*1i - 3)^(1/3)*
(3^(1/3) + 3^(5/6)*1i))/36

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