3.1.26 \(\int \frac {x^4 (1-x^3)}{1-x^3+x^6} \, dx\) [26]

Optimal. Leaf size=382 \[ -\frac {x^2}{2}+\frac {i \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{3 \sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}-\frac {i \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{3 \sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}+\frac {i \log \left (\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} x\right )}{3 \sqrt {3} \sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}-\frac {i \log \left (\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} x\right )}{3 \sqrt {3} \sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}-\frac {i \log \left (\left (1-i \sqrt {3}\right )^{2/3}+\sqrt [3]{2 \left (1-i \sqrt {3}\right )} x+2^{2/3} x^2\right )}{3\ 2^{2/3} \sqrt {3} \sqrt [3]{1-i \sqrt {3}}}+\frac {i \log \left (\left (1+i \sqrt {3}\right )^{2/3}+\sqrt [3]{2 \left (1+i \sqrt {3}\right )} x+2^{2/3} x^2\right )}{3\ 2^{2/3} \sqrt {3} \sqrt [3]{1+i \sqrt {3}}} \]

[Out]

-1/2*x^2+1/3*I*2^(1/3)*arctan(1/3*(1+2*2^(1/3)*x/(1-I*3^(1/2))^(1/3))*3^(1/2))/(1-I*3^(1/2))^(1/3)-1/3*I*2^(1/
3)*arctan(1/3*(1+2*2^(1/3)*x/(1+I*3^(1/2))^(1/3))*3^(1/2))/(1+I*3^(1/2))^(1/3)+1/9*I*2^(1/3)*ln(-2^(1/3)*x+(1-
I*3^(1/2))^(1/3))/(1-I*3^(1/2))^(1/3)*3^(1/2)-1/18*I*ln(2^(2/3)*x^2+2^(1/3)*x*(1-I*3^(1/2))^(1/3)+(1-I*3^(1/2)
)^(2/3))*2^(1/3)/(1-I*3^(1/2))^(1/3)*3^(1/2)-1/9*I*2^(1/3)*ln(-2^(1/3)*x+(1+I*3^(1/2))^(1/3))/(1+I*3^(1/2))^(1
/3)*3^(1/2)+1/18*I*ln(2^(2/3)*x^2+2^(1/3)*x*(1+I*3^(1/2))^(1/3)+(1+I*3^(1/2))^(2/3))*2^(1/3)/(1+I*3^(1/2))^(1/
3)*3^(1/2)

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Rubi [A]
time = 0.22, antiderivative size = 382, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {1516, 12, 1389, 298, 31, 648, 631, 210, 642} \begin {gather*} \frac {i \text {ArcTan}\left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{3 \sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}-\frac {i \text {ArcTan}\left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{3 \sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}-\frac {x^2}{2}-\frac {i \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 )}{3\ 2^{2/3} \sqrt {3} \sqrt [3]{1-i \sqrt {3}}}+\frac {i \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 )}{3\ 2^{2/3} \sqrt {3} \sqrt [3]{1+i \sqrt {3}}}+\frac {i \log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}-\frac {i \log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^4*(1 - x^3))/(1 - x^3 + x^6),x]

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 1389

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

Rule 1516

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :>
Simp[e*f^(n - 1)*(f*x)^(m - n + 1)*((a + b*x^n + c*x^(2*n))^(p + 1)/(c*(m + n*(2*p + 1) + 1))), x] - Dist[f^n
/(c*(m + n*(2*p + 1) + 1)), Int[(f*x)^(m - n)*(a + b*x^n + c*x^(2*n))^p*Simp[a*e*(m - n + 1) + (b*e*(m + n*p +
1) - c*d*(m + n*(2*p + 1) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[n2, 2*n] && NeQ[b^2
- 4*a*c, 0] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*(2*p + 1) + 1, 0] && IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {x^4 \left (1-x^3\right )}{1-x^3+x^6} \, dx &=-\frac {x^2}{2}-\frac {1}{2} \int -\frac {2 x}{1-x^3+x^6} \, dx\\ &=-\frac {x^2}{2}+\int \frac {x}{1-x^3+x^6} \, dx\\ &=-\frac {x^2}{2}-\frac {i \int \frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}+x^3} \, dx}{\sqrt {3}}+\frac {i \int \frac {x}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}+x^3} \, dx}{\sqrt {3}}\\ &=-\frac {x^2}{2}+\frac {i \int \frac {1}{-\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}+x} \, dx}{3 \sqrt {3} \sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}-\frac {i \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}{3 \sqrt {3} \sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}-\frac {i \int \frac {1}{-\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}+x} \, dx}{3 \sqrt {3} \sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}+\frac {i \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}{3 \sqrt {3} \sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}\\ &=-\frac {x^2}{2}+\frac {i \log \left (\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} x\right )}{3 \sqrt {3} \sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}-\frac {i \log \left (\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} x\right )}{3 \sqrt {3} \sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}+\frac {i \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}{2 \sqrt {3}}-\frac {i \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}{2 \sqrt {3}}-\frac {i \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}{3\ 2^{2/3} \sqrt {3} \sqrt [3]{1-i \sqrt {3}}}+\frac {i \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}{3\ 2^{2/3} \sqrt {3} \sqrt [3]{1+i \sqrt {3}}}\\ &=-\frac {x^2}{2}+\frac {i \log \left (\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} x\right )}{3 \sqrt {3} \sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}-\frac {i \log \left (\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} x\right )}{3 \sqrt {3} \sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}-\frac {i \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 )}{3\ 2^{2/3} \sqrt {3} \sqrt [3]{1-i \sqrt {3}}}+\frac {i \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 )}{3\ 2^{2/3} \sqrt {3} \sqrt [3]{1+i \sqrt {3}}}-\frac {i \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 )}{\sqrt {3} \sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}+\frac {i \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 )}{\sqrt {3} \sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}\\ &=-\frac {x^2}{2}+\frac {i \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{3 \sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}-\frac {i \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}}{\sqrt {3}}\right )}{3 \sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}+\frac {i \log \left (\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} x\right )}{3 \sqrt {3} \sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}-\frac {i \log \left (\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} x\right )}{3 \sqrt {3} \sqrt [3]{\frac {1}{2} \left (1+i \sqrt {3}\right )}}-\frac {i \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 )}{3\ 2^{2/3} \sqrt {3} \sqrt [3]{1-i \sqrt {3}}}+\frac {i \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 )}{3\ 2^{2/3} \sqrt {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 = 48, normalized size = 0.13 \begin {gather*} -\frac {x^2}{2}+\frac {1}{3} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {\log (x-\text {$\#$1})}{-\text {$\#$1}+2 \text {$\#$1}^4}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^4*(1 - x^3))/(1 - x^3 + x^6),x]

[Out]

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

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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 = 0.12

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(-x^3+1)/(x^6-x^3+1),x,method=_RETURNVERBOSE)

[Out]

-1/2*x^2+1/3*sum(_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^4*(-x^3+1)/(x^6-x^3+1),x, algorithm="maxima")

[Out]

-1/2*x^2 + integrate(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. 999 vs. \(2 (246) = 492\).
time = 0.50, size = 999, normalized size = 2.62 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(-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(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)) + 2/27*18^(2/3)*12^(1/6)*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(sq
rt(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*arcta
n(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)*1
2^(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))*s
in(2/3*arctan(sqrt(3) - 2)) - 1/2*x^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(1/432*(48*18^(2/3)*12^(2/3)*sqrt(3)*x*cos(2/3*arctan(sqrt(3) -
2))^2 - 24*18^(2/3)*12^(2/3)*sqrt(3)*x - 48*(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(-192*18^(1/3)*12^(1/3)*sqrt(3)*x*cos(2/3*arc
tan(sqrt(3) - 2))*sin(2/3*arctan(sqrt(3) - 2)) + 192*18^(1/3)*12^(1/3)*x*cos(2/3*arctan(sqrt(3) - 2))^2 + 576*
x^2 - 96*18^(1/3)*12^(1/3)*x + 16*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)
) + 432*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(-1/172
8*(24*18^(2/3)*12^(2/3)*x - 1728*cos(2/3*arctan(sqrt(3) - 2))^2 - 18^(2/3)*12^(2/3)*sqrt(-384*18^(1/3)*12^(1/3
)*x*cos(2/3*arctan(sqrt(3) - 2))^2 + 576*x^2 + 192*18^(1/3)*12^(1/3)*x + 16*18^(2/3)*12^(2/3)) + 864)/(cos(2/3
*arctan(sqrt(3) - 2))*sin(2/3*arctan(sqrt(3) - 2)))) + 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(-192*18^(1/3)*12^(1/3)*sqrt(3)*x*cos(2/3*arctan(s
qrt(3) - 2))*sin(2/3*arctan(sqrt(3) - 2)) + 192*18^(1/3)*12^(1/3)*x*cos(2/3*arctan(sqrt(3) - 2))^2 + 576*x^2 -
96*18^(1/3)*12^(1/3)*x + 16*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(-384*18^(1/3)*12^(1/3)*x*cos(2/3*arctan(sqrt(3) - 2))^
2 + 576*x^2 + 192*18^(1/3)*12^(1/3)*x + 16*18^(2/3)*12^(2/3))

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Sympy [A]
time = 0.07, size = 32, normalized size = 0.08 \begin {gather*} - \frac {x^{2}}{2} - \operatorname {RootSum} {\left (19683 t^{6} + 243 t^{3} + 1, \left ( t \mapsto t \log {\left (- 6561 t^{5} - 27 t^{2} + x \right )} \right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(-x**3+1)/(x**6-x**3+1),x)

[Out]

-x**2/2 - RootSum(19683*_t**6 + 243*_t**3 + 1, Lambda(_t, _t*log(-6561*_t**5 - 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. 820 vs. \(2 (246) = 492\).
time = 3.83, size = 820, normalized size = 2.15 \begin {gather*} -\frac {1}{2} \, x^{2} - \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} - \sqrt {3} \cos \left (\frac {4}{9} \, \pi \right )^{2} + \sqrt {3} \sin \left (\frac {4}{9} \, \pi \right )^{2} + 2 \, \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} - \sqrt {3} \cos \left (\frac {2}{9} \, \pi \right )^{2} + \sqrt {3} \sin \left (\frac {2}{9} \, \pi \right )^{2} + 2 \, \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} + \sqrt {3} \cos \left (\frac {1}{9} \, \pi \right )^{2} - \sqrt {3} \sin \left (\frac {1}{9} \, \pi \right )^{2} + 2 \, \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} - 2 \, \sqrt {3} \cos \left (\frac {4}{9} \, \pi \right ) \sin \left (\frac {4}{9} \, \pi \right ) - \cos \left (\frac {4}{9} \, \pi \right )^{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} - 2 \, \sqrt {3} \cos \left (\frac {2}{9} \, \pi \right ) \sin \left (\frac {2}{9} \, \pi \right ) - \cos \left (\frac {2}{9} \, \pi \right )^{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} + 2 \, \sqrt {3} \cos \left (\frac {1}{9} \, \pi \right ) \sin \left (\frac {1}{9} \, \pi \right ) - \cos \left (\frac {1}{9} \, \pi \right )^{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^4*(-x^3+1)/(x^6-x^3+1),x, algorithm="giac")

[Out]

-1/2*x^2 - 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 - 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*x)/((1/2*I*s
qrt(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 - sqr
t(3)*cos(2/9*pi)^2 + sqrt(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*p
i) + 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 + 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*sq
rt(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*p
i)^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))*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 - 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))*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
+ 2*sqrt(3)*cos(1/9*pi)*sin(1/9*pi) - cos(1/9*pi)^2 + sin(1/9*pi)^2)*log((I*sqrt(3)*cos(1/9*pi) + cos(1/9*pi)
)*x + x^2 + 1)

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Mupad [B]
time = 2.28, size = 309, normalized size = 0.81 \begin {gather*} \frac {\ln \left (x+\left (81\,x-\frac {27\,{\left (36-\sqrt {3}\,12{}\mathrm {i}\right )}^{2/3}}{4}\right )\,\left (-\frac {1}{162}+\frac {\sqrt {3}\,1{}\mathrm {i}}{486}\right )\right )\,{\left (36-\sqrt {3}\,12{}\mathrm {i}\right )}^{1/3}}{18}+\frac {\ln \left (x-\left (81\,x-\frac {27\,{\left (36+\sqrt {3}\,12{}\mathrm {i}\right )}^{2/3}}{4}\right )\,\left (\frac {1}{162}+\frac {\sqrt {3}\,1{}\mathrm {i}}{486}\right )\right )\,{\left (36+\sqrt {3}\,12{}\mathrm {i}\right )}^{1/3}}{18}-\frac {x^2}{2}-\frac {2^{2/3}\,\ln \left (x+\frac {2^{1/3}\,3^{2/3}\,{\left (3-\sqrt {3}\,1{}\mathrm {i}\right )}^{2/3}}{12}+\frac {2^{1/3}\,3^{1/6}\,{\left (3-\sqrt {3}\,1{}\mathrm {i}\right )}^{2/3}\,1{}\mathrm {i}}{4}\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}\,3^{2/3}\,{\left (3+\sqrt {3}\,1{}\mathrm {i}\right )}^{2/3}}{12}-\frac {2^{1/3}\,3^{1/6}\,{\left (3+\sqrt {3}\,1{}\mathrm {i}\right )}^{2/3}\,1{}\mathrm {i}}{4}\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}\,3^{2/3}\,{\left (3-\sqrt {3}\,1{}\mathrm {i}\right )}^{2/3}}{6}\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}\,3^{2/3}\,{\left (3+\sqrt {3}\,1{}\mathrm {i}\right )}^{2/3}}{6}\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^4*(x^3 - 1))/(x^6 - x^3 + 1),x)

[Out]

(log(x + (81*x - (27*(36 - 3^(1/2)*12i)^(2/3))/4)*((3^(1/2)*1i)/486 - 1/162))*(36 - 3^(1/2)*12i)^(1/3))/18 + (
log(x - (81*x - (27*(3^(1/2)*12i + 36)^(2/3))/4)*((3^(1/2)*1i)/486 + 1/162))*(3^(1/2)*12i + 36)^(1/3))/18 - x^
2/2 - (2^(2/3)*log(x + (2^(1/3)*3^(2/3)*(3 - 3^(1/2)*1i)^(2/3))/12 + (2^(1/3)*3^(1/6)*(3 - 3^(1/2)*1i)^(2/3)*1
i)/4)*(3 - 3^(1/2)*1i)^(1/3)*(3^(1/3) + 3^(5/6)*1i))/36 - (2^(2/3)*log(x + (2^(1/3)*3^(2/3)*(3^(1/2)*1i + 3)^(
2/3))/12 - (2^(1/3)*3^(1/6)*(3^(1/2)*1i + 3)^(2/3)*1i)/4)*(3^(1/2)*1i + 3)^(1/3)*(3^(1/3) - 3^(5/6)*1i))/36 -
(2^(2/3)*log(x - (2^(1/3)*3^(2/3)*(3 - 3^(1/2)*1i)^(2/3))/6)*(3 - 3^(1/2)*1i)^(1/3)*(3^(1/3) - 3^(5/6)*1i))/36
- (2^(2/3)*log(x - (2^(1/3)*3^(2/3)*(3^(1/2)*1i + 3)^(2/3))/6)*(3^(1/2)*1i + 3)^(1/3)*(3^(1/3) + 3^(5/6)*1i))
/36

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