### 3.26 $$\int \frac{x^4 (1-x^3)}{1-x^3+x^6} \, dx$$

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

[Out]

-x^2/2 + ((I/3)*ArcTan[(1 + (2*x)/((1 - I*Sqrt[3])/2)^(1/3))/Sqrt[3]])/((1 - I*Sqrt[3])/2)^(1/3) - ((I/3)*ArcT
an[(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))

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Rubi [A]  time = 0.327587, antiderivative size = 382, normalized size of antiderivative = 1., 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 = {1502, 12, 1375, 292, 31, 634, 617, 204, 628} $-\frac{x^2}{2}-\frac{i \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1-i \sqrt{3}\right )} x+\left (1-i \sqrt{3}\right )^{2/3}\right )}{3\ 2^{2/3} \sqrt{3} \sqrt [3]{1-i \sqrt{3}}}+\frac{i \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1+i \sqrt{3}\right )} x+\left (1+i \sqrt{3}\right )^{2/3}\right )}{3\ 2^{2/3} \sqrt{3} \sqrt [3]{1+i \sqrt{3}}}+\frac{i \log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt{3}}\right )}{3 \sqrt{3} \sqrt [3]{\frac{1}{2} \left (1-i \sqrt{3}\right )}}-\frac{i \log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt{3}}\right )}{3 \sqrt{3} \sqrt [3]{\frac{1}{2} \left (1+i \sqrt{3}\right )}}+\frac{i \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1-i \sqrt{3}\right )}}}{\sqrt{3}}\right )}{3 \sqrt [3]{\frac{1}{2} \left (1-i \sqrt{3}\right )}}-\frac{i \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1+i \sqrt{3}\right )}}}{\sqrt{3}}\right )}{3 \sqrt [3]{\frac{1}{2} \left (1+i \sqrt{3}\right )}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-x^2/2 + ((I/3)*ArcTan[(1 + (2*x)/((1 - I*Sqrt[3])/2)^(1/3))/Sqrt[3]])/((1 - I*Sqrt[3])/2)^(1/3) - ((I/3)*ArcT
an[(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 1502

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]

Rule 12

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

Rule 1375

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 292

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 31

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

Rule 634

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 617

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 204

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

Rule 628

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]

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 \operatorname{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 \operatorname{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*}

Mathematica [C]  time = 0.0135111, size = 48, normalized size = 0.13 $\frac{1}{3} \text{RootSum}\left [\text{\#1}^6-\text{\#1}^3+1\& ,\frac{\log (x-\text{\#1})}{2 \text{\#1}^4-\text{\#1}}\& \right ]-\frac{x^2}{2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [C]  time = 0.004, size = 44, normalized size = 0.1 \begin{align*} -{\frac{{x}^{2}}{2}}+{\frac{1}{3}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}-{{\it \_Z}}^{3}+1 \right ) }{\frac{{\it \_R}\,\ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{5}-{{\it \_R}}^{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[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., size = 0, normalized size = 0. \begin{align*} -\frac{1}{2} \, x^{2} + \int \frac{x}{x^{6} - x^{3} + 1}\,{d x} \end{align*}

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]  time = 1.94217, size = 5949, normalized size = 15.57 \begin{align*} \text{result too large to display} \end{align*}

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(18^(2/3)*12^(2/3)*cos(2/3*arctan(sqrt(3) - 2))^4 + 18^
(2/3)*12^(2/3)*sin(2/3*arctan(sqrt(3) - 2))^4 + 12*18^(1/3)*12^(1/3)*sqrt(3)*x*cos(2/3*arctan(sqrt(3) - 2))*si
n(2/3*arctan(sqrt(3) - 2)) + 6*18^(1/3)*12^(1/3)*x*cos(2/3*arctan(sqrt(3) - 2))^2 + 2*(18^(2/3)*12^(2/3)*cos(2
/3*arctan(sqrt(3) - 2))^2 - 3*18^(1/3)*12^(1/3)*x)*sin(2/3*arctan(sqrt(3) - 2))^2 + 36*x^2) - 2/27*18^(2/3)*12
^(1/6)*arctan(1/108*(6*18^(2/3)*12^(2/3)*sqrt(3)*x*cos(2/3*arctan(sqrt(3) - 2))^2 + 108*sqrt(3)*cos(2/3*arctan
(sqrt(3) - 2))^4 + 108*sqrt(3)*sin(2/3*arctan(sqrt(3) - 2))^4 - 864*cos(2/3*arctan(sqrt(3) - 2))*sin(2/3*arcta
n(sqrt(3) - 2))^3 - 6*(18^(2/3)*12^(2/3)*sqrt(3)*x - 36*sqrt(3)*cos(2/3*arctan(sqrt(3) - 2))^2)*sin(2/3*arctan
(sqrt(3) - 2))^2 + 12*(18^(2/3)*12^(2/3)*x*cos(2/3*arctan(sqrt(3) - 2)) + 72*cos(2/3*arctan(sqrt(3) - 2))^3)*s
in(2/3*arctan(sqrt(3) - 2)) - sqrt(18^(2/3)*12^(2/3)*cos(2/3*arctan(sqrt(3) - 2))^4 + 18^(2/3)*12^(2/3)*sin(2/
3*arctan(sqrt(3) - 2))^4 + 12*18^(1/3)*12^(1/3)*sqrt(3)*x*cos(2/3*arctan(sqrt(3) - 2))*sin(2/3*arctan(sqrt(3)
- 2)) + 6*18^(1/3)*12^(1/3)*x*cos(2/3*arctan(sqrt(3) - 2))^2 + 2*(18^(2/3)*12^(2/3)*cos(2/3*arctan(sqrt(3) - 2
))^2 - 3*18^(1/3)*12^(1/3)*x)*sin(2/3*arctan(sqrt(3) - 2))^2 + 36*x^2)*(18^(2/3)*12^(2/3)*sqrt(3)*cos(2/3*arct
an(sqrt(3) - 2))^2 - 18^(2/3)*12^(2/3)*sqrt(3)*sin(2/3*arctan(sqrt(3) - 2))^2 + 2*18^(2/3)*12^(2/3)*cos(2/3*ar
ctan(sqrt(3) - 2))*sin(2/3*arctan(sqrt(3) - 2))))/(3*cos(2/3*arctan(sqrt(3) - 2))^4 - 10*cos(2/3*arctan(sqrt(3
) - 2))^2*sin(2/3*arctan(sqrt(3) - 2))^2 + 3*sin(2/3*arctan(sqrt(3) - 2))^4))*sin(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(1/108*(6*18^(2/3)*12^(2/3)*sqrt(3)*x*cos(2/3*arctan(sqrt(3) - 2))^2 + 108*sqrt(3)*cos(2/3*arc
tan(sqrt(3) - 2))^4 + 108*sqrt(3)*sin(2/3*arctan(sqrt(3) - 2))^4 + 864*cos(2/3*arctan(sqrt(3) - 2))*sin(2/3*ar
ctan(sqrt(3) - 2))^3 - 6*(18^(2/3)*12^(2/3)*sqrt(3)*x - 36*sqrt(3)*cos(2/3*arctan(sqrt(3) - 2))^2)*sin(2/3*arc
tan(sqrt(3) - 2))^2 - 12*(18^(2/3)*12^(2/3)*x*cos(2/3*arctan(sqrt(3) - 2)) + 72*cos(2/3*arctan(sqrt(3) - 2))^3
)*sin(2/3*arctan(sqrt(3) - 2)) - sqrt(18^(2/3)*12^(2/3)*cos(2/3*arctan(sqrt(3) - 2))^4 + 18^(2/3)*12^(2/3)*sin
(2/3*arctan(sqrt(3) - 2))^4 - 12*18^(1/3)*12^(1/3)*sqrt(3)*x*cos(2/3*arctan(sqrt(3) - 2))*sin(2/3*arctan(sqrt(
3) - 2)) + 6*18^(1/3)*12^(1/3)*x*cos(2/3*arctan(sqrt(3) - 2))^2 + 2*(18^(2/3)*12^(2/3)*cos(2/3*arctan(sqrt(3)
- 2))^2 - 3*18^(1/3)*12^(1/3)*x)*sin(2/3*arctan(sqrt(3) - 2))^2 + 36*x^2)*(18^(2/3)*12^(2/3)*sqrt(3)*cos(2/3*a
rctan(sqrt(3) - 2))^2 - 18^(2/3)*12^(2/3)*sqrt(3)*sin(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))))/(3*cos(2/3*arctan(sqrt(3) - 2))^4 - 10*cos(2/3*arctan(sqr
t(3) - 2))^2*sin(2/3*arctan(sqrt(3) - 2))^2 + 3*sin(2/3*arctan(sqrt(3) - 2))^4)) + 1/27*(18^(2/3)*12^(1/6)*sqr
t(3)*cos(2/3*arctan(sqrt(3) - 2)) - 18^(2/3)*12^(1/6)*sin(2/3*arctan(sqrt(3) - 2)))*arctan(-1/432*(6*18^(2/3)*
12^(2/3)*x - 216*cos(2/3*arctan(sqrt(3) - 2))^2 + 216*sin(2/3*arctan(sqrt(3) - 2))^2 - 18^(2/3)*12^(2/3)*sqrt(
18^(2/3)*12^(2/3)*cos(2/3*arctan(sqrt(3) - 2))^4 + 18^(2/3)*12^(2/3)*sin(2/3*arctan(sqrt(3) - 2))^4 - 12*18^(1
/3)*12^(1/3)*x*cos(2/3*arctan(sqrt(3) - 2))^2 + 2*(18^(2/3)*12^(2/3)*cos(2/3*arctan(sqrt(3) - 2))^2 + 6*18^(1/
3)*12^(1/3)*x)*sin(2/3*arctan(sqrt(3) - 2))^2 + 36*x^2))/(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(sqr
t(3) - 2)))*log(18^(2/3)*12^(2/3)*cos(2/3*arctan(sqrt(3) - 2))^4 + 18^(2/3)*12^(2/3)*sin(2/3*arctan(sqrt(3) -
2))^4 - 12*18^(1/3)*12^(1/3)*sqrt(3)*x*cos(2/3*arctan(sqrt(3) - 2))*sin(2/3*arctan(sqrt(3) - 2)) + 6*18^(1/3)*
12^(1/3)*x*cos(2/3*arctan(sqrt(3) - 2))^2 + 2*(18^(2/3)*12^(2/3)*cos(2/3*arctan(sqrt(3) - 2))^2 - 3*18^(1/3)*1
2^(1/3)*x)*sin(2/3*arctan(sqrt(3) - 2))^2 + 36*x^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(18^(2/3)*12^(2/3)*cos(2/3*arctan(sqrt(3) - 2))^4 +
18^(2/3)*12^(2/3)*sin(2/3*arctan(sqrt(3) - 2))^4 - 12*18^(1/3)*12^(1/3)*x*cos(2/3*arctan(sqrt(3) - 2))^2 + 2*
(18^(2/3)*12^(2/3)*cos(2/3*arctan(sqrt(3) - 2))^2 + 6*18^(1/3)*12^(1/3)*x)*sin(2/3*arctan(sqrt(3) - 2))^2 + 36
*x^2)

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Sympy [A]  time = 0.193264, size = 32, normalized size = 0.08 \begin{align*} - \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{align*}

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]  time = 1.15957, size = 1103, normalized size = 2.89 \begin{align*} \text{result too large to display} \end{align*}

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(-((sqrt(3)*i + 1)*cos(4/9*pi) - 2*x)/((sqrt(3)*i +
1)*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 - sqrt(3)*cos(2
/9*pi)^2 + sqrt(3)*sin(2/9*pi)^2 + 2*cos(2/9*pi)*sin(2/9*pi))*arctan(-((sqrt(3)*i + 1)*cos(2/9*pi) - 2*x)/((sq
rt(3)*i + 1)*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)*c
os(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(((sqrt(3)*i + 1)*cos(1/9*pi) + 2
*x)/((sqrt(3)*i + 1)*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 - 2*sqrt(3)*cos(4/9*pi)*sin(4/9*pi) - cos(4/9*pi)^2 + sin(4/9*pi)^2)*log(-(sqrt(3)*i*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)*co
s(2/9*pi)*sin(2/9*pi) - cos(2/9*pi)^2 + sin(2/9*pi)^2)*log(-(sqrt(3)*i*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((sqrt(3)*i*cos(1/9*pi) + cos(1/9*pi))*x + x^2 + 1)