### 3.25 $$\int \frac{x^6 (1-x^3)}{1-x^3+x^6} \, dx$$

Optimal. Leaf size=418 $-\frac{x^4}{4}-\frac{\left (3+i \sqrt{3}\right ) \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1-i \sqrt{3}\right )} x+\left (1-i \sqrt{3}\right )^{2/3}\right )}{18 \sqrt [3]{2} \left (1-i \sqrt{3}\right )^{2/3}}-\frac{\left (3-i \sqrt{3}\right ) \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1+i \sqrt{3}\right )} x+\left (1+i \sqrt{3}\right )^{2/3}\right )}{18 \sqrt [3]{2} \left (1+i \sqrt{3}\right )^{2/3}}+\frac{\left (3+i \sqrt{3}\right ) \log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt{3}}\right )}{9 \sqrt [3]{2} \left (1-i \sqrt{3}\right )^{2/3}}+\frac{\left (3-i \sqrt{3}\right ) \log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt{3}}\right )}{9 \sqrt [3]{2} \left (1+i \sqrt{3}\right )^{2/3}}-\frac{\left (\sqrt{3}+i\right ) \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1-i \sqrt{3}\right )}}}{\sqrt{3}}\right )}{3 \sqrt [3]{2} \left (1-i \sqrt{3}\right )^{2/3}}+\frac{\left (-\sqrt{3}+i\right ) \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1+i \sqrt{3}\right )}}}{\sqrt{3}}\right )}{3 \sqrt [3]{2} \left (1+i \sqrt{3}\right )^{2/3}}$

[Out]

-x^4/4 - ((I + Sqrt[3])*ArcTan[(1 + (2*x)/((1 - I*Sqrt[3])/2)^(1/3))/Sqrt[3]])/(3*2^(1/3)*(1 - I*Sqrt[3])^(2/3
)) + ((I - Sqrt[3])*ArcTan[(1 + (2*x)/((1 + I*Sqrt[3])/2)^(1/3))/Sqrt[3]])/(3*2^(1/3)*(1 + I*Sqrt[3])^(2/3)) +
((3 + I*Sqrt[3])*Log[(1 - I*Sqrt[3])^(1/3) - 2^(1/3)*x])/(9*2^(1/3)*(1 - I*Sqrt[3])^(2/3)) + ((3 - I*Sqrt[3])
*Log[(1 + I*Sqrt[3])^(1/3) - 2^(1/3)*x])/(9*2^(1/3)*(1 + I*Sqrt[3])^(2/3)) - ((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^(1/3)*(1 - I*Sqrt[3])^(2/3)) - ((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^(1/3)*(1 + I*Sqrt[3])^(2/3))

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Rubi [A]  time = 0.537566, antiderivative size = 418, 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, 1374, 200, 31, 634, 617, 204, 628} $-\frac{x^4}{4}-\frac{\left (3+i \sqrt{3}\right ) \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1-i \sqrt{3}\right )} x+\left (1-i \sqrt{3}\right )^{2/3}\right )}{18 \sqrt [3]{2} \left (1-i \sqrt{3}\right )^{2/3}}-\frac{\left (3-i \sqrt{3}\right ) \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1+i \sqrt{3}\right )} x+\left (1+i \sqrt{3}\right )^{2/3}\right )}{18 \sqrt [3]{2} \left (1+i \sqrt{3}\right )^{2/3}}+\frac{\left (3+i \sqrt{3}\right ) \log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt{3}}\right )}{9 \sqrt [3]{2} \left (1-i \sqrt{3}\right )^{2/3}}+\frac{\left (3-i \sqrt{3}\right ) \log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt{3}}\right )}{9 \sqrt [3]{2} \left (1+i \sqrt{3}\right )^{2/3}}-\frac{\left (\sqrt{3}+i\right ) \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1-i \sqrt{3}\right )}}}{\sqrt{3}}\right )}{3 \sqrt [3]{2} \left (1-i \sqrt{3}\right )^{2/3}}+\frac{\left (-\sqrt{3}+i\right ) \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1+i \sqrt{3}\right )}}}{\sqrt{3}}\right )}{3 \sqrt [3]{2} \left (1+i \sqrt{3}\right )^{2/3}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-x^4/4 - ((I + Sqrt[3])*ArcTan[(1 + (2*x)/((1 - I*Sqrt[3])/2)^(1/3))/Sqrt[3]])/(3*2^(1/3)*(1 - I*Sqrt[3])^(2/3
)) + ((I - Sqrt[3])*ArcTan[(1 + (2*x)/((1 + I*Sqrt[3])/2)^(1/3))/Sqrt[3]])/(3*2^(1/3)*(1 + I*Sqrt[3])^(2/3)) +
((3 + I*Sqrt[3])*Log[(1 - I*Sqrt[3])^(1/3) - 2^(1/3)*x])/(9*2^(1/3)*(1 - I*Sqrt[3])^(2/3)) + ((3 - I*Sqrt[3])
*Log[(1 + I*Sqrt[3])^(1/3) - 2^(1/3)*x])/(9*2^(1/3)*(1 + I*Sqrt[3])^(2/3)) - ((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^(1/3)*(1 - I*Sqrt[3])^(2/3)) - ((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^(1/3)*(1 + I*Sqrt[3])^(2/3))

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 1374

Int[((d_.)*(x_))^(m_)/((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}
, Dist[(d^n*(b/q + 1))/2, Int[(d*x)^(m - n)/(b/2 + q/2 + c*x^n), x], x] - Dist[(d^n*(b/q - 1))/2, Int[(d*x)^(m
- n)/(b/2 - q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n,
0] && GeQ[m, n]

Rule 200

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 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^6 \left (1-x^3\right )}{1-x^3+x^6} \, dx &=-\frac{x^4}{4}-\frac{1}{4} \int -\frac{4 x^3}{1-x^3+x^6} \, dx\\ &=-\frac{x^4}{4}+\int \frac{x^3}{1-x^3+x^6} \, dx\\ &=-\frac{x^4}{4}-\frac{1}{6} \left (-3+i \sqrt{3}\right ) \int \frac{1}{-\frac{1}{2}-\frac{i \sqrt{3}}{2}+x^3} \, dx+\frac{1}{6} \left (3+i \sqrt{3}\right ) \int \frac{1}{-\frac{1}{2}+\frac{i \sqrt{3}}{2}+x^3} \, dx\\ &=-\frac{x^4}{4}+\frac{\left (3-i \sqrt{3}\right ) \int \frac{1}{-\sqrt [3]{\frac{1}{2} \left (1+i \sqrt{3}\right )}+x} \, dx}{9 \sqrt [3]{2} \left (1+i \sqrt{3}\right )^{2/3}}+\frac{\left (3-i \sqrt{3}\right ) \int \frac{-2^{2/3} \sqrt [3]{1+i \sqrt{3}}-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 \sqrt [3]{2} \left (1+i \sqrt{3}\right )^{2/3}}+\frac{\left (3+i \sqrt{3}\right ) \int \frac{1}{-\sqrt [3]{\frac{1}{2} \left (1-i \sqrt{3}\right )}+x} \, dx}{9 \sqrt [3]{2} \left (1-i \sqrt{3}\right )^{2/3}}+\frac{\left (3+i \sqrt{3}\right ) \int \frac{-2^{2/3} \sqrt [3]{1-i \sqrt{3}}-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 \sqrt [3]{2} \left (1-i \sqrt{3}\right )^{2/3}}\\ &=-\frac{x^4}{4}+\frac{\left (3+i \sqrt{3}\right ) \log \left (\sqrt [3]{1-i \sqrt{3}}-\sqrt [3]{2} x\right )}{9 \sqrt [3]{2} \left (1-i \sqrt{3}\right )^{2/3}}+\frac{\left (3-i \sqrt{3}\right ) \log \left (\sqrt [3]{1+i \sqrt{3}}-\sqrt [3]{2} x\right )}{9 \sqrt [3]{2} \left (1+i \sqrt{3}\right )^{2/3}}-\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 \sqrt [3]{2} \left (1+i \sqrt{3}\right )^{2/3}}-\frac{\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}{6\ 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 \sqrt [3]{2} \left (1-i \sqrt{3}\right )^{2/3}}-\frac{\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}{6\ 2^{2/3} \sqrt [3]{1-i \sqrt{3}}}\\ &=-\frac{x^4}{4}+\frac{\left (3+i \sqrt{3}\right ) \log \left (\sqrt [3]{1-i \sqrt{3}}-\sqrt [3]{2} x\right )}{9 \sqrt [3]{2} \left (1-i \sqrt{3}\right )^{2/3}}+\frac{\left (3-i \sqrt{3}\right ) \log \left (\sqrt [3]{1+i \sqrt{3}}-\sqrt [3]{2} x\right )}{9 \sqrt [3]{2} \left (1+i \sqrt{3}\right )^{2/3}}-\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 \sqrt [3]{2} \left (1-i \sqrt{3}\right )^{2/3}}-\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 \sqrt [3]{2} \left (1+i \sqrt{3}\right )^{2/3}}+\frac{\left (3-i \sqrt{3}\right ) \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 )}{3 \sqrt [3]{2} \left (1+i \sqrt{3}\right )^{2/3}}+\frac{\left (3+i \sqrt{3}\right ) \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 )}{3 \sqrt [3]{2} \left (1-i \sqrt{3}\right )^{2/3}}\\ &=-\frac{x^4}{4}-\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 \sqrt [3]{2} \left (1-i \sqrt{3}\right )^{2/3}}+\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 \sqrt [3]{2} \left (1+i \sqrt{3}\right )^{2/3}}+\frac{\left (3+i \sqrt{3}\right ) \log \left (\sqrt [3]{1-i \sqrt{3}}-\sqrt [3]{2} x\right )}{9 \sqrt [3]{2} \left (1-i \sqrt{3}\right )^{2/3}}+\frac{\left (3-i \sqrt{3}\right ) \log \left (\sqrt [3]{1+i \sqrt{3}}-\sqrt [3]{2} x\right )}{9 \sqrt [3]{2} \left (1+i \sqrt{3}\right )^{2/3}}-\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 \sqrt [3]{2} \left (1-i \sqrt{3}\right )^{2/3}}-\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 \sqrt [3]{2} \left (1+i \sqrt{3}\right )^{2/3}}\\ \end{align*}

Mathematica [C]  time = 0.0113329, size = 47, normalized size = 0.11 $\frac{1}{3} \text{RootSum}\left [\text{\#1}^6-\text{\#1}^3+1\& ,\frac{\text{\#1} \log (x-\text{\#1})}{2 \text{\#1}^3-1}\& \right ]-\frac{x^4}{4}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [C]  time = 0.006, size = 46, normalized size = 0.1 \begin{align*} -{\frac{{x}^{4}}{4}}+{\frac{1}{3}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}-{{\it \_Z}}^{3}+1 \right ) }{\frac{{{\it \_R}}^{3}\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^6*(-x^3+1)/(x^6-x^3+1),x)

[Out]

-1/4*x^4+1/3*sum(_R^3/(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}{4} \, x^{4} + \int \frac{x^{3}}{x^{6} - x^{3} + 1}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.60718, size = 3918, normalized size = 9.37 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*x^4 + 1/54*18^(2/3)*12^(1/6)*cos(2/3*arctan(sqrt(3) + 2))*log(2*18^(2/3)*12^(1/6)*sqrt(3)*x*sin(2/3*arcta
n(sqrt(3) + 2)) + 3*18^(1/3)*12^(1/3)*cos(2/3*arctan(sqrt(3) + 2))^2 + 3*18^(1/3)*12^(1/3)*sin(2/3*arctan(sqrt
(3) + 2))^2 + 18*x^2) + 2/27*18^(2/3)*12^(1/6)*arctan(1/216*(18^(1/3)*12^(5/6)*sqrt(3)*sqrt(2)*sqrt(2*18^(2/3)
*12^(1/6)*sqrt(3)*x*sin(2/3*arctan(sqrt(3) + 2)) + 3*18^(1/3)*12^(1/3)*cos(2/3*arctan(sqrt(3) + 2))^2 + 3*18^(
1/3)*12^(1/3)*sin(2/3*arctan(sqrt(3) + 2))^2 + 18*x^2) - 6*18^(1/3)*12^(5/6)*sqrt(3)*x - 216*sin(2/3*arctan(sq
rt(3) + 2)))/cos(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(-1/108*(6*18^(1/3)*12^(5/6)*
sqrt(3)*x*cos(2/3*arctan(sqrt(3) + 2)) + 108*sqrt(3)*cos(2/3*arctan(sqrt(3) + 2))^2 + 108*sqrt(3)*sin(2/3*arct
an(sqrt(3) + 2))^2 - 18*(18^(1/3)*12^(5/6)*x + 24*cos(2/3*arctan(sqrt(3) + 2)))*sin(2/3*arctan(sqrt(3) + 2)) -
sqrt(-18^(2/3)*12^(1/6)*sqrt(3)*x*sin(2/3*arctan(sqrt(3) + 2)) + 3*18^(2/3)*12^(1/6)*x*cos(2/3*arctan(sqrt(3)
+ 2)) + 3*18^(1/3)*12^(1/3)*cos(2/3*arctan(sqrt(3) + 2))^2 + 3*18^(1/3)*12^(1/3)*sin(2/3*arctan(sqrt(3) + 2))
^2 + 18*x^2)*(18^(1/3)*12^(5/6)*sqrt(3)*sqrt(2)*cos(2/3*arctan(sqrt(3) + 2)) - 3*18^(1/3)*12^(5/6)*sqrt(2)*sin
(2/3*arctan(sqrt(3) + 2))))/(cos(2/3*arctan(sqrt(3) + 2))^2 - 3*sin(2/3*arctan(sqrt(3) + 2))^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(1/1
08*(6*18^(1/3)*12^(5/6)*sqrt(3)*x*cos(2/3*arctan(sqrt(3) + 2)) - 108*sqrt(3)*cos(2/3*arctan(sqrt(3) + 2))^2 -
108*sqrt(3)*sin(2/3*arctan(sqrt(3) + 2))^2 + 18*(18^(1/3)*12^(5/6)*x - 24*cos(2/3*arctan(sqrt(3) + 2)))*sin(2/
3*arctan(sqrt(3) + 2)) - sqrt(-18^(2/3)*12^(1/6)*sqrt(3)*x*sin(2/3*arctan(sqrt(3) + 2)) - 3*18^(2/3)*12^(1/6)*
x*cos(2/3*arctan(sqrt(3) + 2)) + 3*18^(1/3)*12^(1/3)*cos(2/3*arctan(sqrt(3) + 2))^2 + 3*18^(1/3)*12^(1/3)*sin(
2/3*arctan(sqrt(3) + 2))^2 + 18*x^2)*(18^(1/3)*12^(5/6)*sqrt(3)*sqrt(2)*cos(2/3*arctan(sqrt(3) + 2)) + 3*18^(1
/3)*12^(5/6)*sqrt(2)*sin(2/3*arctan(sqrt(3) + 2))))/(cos(2/3*arctan(sqrt(3) + 2))^2 - 3*sin(2/3*arctan(sqrt(3)
+ 2))^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^(1/6)*sqrt(3)*x*sin(2/3*arctan(sqrt(3) + 2)) + 3*18^(2/3)*12^(1/6)*x*cos(2/3*a
rctan(sqrt(3) + 2)) + 3*18^(1/3)*12^(1/3)*cos(2/3*arctan(sqrt(3) + 2))^2 + 3*18^(1/3)*12^(1/3)*sin(2/3*arctan(
sqrt(3) + 2))^2 + 18*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^(1/6)*sqrt(3)*x*sin(2/3*arctan(sqrt(3) + 2)) - 3*18^(2/3)*12^(1
/6)*x*cos(2/3*arctan(sqrt(3) + 2)) + 3*18^(1/3)*12^(1/3)*cos(2/3*arctan(sqrt(3) + 2))^2 + 3*18^(1/3)*12^(1/3)*
sin(2/3*arctan(sqrt(3) + 2))^2 + 18*x^2)

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Sympy [A]  time = 0.176743, size = 31, normalized size = 0.07 \begin{align*} - \frac{x^{4}}{4} - \operatorname{RootSum}{\left (19683 t^{6} - 243 t^{3} + 1, \left ( t \mapsto t \log{\left (- 1458 t^{4} + 9 t + x \right )} \right )\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x**4/4 - RootSum(19683*_t**6 - 243*_t**3 + 1, Lambda(_t, _t*log(-1458*_t**4 + 9*_t + x)))

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Giac [B]  time = 1.1675, size = 867, normalized size = 2.07 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*x^4 - 1/9*(2*sqrt(3)*cos(4/9*pi)^4 - 12*sqrt(3)*cos(4/9*pi)^2*sin(4/9*pi)^2 + 2*sqrt(3)*sin(4/9*pi)^4 + 8
*cos(4/9*pi)^3*sin(4/9*pi) - 8*cos(4/9*pi)*sin(4/9*pi)^3 + sqrt(3)*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*(2*sqrt(3)*cos(2/9*pi)^4 - 12*sqrt(3)*cos(2/9
*pi)^2*sin(2/9*pi)^2 + 2*sqrt(3)*sin(2/9*pi)^4 + 8*cos(2/9*pi)^3*sin(2/9*pi) - 8*cos(2/9*pi)*sin(2/9*pi)^3 + s
qrt(3)*cos(2/9*pi) + sin(2/9*pi))*arctan(-((sqrt(3)*i + 1)*cos(2/9*pi) - 2*x)/((sqrt(3)*i + 1)*sin(2/9*pi))) -
1/9*(2*sqrt(3)*cos(1/9*pi)^4 - 12*sqrt(3)*cos(1/9*pi)^2*sin(1/9*pi)^2 + 2*sqrt(3)*sin(1/9*pi)^4 - 8*cos(1/9*p
i)^3*sin(1/9*pi) + 8*cos(1/9*pi)*sin(1/9*pi)^3 - sqrt(3)*cos(1/9*pi) + sin(1/9*pi))*arctan(((sqrt(3)*i + 1)*co
s(1/9*pi) + 2*x)/((sqrt(3)*i + 1)*sin(1/9*pi))) - 1/18*(8*sqrt(3)*cos(4/9*pi)^3*sin(4/9*pi) - 8*sqrt(3)*cos(4/
9*pi)*sin(4/9*pi)^3 - 2*cos(4/9*pi)^4 + 12*cos(4/9*pi)^2*sin(4/9*pi)^2 - 2*sin(4/9*pi)^4 + sqrt(3)*sin(4/9*pi)
- cos(4/9*pi))*log(-(sqrt(3)*i*cos(4/9*pi) + cos(4/9*pi))*x + x^2 + 1) - 1/18*(8*sqrt(3)*cos(2/9*pi)^3*sin(2/
9*pi) - 8*sqrt(3)*cos(2/9*pi)*sin(2/9*pi)^3 - 2*cos(2/9*pi)^4 + 12*cos(2/9*pi)^2*sin(2/9*pi)^2 - 2*sin(2/9*pi)
^4 + sqrt(3)*sin(2/9*pi) - cos(2/9*pi))*log(-(sqrt(3)*i*cos(2/9*pi) + cos(2/9*pi))*x + x^2 + 1) + 1/18*(8*sqrt
(3)*cos(1/9*pi)^3*sin(1/9*pi) - 8*sqrt(3)*cos(1/9*pi)*sin(1/9*pi)^3 + 2*cos(1/9*pi)^4 - 12*cos(1/9*pi)^2*sin(1
/9*pi)^2 + 2*sin(1/9*pi)^4 - sqrt(3)*sin(1/9*pi) - cos(1/9*pi))*log((sqrt(3)*i*cos(1/9*pi) + cos(1/9*pi))*x +
x^2 + 1)