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

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

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Rubi [A]  time = 0.28, antiderivative size = 411, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1422, 200, 31, 634, 617, 204, 628} \begin {gather*} \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}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(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^(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 31

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

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 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 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 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]

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 1422

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

Rubi steps

\begin {align*} \int \frac {1-x^3}{1-x^3+x^6} \, 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 {1}{6} \left (3+i \sqrt {3}\right ) \int \frac {1}{-\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 \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 {\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 {\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 {\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*}

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Mathematica [C]  time = 0.01, size = 57, normalized size = 0.14 \begin {gather*} -\frac {1}{3} \text {RootSum}\left [\text {$\#$1}^6-\text {$\#$1}^3+1\&,\frac {\text {$\#$1}^3 \log (x-\text {$\#$1})-\log (x-\text {$\#$1})}{2 \text {$\#$1}^5-\text {$\#$1}^2}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(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 + 2*#1^5) & ]

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1-x^3}{1-x^3+x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 1.27, size = 1031, normalized size = 2.51

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-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^(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) - 2/27*18^(2/3)*12^(1/6)*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*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(sq
rt(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*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)*(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))*
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*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*a
rctan(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(1/216*(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(sqrt(3) + 2)))/cos(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(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) - 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*arc
tan(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*arc
tan(sqrt(3) + 2))^2 + 18*x^2)

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giac [B]  time = 0.72, size = 637, normalized size = 1.55

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*(sqrt(3)*cos(4/9*pi)^4 - 6*sqrt(3)*cos(4/9*pi)^2*sin(4/9*pi)^2 + sqrt(3)*sin(4/9*pi)^4 + 4*cos(4/9*pi)^3*s
in(4/9*pi) - 4*cos(4/9*pi)*sin(4/9*pi)^3 + 2*sqrt(3)*cos(4/9*pi) + 2*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)^4 - 6*sqrt(3)*cos(2/9*pi)^2*sin(2/9*
pi)^2 + sqrt(3)*sin(2/9*pi)^4 + 4*cos(2/9*pi)^3*sin(2/9*pi) - 4*cos(2/9*pi)*sin(2/9*pi)^3 + 2*sqrt(3)*cos(2/9*
pi) + 2*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*(sqrt(3)
*cos(1/9*pi)^4 - 6*sqrt(3)*cos(1/9*pi)^2*sin(1/9*pi)^2 + sqrt(3)*sin(1/9*pi)^4 - 4*cos(1/9*pi)^3*sin(1/9*pi) +
4*cos(1/9*pi)*sin(1/9*pi)^3 - 2*sqrt(3)*cos(1/9*pi) + 2*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*(4*sqrt(3)*cos(4/9*pi)^3*sin(4/9*pi) - 4*sqrt(3)*cos(4/9*pi)*sin(4/9*
pi)^3 - cos(4/9*pi)^4 + 6*cos(4/9*pi)^2*sin(4/9*pi)^2 - sin(4/9*pi)^4 + 2*sqrt(3)*sin(4/9*pi) - 2*cos(4/9*pi))
*log(-(sqrt(3)*i*cos(4/9*pi) + cos(4/9*pi))*x + x^2 + 1) + 1/18*(4*sqrt(3)*cos(2/9*pi)^3*sin(2/9*pi) - 4*sqrt(
3)*cos(2/9*pi)*sin(2/9*pi)^3 - cos(2/9*pi)^4 + 6*cos(2/9*pi)^2*sin(2/9*pi)^2 - sin(2/9*pi)^4 + 2*sqrt(3)*sin(2
/9*pi) - 2*cos(2/9*pi))*log(-(sqrt(3)*i*cos(2/9*pi) + cos(2/9*pi))*x + x^2 + 1) - 1/18*(4*sqrt(3)*cos(1/9*pi)^
3*sin(1/9*pi) - 4*sqrt(3)*cos(1/9*pi)*sin(1/9*pi)^3 + cos(1/9*pi)^4 - 6*cos(1/9*pi)^2*sin(1/9*pi)^2 + sin(1/9*
pi)^4 - 2*sqrt(3)*sin(1/9*pi) - 2*cos(1/9*pi))*log((sqrt(3)*i*cos(1/9*pi) + cos(1/9*pi))*x + x^2 + 1)

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maple [C]  time = 0.00, size = 44, normalized size = 0.11 \begin {gather*} \frac {\left (-\RootOf \left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )^{3}+1\right ) \ln \left (-\RootOf \left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )+x \right )}{6 \RootOf \left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )^{5}-3 \RootOf \left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {x^{3} - 1}{x^{6} - x^{3} + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 2.30, size = 319, normalized size = 0.78 \begin {gather*} \frac {\ln \left (x-\frac {\left (-\frac {27}{2}+\frac {\sqrt {3}\,9{}\mathrm {i}}{2}\right )\,{\left (-36-\sqrt {3}\,12{}\mathrm {i}\right )}^{1/3}}{54}\right )\,{\left (-36-\sqrt {3}\,12{}\mathrm {i}\right )}^{1/3}}{18}+\frac {\ln \left (x+\frac {\left (\frac {27}{2}+\frac {\sqrt {3}\,9{}\mathrm {i}}{2}\right )\,{\left (-36+\sqrt {3}\,12{}\mathrm {i}\right )}^{1/3}}{54}\right )\,{\left (-36+\sqrt {3}\,12{}\mathrm {i}\right )}^{1/3}}{18}-\frac {2^{2/3}\,\ln \left (x-\frac {2^{2/3}\,{\left (-3-\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,\left (3^{1/3}+3^{5/6}\,1{}\mathrm {i}\right )\,\left (\frac {3\,\left (3+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (3^{1/3}+3^{5/6}\,1{}\mathrm {i}\right )}^3}{16}+27\right )}{108}\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^{2/3}\,{\left (-3+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,\left (3^{1/3}-3^{5/6}\,1{}\mathrm {i}\right )\,\left (\frac {3\,\left (-3+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (3^{1/3}-3^{5/6}\,1{}\mathrm {i}\right )}^3}{16}-27\right )}{108}\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^{2/3}\,3^{5/6}\,{\left (-3-\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,1{}\mathrm {i}}{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^{2/3}\,3^{5/6}\,{\left (-3+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/3}\,1{}\mathrm {i}}{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^3 - 1)/(x^6 - x^3 + 1),x)

[Out]

(log(x - (((3^(1/2)*9i)/2 - 27/2)*(- 3^(1/2)*12i - 36)^(1/3))/54)*(- 3^(1/2)*12i - 36)^(1/3))/18 + (log(x + ((
(3^(1/2)*9i)/2 + 27/2)*(3^(1/2)*12i - 36)^(1/3))/54)*(3^(1/2)*12i - 36)^(1/3))/18 - (2^(2/3)*log(x - (2^(2/3)*
(- 3^(1/2)*1i - 3)^(1/3)*(3^(1/3) + 3^(5/6)*1i)*((3*(3^(1/2)*1i + 3)*(3^(1/3) + 3^(5/6)*1i)^3)/16 + 27))/108)*
(- 3^(1/2)*1i - 3)^(1/3)*(3^(1/3) + 3^(5/6)*1i))/36 - (2^(2/3)*log(x + (2^(2/3)*(3^(1/2)*1i - 3)^(1/3)*(3^(1/3
) - 3^(5/6)*1i)*((3*(3^(1/2)*1i - 3)*(3^(1/3) - 3^(5/6)*1i)^3)/16 - 27))/108)*(3^(1/2)*1i - 3)^(1/3)*(3^(1/3)
- 3^(5/6)*1i))/36 - (2^(2/3)*log(x + (2^(2/3)*3^(5/6)*(- 3^(1/2)*1i - 3)^(1/3)*1i)/6)*(- 3^(1/2)*1i - 3)^(1/3)
*(3^(1/3) - 3^(5/6)*1i))/36 - (2^(2/3)*log(x - (2^(2/3)*3^(5/6)*(3^(1/2)*1i - 3)^(1/3)*1i)/6)*(3^(1/2)*1i - 3)
^(1/3)*(3^(1/3) + 3^(5/6)*1i))/36

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sympy [A]  time = 0.18, size = 26, normalized size = 0.06 \begin {gather*} - \operatorname {RootSum} {\left (19683 t^{6} - 243 t^{3} + 1, \left (t \mapsto t \log {\left (729 t^{4} - 9 t + x \right )} \right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-RootSum(19683*_t**6 - 243*_t**3 + 1, Lambda(_t, _t*log(729*_t**4 - 9*_t + x)))

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