3.1.20 \(\int \frac {x^8 (1-x^3)}{1-x^3+x^6} \, dx\)

Optimal. Leaf size=46 \[ -\frac {x^6}{6}-\frac {\tan ^{-1}\left (\frac {1-2 x^3}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {1}{6} \log \left (x^6-x^3+1\right ) \]

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Rubi [A]  time = 0.06, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {1474, 800, 634, 618, 204, 628} \begin {gather*} -\frac {x^6}{6}+\frac {1}{6} \log \left (x^6-x^3+1\right )-\frac {\tan ^{-1}\left (\frac {1-2 x^3}{\sqrt {3}}\right )}{3 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^8*(1 - x^3))/(1 - x^3 + x^6),x]

[Out]

-x^6/6 - ArcTan[(1 - 2*x^3)/Sqrt[3]]/(3*Sqrt[3]) + Log[1 - x^3 + x^6]/6

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 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; 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 800

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[((d + e*x)^m*(f + g*x))/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

Rule 1474

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :>
Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a,
b, c, d, e, m, n, p, q}, x] && EqQ[n2, 2*n] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {x^8 \left (1-x^3\right )}{1-x^3+x^6} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {(1-x) x^2}{1-x+x^2} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (-x+\frac {x}{1-x+x^2}\right ) \, dx,x,x^3\right )\\ &=-\frac {x^6}{6}+\frac {1}{3} \operatorname {Subst}\left (\int \frac {x}{1-x+x^2} \, dx,x,x^3\right )\\ &=-\frac {x^6}{6}+\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,x^3\right )+\frac {1}{6} \operatorname {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,x^3\right )\\ &=-\frac {x^6}{6}+\frac {1}{6} \log \left (1-x^3+x^6\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x^3\right )\\ &=-\frac {x^6}{6}-\frac {\tan ^{-1}\left (\frac {1-2 x^3}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {1}{6} \log \left (1-x^3+x^6\right )\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 46, normalized size = 1.00 \begin {gather*} -\frac {x^6}{6}+\frac {\tan ^{-1}\left (\frac {2 x^3-1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {1}{6} \log \left (x^6-x^3+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^8*(1 - x^3))/(1 - x^3 + x^6),x]

[Out]

-1/6*x^6 + ArcTan[(-1 + 2*x^3)/Sqrt[3]]/(3*Sqrt[3]) + Log[1 - x^3 + x^6]/6

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^8 \left (1-x^3\right )}{1-x^3+x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(x^8*(1 - x^3))/(1 - x^3 + x^6),x]

[Out]

IntegrateAlgebraic[(x^8*(1 - x^3))/(1 - x^3 + x^6), x]

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fricas [A]  time = 1.46, size = 37, normalized size = 0.80 \begin {gather*} -\frac {1}{6} \, x^{6} + \frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{3} - 1\right )}\right ) + \frac {1}{6} \, \log \left (x^{6} - x^{3} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(-x^3+1)/(x^6-x^3+1),x, algorithm="fricas")

[Out]

-1/6*x^6 + 1/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^3 - 1)) + 1/6*log(x^6 - x^3 + 1)

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giac [A]  time = 0.42, size = 37, normalized size = 0.80 \begin {gather*} -\frac {1}{6} \, x^{6} + \frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{3} - 1\right )}\right ) + \frac {1}{6} \, \log \left (x^{6} - x^{3} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(-x^3+1)/(x^6-x^3+1),x, algorithm="giac")

[Out]

-1/6*x^6 + 1/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^3 - 1)) + 1/6*log(x^6 - x^3 + 1)

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maple [A]  time = 0.00, size = 38, normalized size = 0.83 \begin {gather*} -\frac {x^{6}}{6}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x^{3}-1\right ) \sqrt {3}}{3}\right )}{9}+\frac {\ln \left (x^{6}-x^{3}+1\right )}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8*(-x^3+1)/(x^6-x^3+1),x)

[Out]

-1/6*x^6+1/6*ln(x^6-x^3+1)+1/9*3^(1/2)*arctan(1/3*(2*x^3-1)*3^(1/2))

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maxima [A]  time = 0.99, size = 37, normalized size = 0.80 \begin {gather*} -\frac {1}{6} \, x^{6} + \frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{3} - 1\right )}\right ) + \frac {1}{6} \, \log \left (x^{6} - x^{3} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(-x^3+1)/(x^6-x^3+1),x, algorithm="maxima")

[Out]

-1/6*x^6 + 1/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^3 - 1)) + 1/6*log(x^6 - x^3 + 1)

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mupad [B]  time = 0.06, size = 39, normalized size = 0.85 \begin {gather*} \frac {\ln \left (x^6-x^3+1\right )}{6}-\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}}{3}-\frac {2\,\sqrt {3}\,x^3}{3}\right )}{9}-\frac {x^6}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^8*(x^3 - 1))/(x^6 - x^3 + 1),x)

[Out]

log(x^6 - x^3 + 1)/6 - (3^(1/2)*atan(3^(1/2)/3 - (2*3^(1/2)*x^3)/3))/9 - x^6/6

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sympy [A]  time = 0.14, size = 42, normalized size = 0.91 \begin {gather*} - \frac {x^{6}}{6} + \frac {\log {\left (x^{6} - x^{3} + 1 \right )}}{6} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x^{3}}{3} - \frac {\sqrt {3}}{3} \right )}}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8*(-x**3+1)/(x**6-x**3+1),x)

[Out]

-x**6/6 + log(x**6 - x**3 + 1)/6 + sqrt(3)*atan(2*sqrt(3)*x**3/3 - sqrt(3)/3)/9

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