3.624 \(\int \frac{x^8}{(1-x^3)^{2/3} (1+x^3)} \, dx\)

Optimal. Leaf size=98 \[ \frac{1}{4} \left (1-x^3\right )^{4/3}-\frac{\log \left (x^3+1\right )}{6\ 2^{2/3}}+\frac{\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}-\frac{\tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3}} \]

[Out]

(1 - x^3)^(4/3)/4 - ArcTan[(1 + 2^(2/3)*(1 - x^3)^(1/3))/Sqrt[3]]/(2^(2/3)*Sqrt[3]) - Log[1 + x^3]/(6*2^(2/3))
+ Log[2^(1/3) - (1 - x^3)^(1/3)]/(2*2^(2/3))

________________________________________________________________________________________

Rubi [A]  time = 0.0749959, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {446, 88, 57, 617, 204, 31} \[ \frac{1}{4} \left (1-x^3\right )^{4/3}-\frac{\log \left (x^3+1\right )}{6\ 2^{2/3}}+\frac{\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}-\frac{\tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[x^8/((1 - x^3)^(2/3)*(1 + x^3)),x]

[Out]

(1 - x^3)^(4/3)/4 - ArcTan[(1 + 2^(2/3)*(1 - x^3)^(1/3))/Sqrt[3]]/(2^(2/3)*Sqrt[3]) - Log[1 + x^3]/(6*2^(2/3))
+ Log[2^(1/3) - (1 - x^3)^(1/3)]/(2*2^(2/3))

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 57

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, -Simp[L
og[RemoveContent[a + b*x, x]]/(2*b*q^2), x] + (-Dist[3/(2*b*q), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x
)^(1/3)], x] - Dist[3/(2*b*q^2), Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x]
&& PosQ[(b*c - a*d)/b]

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 31

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

Rubi steps

\begin{align*} \int \frac{x^8}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^2}{(1-x)^{2/3} (1+x)} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (-\sqrt [3]{1-x}+\frac{1}{(1-x)^{2/3} (1+x)}\right ) \, dx,x,x^3\right )\\ &=\frac{1}{4} \left (1-x^3\right )^{4/3}+\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{(1-x)^{2/3} (1+x)} \, dx,x,x^3\right )\\ &=\frac{1}{4} \left (1-x^3\right )^{4/3}-\frac{\log \left (1+x^3\right )}{6\ 2^{2/3}}-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{2}-x} \, dx,x,\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}-\frac{\operatorname{Subst}\left (\int \frac{1}{2^{2/3}+\sqrt [3]{2} x+x^2} \, dx,x,\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}\\ &=\frac{1}{4} \left (1-x^3\right )^{4/3}-\frac{\log \left (1+x^3\right )}{6\ 2^{2/3}}+\frac{\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2^{2/3} \sqrt [3]{1-x^3}\right )}{2^{2/3}}\\ &=\frac{1}{4} \left (1-x^3\right )^{4/3}-\frac{\tan ^{-1}\left (\frac{1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3}}-\frac{\log \left (1+x^3\right )}{6\ 2^{2/3}}+\frac{\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}\\ \end{align*}

Mathematica [A]  time = 0.036311, size = 135, normalized size = 1.38 \[ \frac{1}{12} \left (-3 \sqrt [3]{1-x^3} x^3+3 \sqrt [3]{1-x^3}+2 \sqrt [3]{2} \log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )-\sqrt [3]{2} \log \left (\left (1-x^3\right )^{2/3}+\sqrt [3]{2-2 x^3}+2^{2/3}\right )-2 \sqrt [3]{2} \sqrt{3} \tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^8/((1 - x^3)^(2/3)*(1 + x^3)),x]

[Out]

(3*(1 - x^3)^(1/3) - 3*x^3*(1 - x^3)^(1/3) - 2*2^(1/3)*Sqrt[3]*ArcTan[(1 + 2^(2/3)*(1 - x^3)^(1/3))/Sqrt[3]] +
2*2^(1/3)*Log[2^(1/3) - (1 - x^3)^(1/3)] - 2^(1/3)*Log[2^(2/3) + (2 - 2*x^3)^(1/3) + (1 - x^3)^(2/3)])/12

________________________________________________________________________________________

Maple [F]  time = 0.046, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{8}}{{x}^{3}+1} \left ( -{x}^{3}+1 \right ) ^{-{\frac{2}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8/(-x^3+1)^(2/3)/(x^3+1),x)

[Out]

int(x^8/(-x^3+1)^(2/3)/(x^3+1),x)

________________________________________________________________________________________

Maxima [A]  time = 1.41675, size = 131, normalized size = 1.34 \begin{align*} -\frac{1}{6} \, \sqrt{3} 2^{\frac{1}{3}} \arctan \left (\frac{1}{6} \, \sqrt{3} 2^{\frac{2}{3}}{\left (2^{\frac{1}{3}} + 2 \,{\left (-x^{3} + 1\right )}^{\frac{1}{3}}\right )}\right ) + \frac{1}{4} \,{\left (-x^{3} + 1\right )}^{\frac{4}{3}} - \frac{1}{12} \cdot 2^{\frac{1}{3}} \log \left (2^{\frac{2}{3}} + 2^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} +{\left (-x^{3} + 1\right )}^{\frac{2}{3}}\right ) + \frac{1}{6} \cdot 2^{\frac{1}{3}} \log \left (-2^{\frac{1}{3}} +{\left (-x^{3} + 1\right )}^{\frac{1}{3}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(-x^3+1)^(2/3)/(x^3+1),x, algorithm="maxima")

[Out]

-1/6*sqrt(3)*2^(1/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3) + 2*(-x^3 + 1)^(1/3))) + 1/4*(-x^3 + 1)^(4/3) - 1/12*
2^(1/3)*log(2^(2/3) + 2^(1/3)*(-x^3 + 1)^(1/3) + (-x^3 + 1)^(2/3)) + 1/6*2^(1/3)*log(-2^(1/3) + (-x^3 + 1)^(1/
3))

________________________________________________________________________________________

Fricas [A]  time = 1.53393, size = 346, normalized size = 3.53 \begin{align*} -\frac{1}{6} \cdot 4^{\frac{1}{6}} \sqrt{3} \arctan \left (\frac{1}{6} \cdot 4^{\frac{1}{6}}{\left (4^{\frac{2}{3}} \sqrt{3}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 4^{\frac{1}{3}} \sqrt{3}\right )}\right ) - \frac{1}{24} \cdot 4^{\frac{2}{3}} \log \left (4^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 2 \,{\left (-x^{3} + 1\right )}^{\frac{2}{3}} + 2 \cdot 4^{\frac{1}{3}}\right ) + \frac{1}{12} \cdot 4^{\frac{2}{3}} \log \left (-4^{\frac{2}{3}} + 2 \,{\left (-x^{3} + 1\right )}^{\frac{1}{3}}\right ) - \frac{1}{4} \,{\left (x^{3} - 1\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(-x^3+1)^(2/3)/(x^3+1),x, algorithm="fricas")

[Out]

-1/6*4^(1/6)*sqrt(3)*arctan(1/6*4^(1/6)*(4^(2/3)*sqrt(3)*(-x^3 + 1)^(1/3) + 4^(1/3)*sqrt(3))) - 1/24*4^(2/3)*l
og(4^(2/3)*(-x^3 + 1)^(1/3) + 2*(-x^3 + 1)^(2/3) + 2*4^(1/3)) + 1/12*4^(2/3)*log(-4^(2/3) + 2*(-x^3 + 1)^(1/3)
) - 1/4*(x^3 - 1)*(-x^3 + 1)^(1/3)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{8}}{\left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac{2}{3}} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8/(-x**3+1)**(2/3)/(x**3+1),x)

[Out]

Integral(x**8/((-(x - 1)*(x**2 + x + 1))**(2/3)*(x + 1)*(x**2 - x + 1)), x)

________________________________________________________________________________________

Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(-x^3+1)^(2/3)/(x^3+1),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError