3.611 \(\int \frac{1}{x^4 \sqrt [3]{1-x^3} (1+x^3)} \, dx\)

Optimal. Leaf size=157 \[ -\frac{\left (1-x^3\right )^{2/3}}{3 x^3}-\frac{\log \left (x^3+1\right )}{6 \sqrt [3]{2}}-\frac{1}{3} \log \left (1-\sqrt [3]{1-x^3}\right )+\frac{\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}-\frac{2 \tan ^{-1}\left (\frac{2 \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{3 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3}}+\frac{\log (x)}{3} \]

[Out]

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

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Rubi [A]  time = 0.103411, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {446, 103, 156, 55, 618, 204, 31, 617} \[ -\frac{\left (1-x^3\right )^{2/3}}{3 x^3}-\frac{\log \left (x^3+1\right )}{6 \sqrt [3]{2}}-\frac{1}{3} \log \left (1-\sqrt [3]{1-x^3}\right )+\frac{\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}-\frac{2 \tan ^{-1}\left (\frac{2 \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{3 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3}}+\frac{\log (x)}{3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 103

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])

Rule 156

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]

Rule 55

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

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]

Rubi steps

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

Mathematica [A]  time = 0.148786, size = 153, normalized size = 0.97 \[ \frac{1}{36} \left (3 \left (-\frac{4 \left (1-x^3\right )^{2/3}}{x^3}-2^{2/3} \log \left (x^3+1\right )-4 \log \left (1-\sqrt [3]{1-x^3}\right )+3\ 2^{2/3} \log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )+2\ 2^{2/3} \sqrt{3} \tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )+4 \log (x)\right )-8 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*Sqrt[3]*ArcTan[(1 + 2*(1 - x^3)^(1/3))/Sqrt[3]] + 3*((-4*(1 - x^3)^(2/3))/x^3 + 2*2^(2/3)*Sqrt[3]*ArcTan[(
1 + 2^(2/3)*(1 - x^3)^(1/3))/Sqrt[3]] + 4*Log[x] - 2^(2/3)*Log[1 + x^3] - 4*Log[1 - (1 - x^3)^(1/3)] + 3*2^(2/
3)*Log[2^(1/3) - (1 - x^3)^(1/3)]))/36

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Maple [F]  time = 0.05, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4} \left ({x}^{3}+1 \right ) }{\frac{1}{\sqrt [3]{-{x}^{3}+1}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (x^{3} + 1\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(-x^3+1)^(1/3)/(x^3+1),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.7055, size = 541, normalized size = 3.45 \begin{align*} \frac{6 \, \sqrt{6} 2^{\frac{1}{6}} x^{3} \arctan \left (\frac{1}{6} \cdot 2^{\frac{1}{6}}{\left (\sqrt{6} 2^{\frac{1}{3}} + 2 \, \sqrt{6}{\left (-x^{3} + 1\right )}^{\frac{1}{3}}\right )}\right ) - 3 \cdot 2^{\frac{2}{3}} x^{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 ) + 6 \cdot 2^{\frac{2}{3}} x^{3} \log \left (-2^{\frac{1}{3}} +{\left (-x^{3} + 1\right )}^{\frac{1}{3}}\right ) - 8 \, \sqrt{3} x^{3} \arctan \left (\frac{2}{3} \, \sqrt{3}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + \frac{1}{3} \, \sqrt{3}\right ) + 4 \, x^{3} \log \left ({\left (-x^{3} + 1\right )}^{\frac{2}{3}} +{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 1\right ) - 8 \, x^{3} \log \left ({\left (-x^{3} + 1\right )}^{\frac{1}{3}} - 1\right ) - 12 \,{\left (-x^{3} + 1\right )}^{\frac{2}{3}}}{36 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(-x^3+1)^(1/3)/(x^3+1),x, algorithm="fricas")

[Out]

1/36*(6*sqrt(6)*2^(1/6)*x^3*arctan(1/6*2^(1/6)*(sqrt(6)*2^(1/3) + 2*sqrt(6)*(-x^3 + 1)^(1/3))) - 3*2^(2/3)*x^3
*log(2^(2/3) + 2^(1/3)*(-x^3 + 1)^(1/3) + (-x^3 + 1)^(2/3)) + 6*2^(2/3)*x^3*log(-2^(1/3) + (-x^3 + 1)^(1/3)) -
8*sqrt(3)*x^3*arctan(2/3*sqrt(3)*(-x^3 + 1)^(1/3) + 1/3*sqrt(3)) + 4*x^3*log((-x^3 + 1)^(2/3) + (-x^3 + 1)^(1
/3) + 1) - 8*x^3*log((-x^3 + 1)^(1/3) - 1) - 12*(-x^3 + 1)^(2/3))/x^3

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{4} \sqrt [3]{- \left (x - 1\right ) \left (x^{2} + x + 1\right )} \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(1/x**4/(-x**3+1)**(1/3)/(x**3+1),x)

[Out]

Integral(1/(x**4*(-(x - 1)*(x**2 + x + 1))**(1/3)*(x + 1)*(x**2 - x + 1)), x)

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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(1/x^4/(-x^3+1)^(1/3)/(x^3+1),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError