3.7.30 \(\int \frac {x^4}{(1-x^3)^{2/3} (1+x^3)} \, dx\) [630]

Optimal. Leaf size=139 \[ -\frac {\tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}-\frac {\log \left (1+x^3\right )}{6\ 2^{2/3}}-\frac {1}{2} \log \left (-x-\sqrt [3]{1-x^3}\right )+\frac {\log \left (-\sqrt [3]{2} x-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {494, 337, 503} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\text {ArcTan}\left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}-\frac {\log \left (x^3+1\right )}{6\ 2^{2/3}}-\frac {1}{2} \log \left (-\sqrt [3]{1-x^3}-x\right )+\frac {\log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )}{2\ 2^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4/((1 - x^3)^(2/3)*(1 + x^3)),x]

[Out]

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

Rule 337

Int[(x_)/((a_) + (b_.)*(x_)^3)^(2/3), x_Symbol] :> With[{q = Rt[b, 3]}, Simp[-ArcTan[(1 + 2*q*(x/(a + b*x^3)^(
1/3)))/Sqrt[3]]/(Sqrt[3]*q^2), x] - Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*q^2), x]] /; FreeQ[{a, b}, x]

Rule 494

Int[(((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_)^(n_))^(q_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Dist[e^n/b, Int[
(e*x)^(m - n)*(c + d*x^n)^q, x], x] - Dist[a*(e^n/b), Int[(e*x)^(m - n)*((c + d*x^n)^q/(a + b*x^n)), x], x] /;
FreeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LeQ[n, m, 2*n - 1] && IntBinomialQ[a, b
, c, d, e, m, n, -1, q, x]

Rule 503

Int[(x_)/(((a_) + (b_.)*(x_)^3)^(2/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/c, 3]}, Si
mp[-ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sqrt[3]*c*q^2), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/
(2*c*q^2), x] + Simp[Log[c + d*x^3]/(6*c*q^2), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rubi steps

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

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Mathematica [A]
time = 0.27, size = 206, normalized size = 1.48 \begin {gather*} \frac {1}{12} \left (-4 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x-2 \sqrt [3]{1-x^3}}\right )+2 \sqrt [3]{2} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x-2^{2/3} \sqrt [3]{1-x^3}}\right )-4 \log \left (x+\sqrt [3]{1-x^3}\right )+2 \sqrt [3]{2} \log \left (2 x+2^{2/3} \sqrt [3]{1-x^3}\right )+2 \log \left (x^2-x \sqrt [3]{1-x^3}+\left (1-x^3\right )^{2/3}\right )-\sqrt [3]{2} \log \left (-2 x^2+2^{2/3} x \sqrt [3]{1-x^3}-\sqrt [3]{2} \left (1-x^3\right )^{2/3}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4/((1 - x^3)^(2/3)*(1 + x^3)),x]

[Out]

(-4*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x - 2*(1 - x^3)^(1/3))] + 2*2^(1/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x - 2^(2/3)*(1
- x^3)^(1/3))] - 4*Log[x + (1 - x^3)^(1/3)] + 2*2^(1/3)*Log[2*x + 2^(2/3)*(1 - x^3)^(1/3)] + 2*Log[x^2 - x*(1
- x^3)^(1/3) + (1 - x^3)^(2/3)] - 2^(1/3)*Log[-2*x^2 + 2^(2/3)*x*(1 - x^3)^(1/3) - 2^(1/3)*(1 - x^3)^(2/3)])/
12

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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {x^{4}}{\left (-x^{3}+1\right )^{\frac {2}{3}} \left (x^{3}+1\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(-x^3+1)^(2/3)/(x^3+1),x)

[Out]

int(x^4/(-x^3+1)^(2/3)/(x^3+1),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(-x^3+1)^(2/3)/(x^3+1),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 2.70, size = 197, normalized size = 1.42 \begin {gather*} \frac {1}{6} \cdot 4^{\frac {1}{6}} \sqrt {3} \arctan \left (-\frac {4^{\frac {1}{6}} {\left (4^{\frac {1}{3}} \sqrt {3} x - 4^{\frac {2}{3}} \sqrt {3} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}}{6 \, x}\right ) + \frac {1}{12} \cdot 4^{\frac {2}{3}} \log \left (\frac {4^{\frac {2}{3}} x + 2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{24} \cdot 4^{\frac {2}{3}} \log \left (\frac {2 \cdot 4^{\frac {1}{3}} x^{2} - 4^{\frac {2}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x + 2 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2}}\right ) - \frac {1}{3} \, \sqrt {3} \arctan \left (-\frac {\sqrt {3} x - 2 \, \sqrt {3} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{3 \, x}\right ) - \frac {1}{3} \, \log \left (\frac {x + {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{6} \, \log \left (\frac {x^{2} - {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x + {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4}}{\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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(-x^3+1)^(2/3)/(x^3+1),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^4}{{\left (1-x^3\right )}^{2/3}\,\left (x^3+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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