3.7.12 \(\int \frac {x^6}{\sqrt [3]{1-x^3} (1+x^3)} \, dx\) [612]

Optimal. Leaf size=154 \[ -\frac {1}{3} x \left (1-x^3\right )^{2/3}+\frac {2 \tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {\log \left (1+x^3\right )}{6 \sqrt [3]{2}}+\frac {\log \left (-\sqrt [3]{2} x-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}-\frac {1}{3} \log \left (x+\sqrt [3]{1-x^3}\right ) \]

[Out]

-1/3*x*(-x^3+1)^(2/3)-1/12*ln(x^3+1)*2^(2/3)+1/4*ln(-2^(1/3)*x-(-x^3+1)^(1/3))*2^(2/3)-1/3*ln(x+(-x^3+1)^(1/3)
)+2/9*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^(2/3)*3^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {490, 544, 245, 384} \begin {gather*} \frac {2 \text {ArcTan}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\text {ArcTan}\left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {1}{3} \left (1-x^3\right )^{2/3} x-\frac {\log \left (x^3+1\right )}{6 \sqrt [3]{2}}+\frac {\log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )}{2 \sqrt [3]{2}}-\frac {1}{3} \log \left (\sqrt [3]{1-x^3}+x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 245

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

Rule 384

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

Rule 490

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

Rule 544

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

Rubi steps

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

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Mathematica [A]
time = 0.39, size = 220, normalized size = 1.43 \begin {gather*} \frac {1}{36} \left (-12 x \left (1-x^3\right )^{2/3}+8 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x-2 \sqrt [3]{1-x^3}}\right )-6\ 2^{2/3} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x-2^{2/3} \sqrt [3]{1-x^3}}\right )-8 \log \left (x+\sqrt [3]{1-x^3}\right )+6\ 2^{2/3} \log \left (2 x+2^{2/3} \sqrt [3]{1-x^3}\right )+4 \log \left (x^2-x \sqrt [3]{1-x^3}+\left (1-x^3\right )^{2/3}\right )-3\ 2^{2/3} \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^6/((1 - x^3)^(1/3)*(1 + x^3)),x]

[Out]

(-12*x*(1 - x^3)^(2/3) + 8*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x - 2*(1 - x^3)^(1/3))] - 6*2^(2/3)*Sqrt[3]*ArcTan[(Sqr
t[3]*x)/(x - 2^(2/3)*(1 - x^3)^(1/3))] - 8*Log[x + (1 - x^3)^(1/3)] + 6*2^(2/3)*Log[2*x + 2^(2/3)*(1 - x^3)^(1
/3)] + 4*Log[x^2 - x*(1 - x^3)^(1/3) + (1 - x^3)^(2/3)] - 3*2^(2/3)*Log[-2*x^2 + 2^(2/3)*x*(1 - x^3)^(1/3) - 2
^(1/3)*(1 - x^3)^(2/3)])/36

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {x^{6}}{\left (-x^{3}+1\right )^{\frac {1}{3}} \left (x^{3}+1\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Fricas [A]
time = 2.20, size = 201, normalized size = 1.31 \begin {gather*} -\frac {1}{3} \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} x - \frac {1}{6} \, \sqrt {6} 2^{\frac {1}{6}} \arctan \left (-\frac {2^{\frac {1}{6}} {\left (\sqrt {6} 2^{\frac {1}{3}} x - 2 \, \sqrt {6} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}}{6 \, x}\right ) + \frac {1}{6} \cdot 2^{\frac {2}{3}} \log \left (\frac {2^{\frac {1}{3}} x + {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{12} \cdot 2^{\frac {2}{3}} \log \left (\frac {2^{\frac {2}{3}} x^{2} - 2^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x + {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2}}\right ) + \frac {2}{9} \, \sqrt {3} \arctan \left (-\frac {\sqrt {3} x - 2 \, \sqrt {3} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{3 \, x}\right ) - \frac {2}{9} \, \log \left (\frac {x + {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{9} \, \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^6/(-x^3+1)^(1/3)/(x^3+1),x, algorithm="fricas")

[Out]

-1/3*(-x^3 + 1)^(2/3)*x - 1/6*sqrt(6)*2^(1/6)*arctan(-1/6*2^(1/6)*(sqrt(6)*2^(1/3)*x - 2*sqrt(6)*(-x^3 + 1)^(1
/3))/x) + 1/6*2^(2/3)*log((2^(1/3)*x + (-x^3 + 1)^(1/3))/x) - 1/12*2^(2/3)*log((2^(2/3)*x^2 - 2^(1/3)*(-x^3 +
1)^(1/3)*x + (-x^3 + 1)^(2/3))/x^2) + 2/9*sqrt(3)*arctan(-1/3*(sqrt(3)*x - 2*sqrt(3)*(-x^3 + 1)^(1/3))/x) - 2/
9*log((x + (-x^3 + 1)^(1/3))/x) + 1/9*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^{6}}{\sqrt [3]{- \left (x - 1\right ) \left (x^{2} + x + 1\right )} \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**6/(-x**3+1)**(1/3)/(x**3+1),x)

[Out]

Integral(x**6/((-(x - 1)*(x**2 + x + 1))**(1/3)*(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^6/(-x^3+1)^(1/3)/(x^3+1),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^6}{{\left (1-x^3\right )}^{1/3}\,\left (x^3+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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