3.2.15 \(\int \frac {(1-x) (1-x^3)^{2/3}}{1+x^3} \, dx\) [115]

Optimal. Leaf size=383 \[ -\frac {2^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\tan ^{-1}\left (\frac {1+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {2^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} x^2 \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};x^3\right )-\frac {\log \left ((1-x) (1+x)^2\right )}{6 \sqrt [3]{2}}-\frac {\log \left (1+x^3\right )}{3 \sqrt [3]{2}}-\frac {\log \left (1+\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )}{3 \sqrt [3]{2}}+\frac {1}{3} 2^{2/3} \log \left (1+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )+\frac {\log \left (-\sqrt [3]{2} x-\sqrt [3]{1-x^3}\right )}{\sqrt [3]{2}}-\frac {1}{2} \log \left (x+\sqrt [3]{1-x^3}\right )+\frac {\log \left (-1+x+2^{2/3} \sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}} \]

[Out]

1/2*x^2*hypergeom([1/3, 2/3],[5/3],x^3)-1/12*ln((1-x)*(1+x)^2)*2^(2/3)-1/6*ln(x^3+1)*2^(2/3)-1/6*ln(1+2^(2/3)*
(1-x)^2/(-x^3+1)^(2/3)-2^(1/3)*(1-x)/(-x^3+1)^(1/3))*2^(2/3)+1/3*ln(1+2^(1/3)*(1-x)/(-x^3+1)^(1/3))*2^(2/3)+1/
2*ln(-2^(1/3)*x-(-x^3+1)^(1/3))*2^(2/3)-1/2*ln(x+(-x^3+1)^(1/3))+1/4*ln(-1+x+2^(2/3)*(-x^3+1)^(1/3))*2^(2/3)-1
/3*arctan(1/3*(1-2*2^(1/3)*(1-x)/(-x^3+1)^(1/3))*3^(1/2))*2^(2/3)*3^(1/2)-1/6*arctan(1/3*(1+2^(1/3)*(1-x)/(-x^
3+1)^(1/3))*3^(1/2))*3^(1/2)*2^(2/3)+1/3*arctan(1/3*(1-2*x/(-x^3+1)^(1/3))*3^(1/2))*3^(1/2)-1/3*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 = 0.58, antiderivative size = 648, normalized size of antiderivative = 1.69, number of steps used = 17, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6857, 2178, 2177, 245, 2174, 371} \begin {gather*} -\frac {2^{2/3} \text {ArcTan}\left (\frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\left (1+(-1)^{2/3}\right ) \text {ArcTan}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {\left (1-\sqrt [3]{-1}\right ) \text {ArcTan}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {2 \text {ArcTan}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\left (1-\sqrt [3]{-1}\right ) \text {ArcTan}\left (\frac {1-\frac {\sqrt [3]{2} \left (x+\sqrt [3]{-1}\right )}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {\left (1+(-1)^{2/3}\right ) \text {ArcTan}\left (\frac {\frac {(-1)^{2/3} \sqrt [3]{2} \left (\sqrt [3]{-1} x+1\right )}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {1}{6} \left (1+(-1)^{2/3}\right ) x^2 \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};x^3\right )+\frac {1}{6} \left (1-\sqrt [3]{-1}\right ) x^2 \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};x^3\right )+\frac {1}{3} x^2 \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};x^3\right )-\frac {1}{6} \left (1+(-1)^{2/3}\right ) \log \left (\sqrt [3]{1-x^3}+x\right )-\frac {1}{6} \left (1-\sqrt [3]{-1}\right ) \log \left (\sqrt [3]{1-x^3}+x\right )-\frac {1}{3} \log \left (\sqrt [3]{1-x^3}+x\right )+\frac {\left (1-\sqrt [3]{-1}\right ) \log \left (-(-2)^{2/3} \sqrt [3]{1-x^3}-(-1)^{2/3} x+1\right )}{2 \sqrt [3]{2}}+\frac {\log \left (-2^{2/3} \sqrt [3]{1-x^3}-x+1\right )}{\sqrt [3]{2}}+\frac {\left (1+(-1)^{2/3}\right ) \log \left (\sqrt [3]{-1} 2^{2/3} \sqrt [3]{1-x^3}+\sqrt [3]{-1} x+1\right )}{2 \sqrt [3]{2}}-\frac {\log \left (-\left ((1-x) (x+1)^2\right )\right )}{3 \sqrt [3]{2}}-\frac {\left (1+(-1)^{2/3}\right ) \log \left (-(-1)^{2/3} \left (x+(-1)^{2/3}\right )^2 \left (\sqrt [3]{-1} x+1\right )\right )}{6 \sqrt [3]{2}}-\frac {\left (1-\sqrt [3]{-1}\right ) \log \left ((-1)^{2/3} \left (x+\sqrt [3]{-1}\right ) \left ((-1)^{2/3} x+1\right )^2\right )}{6 \sqrt [3]{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((2^(2/3)*ArcTan[(1 + (2^(1/3)*(1 - x))/(1 - x^3)^(1/3))/Sqrt[3]])/Sqrt[3]) + (2*ArcTan[(1 - (2*x)/(1 - x^3)^
(1/3))/Sqrt[3]])/(3*Sqrt[3]) + ((1 - (-1)^(1/3))*ArcTan[(1 - (2*x)/(1 - x^3)^(1/3))/Sqrt[3]])/(3*Sqrt[3]) + ((
1 + (-1)^(2/3))*ArcTan[(1 - (2*x)/(1 - x^3)^(1/3))/Sqrt[3]])/(3*Sqrt[3]) - ((1 - (-1)^(1/3))*ArcTan[(1 - (2^(1
/3)*((-1)^(1/3) + x))/(1 - x^3)^(1/3))/Sqrt[3]])/(2^(1/3)*Sqrt[3]) - ((1 + (-1)^(2/3))*ArcTan[(1 + ((-1)^(2/3)
*2^(1/3)*(1 + (-1)^(1/3)*x))/(1 - x^3)^(1/3))/Sqrt[3]])/(2^(1/3)*Sqrt[3]) + (x^2*Hypergeometric2F1[1/3, 2/3, 5
/3, x^3])/3 + ((1 - (-1)^(1/3))*x^2*Hypergeometric2F1[1/3, 2/3, 5/3, x^3])/6 + ((1 + (-1)^(2/3))*x^2*Hypergeom
etric2F1[1/3, 2/3, 5/3, x^3])/6 - Log[-((1 - x)*(1 + x)^2)]/(3*2^(1/3)) - ((1 + (-1)^(2/3))*Log[-((-1)^(2/3)*(
(-1)^(2/3) + x)^2*(1 + (-1)^(1/3)*x))])/(6*2^(1/3)) - ((1 - (-1)^(1/3))*Log[(-1)^(2/3)*((-1)^(1/3) + x)*(1 + (
-1)^(2/3)*x)^2])/(6*2^(1/3)) - Log[x + (1 - x^3)^(1/3)]/3 - ((1 - (-1)^(1/3))*Log[x + (1 - x^3)^(1/3)])/6 - ((
1 + (-1)^(2/3))*Log[x + (1 - x^3)^(1/3)])/6 + ((1 - (-1)^(1/3))*Log[1 - (-1)^(2/3)*x - (-2)^(2/3)*(1 - x^3)^(1
/3)])/(2*2^(1/3)) + Log[1 - x - 2^(2/3)*(1 - x^3)^(1/3)]/2^(1/3) + ((1 + (-1)^(2/3))*Log[1 + (-1)^(1/3)*x + (-
1)^(1/3)*2^(2/3)*(1 - x^3)^(1/3)])/(2*2^(1/3))

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 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 2174

Int[1/(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^3)^(1/3)), x_Symbol] :> Simp[Sqrt[3]*(ArcTan[(1 - 2^(1/3)*Rt[b,
3]*((c - d*x)/(d*(a + b*x^3)^(1/3))))/Sqrt[3]]/(2^(4/3)*Rt[b, 3]*c)), x] + (Simp[Log[(c + d*x)^2*(c - d*x)]/(2
^(7/3)*Rt[b, 3]*c), x] - Simp[(3*Log[Rt[b, 3]*(c - d*x) + 2^(2/3)*d*(a + b*x^3)^(1/3)])/(2^(7/3)*Rt[b, 3]*c),
x]) /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^3 + a*d^3, 0]

Rule 2177

Int[((e_.) + (f_.)*(x_))/(((c_.) + (d_.)*(x_))*((a_) + (b_.)*(x_)^3)^(1/3)), x_Symbol] :> Dist[f/d, Int[1/(a +
b*x^3)^(1/3), x], x] + Dist[(d*e - c*f)/d, Int[1/((c + d*x)*(a + b*x^3)^(1/3)), x], x] /; FreeQ[{a, b, c, d,
e, f}, x]

Rule 2178

Int[((a_) + (b_.)*(x_)^3)^(2/3)/((c_) + (d_.)*(x_)), x_Symbol] :> Simp[(a + b*x^3)^(2/3)/(2*d), x] + (Dist[1/d
^2, Int[(a*d^2 + b*c^2*x)/((c + d*x)*(a + b*x^3)^(1/3)), x], x] - Dist[b*(c/d^2), Int[x/(a + b*x^3)^(1/3), x],
x]) /; FreeQ[{a, b, c, d}, x]

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps

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

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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
time = 10.17, size = 138, normalized size = 0.36 \begin {gather*} -\frac {1}{2} x^2 F_1\left (\frac {2}{3};-\frac {2}{3},1;\frac {5}{3};x^3,-x^3\right )-\frac {4 x \left (1-x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,-x^3\right )}{\left (1+x^3\right ) \left (-4 F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,-x^3\right )+x^3 \left (3 F_1\left (\frac {4}{3};-\frac {2}{3},2;\frac {7}{3};x^3,-x^3\right )+2 F_1\left (\frac {4}{3};\frac {1}{3},1;\frac {7}{3};x^3,-x^3\right )\right )\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/2*(x^2*AppellF1[2/3, -2/3, 1, 5/3, x^3, -x^3]) - (4*x*(1 - x^3)^(2/3)*AppellF1[1/3, -2/3, 1, 4/3, x^3, -x^3
])/((1 + x^3)*(-4*AppellF1[1/3, -2/3, 1, 4/3, x^3, -x^3] + x^3*(3*AppellF1[4/3, -2/3, 2, 7/3, x^3, -x^3] + 2*A
ppellF1[4/3, 1/3, 1, 7/3, x^3, -x^3])))

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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\left (1-x \right ) \left (-x^{3}+1\right )^{\frac {2}{3}}}{x^{3}+1}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((1-x)*(-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((1-x)*(-x^3+1)^(2/3)/(x^3+1),x, algorithm="maxima")

[Out]

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

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Fricas [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((1-x)*(-x^3+1)^(2/3)/(x^3+1),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- \frac {\left (1 - x^{3}\right )^{\frac {2}{3}}}{x^{3} + 1}\right )\, dx - \int \frac {x \left (1 - x^{3}\right )^{\frac {2}{3}}}{x^{3} + 1}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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