3.2.12 \(\int \frac {(1-x+x^2) (1-x^3)^{2/3}}{1+x^3} \, dx\) [112]

Optimal. Leaf size=177 \[ \frac {1}{2} \left (1-x^3\right )^{2/3}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {1+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2}}+\frac {\tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} x^2 \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};x^3\right )-\frac {\log \left ((1-x) (1+x)^2\right )}{2 \sqrt [3]{2}}-\frac {1}{2} \log \left (x+\sqrt [3]{1-x^3}\right )+\frac {3 \log \left (-1+x+2^{2/3} \sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}} \]

[Out]

1/2*(-x^3+1)^(2/3)+1/2*x^2*hypergeom([1/3, 2/3],[5/3],x^3)-1/4*ln((1-x)*(1+x)^2)*2^(2/3)-1/2*ln(x+(-x^3+1)^(1/
3))+3/4*ln(-1+x+2^(2/3)*(-x^3+1)^(1/3))*2^(2/3)+1/3*arctan(1/3*(1-2*x/(-x^3+1)^(1/3))*3^(1/2))*3^(1/2)-1/2*arc
tan(1/3*(1+2^(1/3)*(1-x)/(-x^3+1)^(1/3))*3^(1/2))*3^(1/2)*2^(2/3)

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Rubi [A]
time = 0.10, antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1600, 2178, 2177, 245, 2174, 371} \begin {gather*} \frac {1}{2} x^2 \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};x^3\right )+\frac {1}{2} \left (1-x^3\right )^{2/3}-\frac {1}{2} \log \left (\sqrt [3]{1-x^3}+x\right )+\frac {3 \log \left (2^{2/3} \sqrt [3]{1-x^3}+x-1\right )}{2 \sqrt [3]{2}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )}{\sqrt [3]{2}}+\frac {\tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\log \left ((1-x) (x+1)^2\right )}{2 \sqrt [3]{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(1 - x^3)^(2/3)/2 - (Sqrt[3]*ArcTan[(1 + (2^(1/3)*(1 - x))/(1 - x^3)^(1/3))/Sqrt[3]])/2^(1/3) + ArcTan[(1 - (2
*x)/(1 - x^3)^(1/3))/Sqrt[3]]/Sqrt[3] + (x^2*Hypergeometric2F1[1/3, 2/3, 5/3, x^3])/2 - Log[(1 - x)*(1 + x)^2]
/(2*2^(1/3)) - Log[x + (1 - x^3)^(1/3)]/2 + (3*Log[-1 + x + 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 1600

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 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]

Rubi steps

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

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Mathematica [F]
time = 20.13, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1-x+x^2\right ) \left (1-x^3\right )^{2/3}}{1+x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Mathics [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Fricas [F]
time = 1.89, 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^2-x+1)*(-x^3+1)^(2/3)/(x^3+1),x, algorithm="fricas")

[Out]

integral((-x^3 + 1)^(2/3)/(x + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F] N/A
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^2-x+1)*(-x^3+1)^(2/3)/(x^3+1),x)

[Out]

Could not integrate

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

[Out]

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

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