3.113 \(\int \frac {(1-x^3)^{2/3}}{1+x^3} \, dx\)

Optimal. Leaf size=132 \[ -\frac {\log \left (x^3+1\right )}{3 \sqrt [3]{2}}+\frac {\log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )}{\sqrt [3]{2}}-\frac {1}{2} \log \left (\sqrt [3]{1-x^3}+x\right )+\frac {\tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {2^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}} \]

[Out]

-1/6*ln(x^3+1)*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))+1/3*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 [C]  time = 0.01, antiderivative size = 21, normalized size of antiderivative = 0.16, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {429} \[ x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,-x^3\right ) \]

Warning: Unable to verify antiderivative.

[In]

Int[(1 - x^3)^(2/3)/(1 + x^3),x]

[Out]

x*AppellF1[1/3, -2/3, 1, 4/3, x^3, -x^3]

Rule 429

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p,
-q, 1 + 1/n, -((b*x^n)/a), -((d*x^n)/c)], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n
, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rubi steps

\begin {align*} \int \frac {\left (1-x^3\right )^{2/3}}{1+x^3} \, dx &=x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,-x^3\right )\\ \end {align*}

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Mathematica [C]  time = 0.10, size = 111, normalized size = 0.84 \[ -\frac {4 x \left (1-x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,-x^3\right )}{\left (x^3+1\right ) \left (x^3 \left (3 F_1\left (\frac {4}{3};-\frac {2}{3},2;\frac {7}{3};x^3,-x^3\right )+2 F_1\left (\frac {4}{3};\frac {1}{3},1;\frac {7}{3};x^3,-x^3\right )\right )-4 F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,-x^3\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-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*AppellF1[4/3, 1/3, 1, 7/3, x^3, -x^3])))

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fricas [A]  time = 0.63, size = 191, normalized size = 1.45 \[ -\frac {1}{3} \cdot 4^{\frac {1}{3}} \sqrt {3} \arctan \left (-\frac {\sqrt {3} x - 4^{\frac {1}{3}} \sqrt {3} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{3 \, x}\right ) + \frac {1}{3} \, \sqrt {3} \arctan \left (-\frac {\sqrt {3} x - 2 \, \sqrt {3} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{3 \, x}\right ) + \frac {1}{3} \cdot 4^{\frac {1}{3}} \log \left (\frac {4^{\frac {2}{3}} x + 2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{6} \cdot 4^{\frac {1}{3}} \log \left (\frac {2 \cdot 4^{\frac {1}{3}} x^{2} - 4^{\frac {2}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x + 2 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2}}\right ) - \frac {1}{3} \, \log \left (\frac {x + {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{6} \, \log \left (\frac {x^{2} - {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x + {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*4^(1/3)*sqrt(3)*arctan(-1/3*(sqrt(3)*x - 4^(1/3)*sqrt(3)*(-x^3 + 1)^(1/3))/x) + 1/3*sqrt(3)*arctan(-1/3*(
sqrt(3)*x - 2*sqrt(3)*(-x^3 + 1)^(1/3))/x) + 1/3*4^(1/3)*log((4^(2/3)*x + 2*(-x^3 + 1)^(1/3))/x) - 1/6*4^(1/3)
*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*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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{3} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.60, size = 0, normalized size = 0.00 \[ \int \frac {\left (-x^{3}+1\right )^{\frac {2}{3}}}{x^{3}+1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - x^3)^(2/3)/(x^3 + 1),x)

[Out]

int((1 - x^3)^(2/3)/(x^3 + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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