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

Optimal. Leaf size=293 \[ \frac {1}{2} x \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};x^3\right )+\frac {\log \left (2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{6\ 2^{2/3}}-\frac {\log \left (\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{6\ 2^{2/3}}+\frac {\log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3\ 2^{2/3}}-\frac {\log \left (\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}\right )}{12\ 2^{2/3}}+\frac {\tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )}{2\ 2^{2/3} \sqrt {3}} \]

[Out]

1/2*x*hypergeom([1/3, 2/3],[4/3],x^3)+1/12*ln(2^(2/3)+(-1+x)/(-x^3+1)^(1/3))*2^(1/3)-1/12*ln(1+2^(2/3)*(1-x)^2
/(-x^3+1)^(2/3)-2^(1/3)*(1-x)/(-x^3+1)^(1/3))*2^(1/3)+1/6*ln(1+2^(1/3)*(1-x)/(-x^3+1)^(1/3))*2^(1/3)-1/24*ln(2
*2^(1/3)+(1-x)^2/(-x^3+1)^(2/3)+2^(2/3)*(1-x)/(-x^3+1)^(1/3))*2^(1/3)+1/6*arctan(1/3*(1-2*2^(1/3)*(1-x)/(-x^3+
1)^(1/3))*3^(1/2))*2^(1/3)*3^(1/2)+1/12*arctan(1/3*(1+2^(1/3)*(1-x)/(-x^3+1)^(1/3))*3^(1/2))*2^(1/3)*3^(1/2)

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Rubi [C]  time = 0.01, antiderivative size = 21, normalized size of antiderivative = 0.07, 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/((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 {1}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, 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 = 0.21, size = 111, normalized size = 0.38 \[ -\frac {4 x F_1\left (\frac {1}{3};\frac {2}{3},1;\frac {4}{3};x^3,-x^3\right )}{\left (1-x^3\right )^{2/3} \left (x^3+1\right ) \left (x^3 \left (3 F_1\left (\frac {4}{3};\frac {2}{3},2;\frac {7}{3};x^3,-x^3\right )-2 F_1\left (\frac {4}{3};\frac {5}{3},1;\frac {7}{3};x^3,-x^3\right )\right )-4 F_1\left (\frac {1}{3};\frac {2}{3},1;\frac {4}{3};x^3,-x^3\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [F]  time = 3.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x^{6} - 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 1.94, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-x^{3}+1\right )^{\frac {2}{3}} \left (x^{3}+1\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(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 \[ \int \frac {1}{{\left (x^{3} + 1\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((1 - x^3)^(2/3)*(x^3 + 1)),x)

[Out]

int(1/((1 - x^3)^(2/3)*(x^3 + 1)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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