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

Optimal. Leaf size=272 \[ \frac{\log \left (2^{2/3}-\frac{1-x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}-\frac{\log \left (\frac{2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac{\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3\ 2^{2/3}}+\frac{1}{3} \sqrt [3]{2} \log \left (\frac{\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )-\frac{\log \left (\frac{(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac{2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}\right )}{6\ 2^{2/3}}+\frac{\sqrt [3]{2} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt{3}}+\frac{\tan ^{-1}\left (\frac{\frac{\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3}} \]

[Out]

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

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Rubi [C]  time = 0.0078488, antiderivative size = 21, normalized size of antiderivative = 0.08, 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} \[ x F_1\left (\frac{1}{3};-\frac{1}{3},1;\frac{4}{3};x^3,-x^3\right ) \]

Warning: Unable to verify antiderivative.

[In]

Int[(1 - x^3)^(1/3)/(1 + x^3),x]

[Out]

x*AppellF1[1/3, -1/3, 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{\sqrt [3]{1-x^3}}{1+x^3} \, dx &=x F_1\left (\frac{1}{3};-\frac{1}{3},1;\frac{4}{3};x^3,-x^3\right )\\ \end{align*}

Mathematica [C]  time = 0.0882679, size = 109, normalized size = 0.4 \[ -\frac{4 x \sqrt [3]{1-x^3} F_1\left (\frac{1}{3};-\frac{1}{3},1;\frac{4}{3};x^3,-x^3\right )}{\left (x^3+1\right ) \left (x^3 \left (3 F_1\left (\frac{4}{3};-\frac{1}{3},2;\frac{7}{3};x^3,-x^3\right )+F_1\left (\frac{4}{3};\frac{2}{3},1;\frac{7}{3};x^3,-x^3\right )\right )-4 F_1\left (\frac{1}{3};-\frac{1}{3},1;\frac{4}{3};x^3,-x^3\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 0.038, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}+1}\sqrt [3]{-{x}^{3}+1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{x^{3} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 21.8302, size = 902, normalized size = 3.32 \begin{align*} \frac{1}{18} \, \sqrt{3} 2^{\frac{1}{3}} \arctan \left (-\frac{6 \, \sqrt{3} 2^{\frac{2}{3}}{\left (x^{16} - 33 \, x^{13} + 110 \, x^{10} - 110 \, x^{7} + 33 \, x^{4} - x\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} - 24 \, \sqrt{3} 2^{\frac{1}{3}}{\left (x^{14} - 2 \, x^{11} - 6 \, x^{8} - 2 \, x^{5} + x^{2}\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} - \sqrt{3}{\left (x^{18} + 42 \, x^{15} - 417 \, x^{12} + 812 \, x^{9} - 417 \, x^{6} + 42 \, x^{3} + 1\right )}}{3 \,{\left (x^{18} - 102 \, x^{15} + 447 \, x^{12} - 628 \, x^{9} + 447 \, x^{6} - 102 \, x^{3} + 1\right )}}\right ) + \frac{1}{18} \cdot 2^{\frac{1}{3}} \log \left (-\frac{12 \,{\left (-x^{3} + 1\right )}^{\frac{2}{3}} x^{2} + 2^{\frac{2}{3}}{\left (x^{6} + 2 \, x^{3} + 1\right )} - 6 \cdot 2^{\frac{1}{3}}{\left (x^{4} - x\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{x^{6} + 2 \, x^{3} + 1}\right ) - \frac{1}{36} \cdot 2^{\frac{1}{3}} \log \left (\frac{12 \cdot 2^{\frac{2}{3}}{\left (x^{8} - 4 \, x^{5} + x^{2}\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} + 2^{\frac{1}{3}}{\left (x^{12} - 32 \, x^{9} + 78 \, x^{6} - 32 \, x^{3} + 1\right )} + 6 \,{\left (x^{10} - 11 \, x^{7} + 11 \, x^{4} - x\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{x^{12} + 4 \, x^{9} + 6 \, x^{6} + 4 \, x^{3} + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*sqrt(3)*2^(1/3)*arctan(-1/3*(6*sqrt(3)*2^(2/3)*(x^16 - 33*x^13 + 110*x^10 - 110*x^7 + 33*x^4 - x)*(-x^3 +
1)^(1/3) - 24*sqrt(3)*2^(1/3)*(x^14 - 2*x^11 - 6*x^8 - 2*x^5 + x^2)*(-x^3 + 1)^(2/3) - sqrt(3)*(x^18 + 42*x^1
5 - 417*x^12 + 812*x^9 - 417*x^6 + 42*x^3 + 1))/(x^18 - 102*x^15 + 447*x^12 - 628*x^9 + 447*x^6 - 102*x^3 + 1)
) + 1/18*2^(1/3)*log(-(12*(-x^3 + 1)^(2/3)*x^2 + 2^(2/3)*(x^6 + 2*x^3 + 1) - 6*2^(1/3)*(x^4 - x)*(-x^3 + 1)^(1
/3))/(x^6 + 2*x^3 + 1)) - 1/36*2^(1/3)*log((12*2^(2/3)*(x^8 - 4*x^5 + x^2)*(-x^3 + 1)^(2/3) + 2^(1/3)*(x^12 -
32*x^9 + 78*x^6 - 32*x^3 + 1) + 6*(x^10 - 11*x^7 + 11*x^4 - x)*(-x^3 + 1)^(1/3))/(x^12 + 4*x^9 + 6*x^6 + 4*x^3
+ 1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{x^{3} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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