3.60 \(\int \frac{\sqrt [3]{1-x^3}}{2+x} \, dx\)

Optimal. Leaf size=232 \[ \frac{1}{2} x F_1\left (\frac{1}{3};-\frac{1}{3},1;\frac{4}{3};x^3,-\frac{x^3}{8}\right )+\sqrt [3]{1-x^3}-\frac{\log \left (x^3+8\right )}{\sqrt [3]{3}}+\frac{1}{2} 3^{2/3} \log \left (3^{2/3}-\sqrt [3]{1-x^3}\right )-\log \left (-\sqrt [3]{1-x^3}-x\right )+\frac{1}{2} 3^{2/3} \log \left (-\sqrt [3]{1-x^3}-\frac{1}{2} 3^{2/3} x\right )-\frac{2 \tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt{3}}+\sqrt [6]{3} \tan ^{-1}\left (\frac{1-\frac{3^{2/3} x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )-\sqrt [6]{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{1-x^3}}{3 \sqrt [6]{3}}+\frac{1}{\sqrt{3}}\right ) \]

[Out]

(1 - x^3)^(1/3) + (x*AppellF1[1/3, -1/3, 1, 4/3, x^3, -x^3/8])/2 - (2*ArcTan[(1 - (2*x)/(1 - x^3)^(1/3))/Sqrt[
3]])/Sqrt[3] + 3^(1/6)*ArcTan[(1 - (3^(2/3)*x)/(1 - x^3)^(1/3))/Sqrt[3]] - 3^(1/6)*ArcTan[1/Sqrt[3] + (2*(1 -
x^3)^(1/3))/(3*3^(1/6))] - Log[8 + x^3]/3^(1/3) + (3^(2/3)*Log[3^(2/3) - (1 - x^3)^(1/3)])/2 - Log[-x - (1 - x
^3)^(1/3)] + (3^(2/3)*Log[-(3^(2/3)*x)/2 - (1 - x^3)^(1/3)])/2

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Rubi [F]  time = 0.0499624, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\sqrt [3]{1-x^3}}{2+x} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(1 - x^3)^(1/3)/(2 + x),x]

[Out]

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

Rubi steps

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

Mathematica [F]  time = 0.343464, size = 0, normalized size = 0. \[ \int \frac{\sqrt [3]{1-x^3}}{2+x} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(1 - x^3)^(1/3)/(2 + x),x]

[Out]

Integrate[(1 - x^3)^(1/3)/(2 + x), x]

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Maple [F]  time = 0.048, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{2+x}\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)/(2+x),x)

[Out]

int((-x^3+1)^(1/3)/(2+x),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 + 2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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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 )}}{x + 2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((-(x - 1)*(x**2 + x + 1))**(1/3)/(x + 2), 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 + 2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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