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

Optimal. Leaf size=482 \[ \sqrt [3]{1-x^3}-\frac{1}{3} \sqrt [3]{2} \log \left (x^3+1\right )+\frac{\log \left (2^{2/3}-\frac{1-x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}-\frac{\log \left (\frac{2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac{\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3\ 2^{2/3}}+\frac{1}{3} \sqrt [3]{2} \log \left (\frac{\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )-\frac{\log \left (\frac{(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac{2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}\right )}{6\ 2^{2/3}}+\frac{\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2^{2/3}}-\frac{1}{2} \log \left (-\sqrt [3]{1-x^3}-x\right )+\frac{\log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )}{2^{2/3}}+\frac{\sqrt [3]{2} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt{3}}+\frac{\tan ^{-1}\left (\frac{\frac{\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt{3}}+\frac{\sqrt [3]{2} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{\sqrt [3]{2} \tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{\sqrt{3}} \]

[Out]

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

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Rubi [F]  time = 0.0520396, 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 = {} \[ \int \frac{\sqrt [3]{1-x^3}}{1+x} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(1 - x^3)^(1/3)/(1 + x),x]

[Out]

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

Rubi steps

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

Mathematica [F]  time = 0.35477, size = 0, normalized size = 0. \[ \int \frac{\sqrt [3]{1-x^3}}{1+x} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(1 - x^3)^(1/3)/(1 + x),x]

[Out]

Integrate[(1 - x^3)^(1/3)/(1 + x), x]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((-x^3 + 1)^(1/3)/(x + 1), 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)/(1+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 + 1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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