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

Optimal. Leaf size=145 \[ \frac{1}{2} \log \left (\sqrt [3]{1-x^3}+x\right )-\frac{3 \log \left (2^{2/3} \sqrt [3]{1-x^3}+x-1\right )}{4 \sqrt [3]{2}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{\frac{\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt{3}}\right )}{2 \sqrt [3]{2}}-\frac{\tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt{3}}+\frac{\log \left ((1-x) (x+1)^2\right )}{4 \sqrt [3]{2}} \]

[Out]

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

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Rubi [A]  time = 0.193547, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{1}{2} \log \left (\sqrt [3]{1-x^3}+x\right )-\frac{3 \log \left (2^{2/3} \sqrt [3]{1-x^3}+x-1\right )}{4 \sqrt [3]{2}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{\frac{\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt{3}}\right )}{2 \sqrt [3]{2}}-\frac{\tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt{3}}+\frac{\log \left ((1-x) (x+1)^2\right )}{4 \sqrt [3]{2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(1+x)/(-x**3+1)**(1/3),x)

[Out]

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

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Mathematica [A]  time = 0.0889764, size = 0, normalized size = 0. \[ \int \frac{x}{(1+x) \sqrt [3]{1-x^3}} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.059, size = 0, normalized size = 0. \[ \int{\frac{x}{1+x}{\frac{1}{\sqrt [3]{-{x}^{3}+1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(1+x)/(-x^3+1)^(1/3),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/((-x^3 + 1)^(1/3)*(x + 1)),x, algorithm="maxima")

[Out]

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

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/((-x^3 + 1)^(1/3)*(x + 1)),x, algorithm="fricas")

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(1+x)/(-x**3+1)**(1/3),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{{\left (-x^{3} + 1\right )}^{\frac{1}{3}}{\left (x + 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/((-x^3 + 1)^(1/3)*(x + 1)),x, algorithm="giac")

[Out]

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