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

Optimal. Leaf size=121 \[ \frac{3 \log \left (2^{2/3} \sqrt [3]{1-x^3}+x-1\right )}{4 \sqrt [3]{2}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x^3}+2 (1-x)}{2^{2/3} \sqrt{3} \sqrt [3]{1-x^3}}\right )}{2 \sqrt [3]{2}}-\frac{\log (1-x)}{4 \sqrt [3]{2}}-\frac{\log (x+1)}{2 \sqrt [3]{2}} \]

[Out]

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

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Rubi [A]  time = 0.0830356, antiderivative size = 97, normalized size of antiderivative = 0.8, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{3 \log \left (2^{2/3} \sqrt [3]{1-x^3}+x-1\right )}{4 \sqrt [3]{2}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\frac{\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt{3}}\right )}{2 \sqrt [3]{2}}-\frac{\log \left ((1-x) (x+1)^2\right )}{4 \sqrt [3]{2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((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)) -
Log[(1 - x)*(1 + x)^2]/(4*2^(1/3)) + (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. \[ \int \frac{1}{\left (x + 1\right ) \sqrt [3]{- x^{3} + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0732422, size = 0, normalized size = 0. \[ \int \frac{1}{(1+x) \sqrt [3]{1-x^3}} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.049, size = 0, normalized size = 0. \[ \int{\frac{1}{1+x}{\frac{1}{\sqrt [3]{-{x}^{3}+1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1+x)/(-x^3+1)^(1/3),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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