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

Optimal. Leaf size=108 \[ \frac{3}{4} \log \left (-\sqrt [3]{x^3+2}+x+2\right )-\frac{1}{4} \log \left (\sqrt [3]{x^3+2}-x\right )+\frac{\tan ^{-1}\left (\frac{\frac{2 x}{\sqrt [3]{x^3+2}}+1}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{1}{2} \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 (x+2)}{\sqrt [3]{x^3+2}}+1}{\sqrt{3}}\right )-\frac{1}{2} \log (x+1) \]

[Out]

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

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Rubi [A]  time = 0.160376, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{3}{4} \log \left (-\sqrt [3]{x^3+2}+x+2\right )-\frac{1}{4} \log \left (\sqrt [3]{x^3+2}-x\right )+\frac{\tan ^{-1}\left (\frac{\frac{2 x}{\sqrt [3]{x^3+2}}+1}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{1}{2} \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 (x+2)}{\sqrt [3]{x^3+2}}+1}{\sqrt{3}}\right )-\frac{1}{2} \log (x+1) \]

Antiderivative was successfully verified.

[In]  Int[1/((1 + x)*(2 + x^3)^(1/3)),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(1+x)/(x**3+2)**(1/3),x)

[Out]

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

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Mathematica [A]  time = 0.0469073, size = 0, normalized size = 0. \[ \int \frac{1}{(1+x) \sqrt [3]{2+x^3}} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[1/((1 + x)*(2 + x^3)^(1/3)),x]

[Out]

Integrate[1/((1 + x)*(2 + x^3)^(1/3)), x]

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Maple [F]  time = 0.039, size = 0, normalized size = 0. \[ \int{\frac{1}{1+x}{\frac{1}{\sqrt [3]{{x}^{3}+2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1+x)/(x^3+2)^(1/3),x)

[Out]

int(1/(1+x)/(x^3+2)^(1/3),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^3 + 2)^(1/3)*(x + 1)),x, algorithm="maxima")

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(1+x)/(x**3+2)**(1/3),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{3} + 2\right )}^{\frac{1}{3}}{\left (x + 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^3 + 2)^(1/3)*(x + 1)),x, algorithm="giac")

[Out]

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