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

Optimal. Leaf size=16 \[ \frac{3}{4} \log \left (1-\left (x^2\right )^{2/3}\right ) \]

[Out]

(3*Log[1 - (x^2)^(2/3)])/4

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Rubi [A]  time = 0.0953562, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{3}{4} \log \left (1-\left (x^2\right )^{2/3}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x/(x^2 - (x^2)^(1/3)),x]

[Out]

(3*Log[1 - (x^2)^(2/3)])/4

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Rubi in Sympy [A]  time = 4.7208, size = 12, normalized size = 0.75 \[ \frac{3 \log{\left (- \left (x^{2}\right )^{\frac{2}{3}} + 1 \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*log(-(x**2)**(2/3) + 1)/4

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Mathematica [A]  time = 0.0129344, size = 14, normalized size = 0.88 \[ \frac{3}{4} \log \left (\left (x^2\right )^{2/3}-1\right ) \]

Antiderivative was successfully verified.

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

[Out]

(3*Log[-1 + (x^2)^(2/3)])/4

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Maple [B]  time = 0.056, size = 70, normalized size = 4.4 \[{\frac{\ln \left ({x}^{2}-1 \right ) }{4}}+{\frac{\ln \left ({x}^{2}+1 \right ) }{4}}-{\frac{1}{4}\ln \left ( \left ({x}^{2} \right ) ^{{\frac{2}{3}}}+\sqrt [3]{{x}^{2}}+1 \right ) }+{\frac{1}{2}\ln \left ( \sqrt [3]{{x}^{2}}-1 \right ) }+{\frac{1}{2}\ln \left ( 1+\sqrt [3]{{x}^{2}} \right ) }-{\frac{1}{4}\ln \left ( \left ({x}^{2} \right ) ^{{\frac{2}{3}}}-\sqrt [3]{{x}^{2}}+1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(x^2-(x^2)^(1/3)),x)

[Out]

1/4*ln(x^2-1)+1/4*ln(x^2+1)-1/4*ln((x^2)^(2/3)+(x^2)^(1/3)+1)+1/2*ln((x^2)^(1/3)
-1)+1/2*ln(1+(x^2)^(1/3))-1/4*ln((x^2)^(2/3)-(x^2)^(1/3)+1)

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Maxima [A]  time = 0.719974, size = 28, normalized size = 1.75 \[ \frac{3}{4} \, \log \left ({\left (x^{2}\right )}^{\frac{1}{3}} + 1\right ) + \frac{3}{4} \, \log \left ({\left (x^{2}\right )}^{\frac{1}{3}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/4*log((x^2)^(1/3) + 1) + 3/4*log((x^2)^(1/3) - 1)

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Fricas [A]  time = 0.266958, size = 43, normalized size = 2.69 \[ -3 \, \log \left (\frac{{\left (x^{2}\right )}^{\frac{1}{3}}}{x}\right ) + \frac{3}{4} \, \log \left (-\frac{x^{2} -{\left (x^{2}\right )}^{\frac{1}{3}}}{x^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(x^2 - (x^2)^(1/3)),x, algorithm="fricas")

[Out]

-3*log((x^2)^(1/3)/x) + 3/4*log(-(x^2 - (x^2)^(1/3))/x^2)

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Sympy [A]  time = 0.584835, size = 19, normalized size = 1.19 \[ - \frac{\log{\left (x \right )}}{2} + \frac{3 \log{\left (x^{2} - \sqrt [3]{x^{2}} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(x)/2 + 3*log(x**2 - (x**2)**(1/3))/4

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GIAC/XCAS [A]  time = 0.266682, size = 22, normalized size = 1.38 \[ \frac{3}{4} \,{\rm ln}\left ({\left | \left (x{\rm sign}\left (x\right )\right )^{\frac{1}{3}} x{\rm sign}\left (x\right ) - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/4*ln(abs((x*sign(x))^(1/3)*x*sign(x) - 1))