3.1010 \(\int \frac{x}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx\)

Optimal. Leaf size=79 \[ -\frac{\log \left (x^2+3\right )}{4\ 2^{2/3}}+\frac{3 \log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{4\ 2^{2/3}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{2-2 x^2}+1}{\sqrt{3}}\right )}{2\ 2^{2/3}} \]

[Out]

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

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Rubi [A]  time = 0.120327, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{\log \left (x^2+3\right )}{4\ 2^{2/3}}+\frac{3 \log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{4\ 2^{2/3}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{2-2 x^2}+1}{\sqrt{3}}\right )}{2\ 2^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.78088, size = 73, normalized size = 0.92 \[ - \frac{\sqrt [3]{2} \log{\left (x^{2} + 3 \right )}}{8} + \frac{3 \sqrt [3]{2} \log{\left (- \sqrt [3]{- x^{2} + 1} + 2^{\frac{2}{3}} \right )}}{8} + \frac{\sqrt [3]{2} \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{\sqrt [3]{2} \sqrt [3]{- x^{2} + 1}}{3} + \frac{1}{3}\right ) \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2**(1/3)*log(x**2 + 3)/8 + 3*2**(1/3)*log(-(-x**2 + 1)**(1/3) + 2**(2/3))/8 + 2
**(1/3)*sqrt(3)*atan(sqrt(3)*(2**(1/3)*(-x**2 + 1)**(1/3)/3 + 1/3))/4

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Mathematica [A]  time = 0.0674892, size = 84, normalized size = 1.06 \[ \frac{2 \log \left (2-\sqrt [3]{2-2 x^2}\right )-\log \left (\left (2-2 x^2\right )^{2/3}+2 \sqrt [3]{2-2 x^2}+4\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{2-2 x^2}+1}{\sqrt{3}}\right )}{4\ 2^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(-x^2+1)^(1/3)/(x^2+3),x)

[Out]

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

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Maxima [A]  time = 1.50847, size = 116, normalized size = 1.47 \[ \frac{1}{8} \cdot 4^{\frac{2}{3}} \sqrt{3} \arctan \left (\frac{1}{12} \cdot 4^{\frac{2}{3}} \sqrt{3}{\left (4^{\frac{1}{3}} + 2 \,{\left (-x^{2} + 1\right )}^{\frac{1}{3}}\right )}\right ) - \frac{1}{16} \cdot 4^{\frac{2}{3}} \log \left (4^{\frac{2}{3}} + 4^{\frac{1}{3}}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} +{\left (-x^{2} + 1\right )}^{\frac{2}{3}}\right ) + \frac{1}{8} \cdot 4^{\frac{2}{3}} \log \left (-4^{\frac{1}{3}} +{\left (-x^{2} + 1\right )}^{\frac{1}{3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*4^(2/3)*sqrt(3)*arctan(1/12*4^(2/3)*sqrt(3)*(4^(1/3) + 2*(-x^2 + 1)^(1/3)))
- 1/16*4^(2/3)*log(4^(2/3) + 4^(1/3)*(-x^2 + 1)^(1/3) + (-x^2 + 1)^(2/3)) + 1/8*
4^(2/3)*log(-4^(1/3) + (-x^2 + 1)^(1/3))

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Fricas [A]  time = 0.236298, size = 109, normalized size = 1.38 \[ \frac{1}{16} \cdot 4^{\frac{2}{3}}{\left (2 \, \sqrt{3} \arctan \left (\frac{1}{6} \, \sqrt{3}{\left (4^{\frac{2}{3}}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + 2\right )}\right ) - \log \left (4^{\frac{2}{3}}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + 4^{\frac{1}{3}}{\left (-x^{2} + 1\right )}^{\frac{2}{3}} + 4\right ) + 2 \, \log \left (4^{\frac{2}{3}}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} - 4\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*4^(2/3)*(2*sqrt(3)*arctan(1/6*sqrt(3)*(4^(2/3)*(-x^2 + 1)^(1/3) + 2)) - log
(4^(2/3)*(-x^2 + 1)^(1/3) + 4^(1/3)*(-x^2 + 1)^(2/3) + 4) + 2*log(4^(2/3)*(-x^2
+ 1)^(1/3) - 4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError