3.1012 \(\int \frac{1}{x^3 \sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx\)

Optimal. Leaf size=97 \[ -\frac{\left (1-x^2\right )^{2/3}}{6 x^2}-\frac{\log \left (x^2+3\right )}{36\ 2^{2/3}}+\frac{\log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{12\ 2^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{2-2 x^2}+1}{\sqrt{3}}\right )}{6\ 2^{2/3} \sqrt{3}} \]

[Out]

-(1 - x^2)^(2/3)/(6*x^2) + ArcTan[(1 + (2 - 2*x^2)^(1/3))/Sqrt[3]]/(6*2^(2/3)*Sq
rt[3]) - Log[3 + x^2]/(36*2^(2/3)) + Log[2^(2/3) - (1 - x^2)^(1/3)]/(12*2^(2/3))

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Rubi [A]  time = 0.183307, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318 \[ -\frac{\left (1-x^2\right )^{2/3}}{6 x^2}-\frac{\log \left (x^2+3\right )}{36\ 2^{2/3}}+\frac{\log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{12\ 2^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{2-2 x^2}+1}{\sqrt{3}}\right )}{6\ 2^{2/3} \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

-(1 - x^2)^(2/3)/(6*x^2) + ArcTan[(1 + (2 - 2*x^2)^(1/3))/Sqrt[3]]/(6*2^(2/3)*Sq
rt[3]) - Log[3 + x^2]/(36*2^(2/3)) + Log[2^(2/3) - (1 - x^2)^(1/3)]/(12*2^(2/3))

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Rubi in Sympy [A]  time = 13.0103, size = 85, normalized size = 0.88 \[ - \frac{\sqrt [3]{2} \log{\left (x^{2} + 3 \right )}}{72} + \frac{\sqrt [3]{2} \log{\left (- \sqrt [3]{- x^{2} + 1} + 2^{\frac{2}{3}} \right )}}{24} + \frac{\sqrt [3]{2} \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{\sqrt [3]{2} \sqrt [3]{- x^{2} + 1}}{3} + \frac{1}{3}\right ) \right )}}{36} - \frac{\left (- x^{2} + 1\right )^{\frac{2}{3}}}{6 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2**(1/3)*log(x**2 + 3)/72 + 2**(1/3)*log(-(-x**2 + 1)**(1/3) + 2**(2/3))/24 + 2
**(1/3)*sqrt(3)*atan(sqrt(3)*(2**(1/3)*(-x**2 + 1)**(1/3)/3 + 1/3))/36 - (-x**2
+ 1)**(2/3)/(6*x**2)

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Mathematica [C]  time = 0.202599, size = 115, normalized size = 1.19 \[ \frac{-\frac{2 x^4 F_1\left (1;\frac{1}{3},1;2;x^2,-\frac{x^2}{3}\right )}{\left (x^2+3\right ) \left (x^2 \left (F_1\left (2;\frac{1}{3},2;3;x^2,-\frac{x^2}{3}\right )-F_1\left (2;\frac{4}{3},1;3;x^2,-\frac{x^2}{3}\right )\right )-6 F_1\left (1;\frac{1}{3},1;2;x^2,-\frac{x^2}{3}\right )\right )}+x^2-1}{6 x^2 \sqrt [3]{1-x^2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-1 + x^2 - (2*x^4*AppellF1[1, 1/3, 1, 2, x^2, -x^2/3])/((3 + x^2)*(-6*AppellF1[
1, 1/3, 1, 2, x^2, -x^2/3] + x^2*(AppellF1[2, 1/3, 2, 3, x^2, -x^2/3] - AppellF1
[2, 4/3, 1, 3, x^2, -x^2/3]))))/(6*x^2*(1 - x^2)^(1/3))

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Maple [F]  time = 0.064, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{3} \left ({x}^{2}+3 \right ) }{\frac{1}{\sqrt [3]{-{x}^{2}+1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.23311, size = 159, normalized size = 1.64 \[ -\frac{4^{\frac{2}{3}} \sqrt{3}{\left (\sqrt{3} x^{2} \log \left (4^{\frac{2}{3}}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + 4^{\frac{1}{3}}{\left (-x^{2} + 1\right )}^{\frac{2}{3}} + 4\right ) - 2 \, \sqrt{3} x^{2} \log \left (4^{\frac{2}{3}}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} - 4\right ) - 6 \, x^{2} \arctan \left (\frac{1}{6} \cdot 4^{\frac{2}{3}} \sqrt{3}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + \frac{1}{3} \, \sqrt{3}\right ) + 6 \cdot 4^{\frac{1}{3}} \sqrt{3}{\left (-x^{2} + 1\right )}^{\frac{2}{3}}\right )}}{432 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/432*4^(2/3)*sqrt(3)*(sqrt(3)*x^2*log(4^(2/3)*(-x^2 + 1)^(1/3) + 4^(1/3)*(-x^2
+ 1)^(2/3) + 4) - 2*sqrt(3)*x^2*log(4^(2/3)*(-x^2 + 1)^(1/3) - 4) - 6*x^2*arcta
n(1/6*4^(2/3)*sqrt(3)*(-x^2 + 1)^(1/3) + 1/3*sqrt(3)) + 6*4^(1/3)*sqrt(3)*(-x^2
+ 1)^(2/3))/x^2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x**3*(-(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(1/((x^2 + 3)*(-x^2 + 1)^(1/3)*x^3),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError