3.1013 \(\int \frac{1}{x^5 \sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx\)

Optimal. Leaf size=172 \[ -\frac{\left (1-x^2\right )^{2/3}}{18 x^2}+\frac{\log \left (x^2+3\right )}{108\ 2^{2/3}}+\frac{1}{18} \log \left (1-\sqrt [3]{1-x^2}\right )-\frac{\log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{36\ 2^{2/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{2-2 x^2}+1}{\sqrt{3}}\right )}{18\ 2^{2/3} \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{1-x^2}+1}{\sqrt{3}}\right )}{9 \sqrt{3}}-\frac{\left (1-x^2\right )^{2/3}}{12 x^4}-\frac{\log (x)}{27} \]

[Out]

-(1 - x^2)^(2/3)/(12*x^4) - (1 - x^2)^(2/3)/(18*x^2) - ArcTan[(1 + (2 - 2*x^2)^(
1/3))/Sqrt[3]]/(18*2^(2/3)*Sqrt[3]) + ArcTan[(1 + 2*(1 - x^2)^(1/3))/Sqrt[3]]/(9
*Sqrt[3]) - Log[x]/27 + Log[3 + x^2]/(108*2^(2/3)) + Log[1 - (1 - x^2)^(1/3)]/18
- Log[2^(2/3) - (1 - x^2)^(1/3)]/(36*2^(2/3))

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Rubi [A]  time = 0.352539, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409 \[ -\frac{\left (1-x^2\right )^{2/3}}{18 x^2}+\frac{\log \left (x^2+3\right )}{108\ 2^{2/3}}+\frac{1}{18} \log \left (1-\sqrt [3]{1-x^2}\right )-\frac{\log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{36\ 2^{2/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{2-2 x^2}+1}{\sqrt{3}}\right )}{18\ 2^{2/3} \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{1-x^2}+1}{\sqrt{3}}\right )}{9 \sqrt{3}}-\frac{\left (1-x^2\right )^{2/3}}{12 x^4}-\frac{\log (x)}{27} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^5*(1 - x^2)^(1/3)*(3 + x^2)),x]

[Out]

-(1 - x^2)^(2/3)/(12*x^4) - (1 - x^2)^(2/3)/(18*x^2) - ArcTan[(1 + (2 - 2*x^2)^(
1/3))/Sqrt[3]]/(18*2^(2/3)*Sqrt[3]) + ArcTan[(1 + 2*(1 - x^2)^(1/3))/Sqrt[3]]/(9
*Sqrt[3]) - Log[x]/27 + Log[3 + x^2]/(108*2^(2/3)) + Log[1 - (1 - x^2)^(1/3)]/18
- Log[2^(2/3) - (1 - x^2)^(1/3)]/(36*2^(2/3))

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Rubi in Sympy [A]  time = 22.7552, size = 148, normalized size = 0.86 \[ - \frac{\log{\left (x^{2} \right )}}{54} + \frac{\sqrt [3]{2} \log{\left (x^{2} + 3 \right )}}{216} + \frac{\log{\left (- \sqrt [3]{- x^{2} + 1} + 1 \right )}}{18} - \frac{\sqrt [3]{2} \log{\left (- \sqrt [3]{- x^{2} + 1} + 2^{\frac{2}{3}} \right )}}{72} - \frac{\sqrt [3]{2} \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{\sqrt [3]{2} \sqrt [3]{- x^{2} + 1}}{3} + \frac{1}{3}\right ) \right )}}{108} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{- x^{2} + 1}}{3} + \frac{1}{3}\right ) \right )}}{27} - \frac{\left (- x^{2} + 1\right )^{\frac{2}{3}}}{18 x^{2}} - \frac{\left (- x^{2} + 1\right )^{\frac{2}{3}}}{12 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**5/(-x**2+1)**(1/3)/(x**2+3),x)

[Out]

-log(x**2)/54 + 2**(1/3)*log(x**2 + 3)/216 + log(-(-x**2 + 1)**(1/3) + 1)/18 - 2
**(1/3)*log(-(-x**2 + 1)**(1/3) + 2**(2/3))/72 - 2**(1/3)*sqrt(3)*atan(sqrt(3)*(
2**(1/3)*(-x**2 + 1)**(1/3)/3 + 1/3))/108 + sqrt(3)*atan(sqrt(3)*(2*(-x**2 + 1)*
*(1/3)/3 + 1/3))/27 - (-x**2 + 1)**(2/3)/(18*x**2) - (-x**2 + 1)**(2/3)/(12*x**4
)

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Mathematica [C]  time = 0.293443, size = 215, normalized size = 1.25 \[ \frac{-\frac{4 x^2 F_1\left (1;\frac{1}{3},1;2;x^2,-\frac{x^2}{3}\right )}{\left (x^2+3\right ) \left (x^2 \left (F_1\left (2;\frac{1}{3},2;3;x^2,-\frac{x^2}{3}\right )-F_1\left (2;\frac{4}{3},1;3;x^2,-\frac{x^2}{3}\right )\right )-6 F_1\left (1;\frac{1}{3},1;2;x^2,-\frac{x^2}{3}\right )\right )}-\frac{21 x^2 F_1\left (\frac{4}{3};\frac{1}{3},1;\frac{7}{3};\frac{1}{x^2},-\frac{3}{x^2}\right )}{\left (x^2+3\right ) \left (7 x^2 F_1\left (\frac{4}{3};\frac{1}{3},1;\frac{7}{3};\frac{1}{x^2},-\frac{3}{x^2}\right )-9 F_1\left (\frac{7}{3};\frac{1}{3},2;\frac{10}{3};\frac{1}{x^2},-\frac{3}{x^2}\right )+F_1\left (\frac{7}{3};\frac{4}{3},1;\frac{10}{3};\frac{1}{x^2},-\frac{3}{x^2}\right )\right )}-\frac{3}{x^4}+\frac{1}{x^2}+2}{36 \sqrt [3]{1-x^2}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/(x^5*(1 - x^2)^(1/3)*(3 + x^2)),x]

[Out]

(2 - 3/x^4 + x^(-2) - (4*x^2*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]))) - (21*x^2*AppellF1[4/3, 1/3, 1, 7/3, x^
(-2), -3/x^2])/((3 + x^2)*(7*x^2*AppellF1[4/3, 1/3, 1, 7/3, x^(-2), -3/x^2] - 9*
AppellF1[7/3, 1/3, 2, 10/3, x^(-2), -3/x^2] + AppellF1[7/3, 4/3, 1, 10/3, x^(-2)
, -3/x^2])))/(36*(1 - x^2)^(1/3))

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Maple [F]  time = 0.081, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{5} \left ({x}^{2}+3 \right ) }{\frac{1}{\sqrt [3]{-{x}^{2}+1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^5/(-x^2+1)^(1/3)/(x^2+3),x)

[Out]

int(1/x^5/(-x^2+1)^(1/3)/(x^2+3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{2} + 3\right )}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^2 + 3)*(-x^2 + 1)^(1/3)*x^5),x, algorithm="maxima")

[Out]

integrate(1/((x^2 + 3)*(-x^2 + 1)^(1/3)*x^5), x)

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Fricas [A]  time = 0.243442, size = 319, normalized size = 1.85 \[ -\frac{4^{\frac{2}{3}} \sqrt{3}{\left (\sqrt{3} \left (-1\right )^{\frac{1}{3}} x^{4} \log \left (4^{\frac{2}{3}} \left (-1\right )^{\frac{2}{3}}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + 4^{\frac{1}{3}}{\left (-x^{2} + 1\right )}^{\frac{2}{3}} - 4 \, \left (-1\right )^{\frac{1}{3}}\right ) - 2 \, \sqrt{3} \left (-1\right )^{\frac{1}{3}} x^{4} \log \left (4^{\frac{2}{3}}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} - 4 \, \left (-1\right )^{\frac{2}{3}}\right ) + 2 \cdot 4^{\frac{1}{3}} \sqrt{3} x^{4} \log \left ({\left (-x^{2} + 1\right )}^{\frac{2}{3}} +{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + 1\right ) - 4 \cdot 4^{\frac{1}{3}} \sqrt{3} x^{4} \log \left ({\left (-x^{2} + 1\right )}^{\frac{1}{3}} - 1\right ) - 6 \, \left (-1\right )^{\frac{1}{3}} x^{4} \arctan \left (-\frac{1}{6} \, \left (-1\right )^{\frac{1}{3}}{\left (4^{\frac{2}{3}} \sqrt{3}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + 2 \, \sqrt{3} \left (-1\right )^{\frac{2}{3}}\right )}\right ) - 12 \cdot 4^{\frac{1}{3}} x^{4} \arctan \left (\frac{2}{3} \, \sqrt{3}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + \frac{1}{3} \, \sqrt{3}\right ) + 3 \cdot 4^{\frac{1}{3}} \sqrt{3}{\left (2 \, x^{2} + 3\right )}{\left (-x^{2} + 1\right )}^{\frac{2}{3}}\right )}}{1296 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^2 + 3)*(-x^2 + 1)^(1/3)*x^5),x, algorithm="fricas")

[Out]

-1/1296*4^(2/3)*sqrt(3)*(sqrt(3)*(-1)^(1/3)*x^4*log(4^(2/3)*(-1)^(2/3)*(-x^2 + 1
)^(1/3) + 4^(1/3)*(-x^2 + 1)^(2/3) - 4*(-1)^(1/3)) - 2*sqrt(3)*(-1)^(1/3)*x^4*lo
g(4^(2/3)*(-x^2 + 1)^(1/3) - 4*(-1)^(2/3)) + 2*4^(1/3)*sqrt(3)*x^4*log((-x^2 + 1
)^(2/3) + (-x^2 + 1)^(1/3) + 1) - 4*4^(1/3)*sqrt(3)*x^4*log((-x^2 + 1)^(1/3) - 1
) - 6*(-1)^(1/3)*x^4*arctan(-1/6*(-1)^(1/3)*(4^(2/3)*sqrt(3)*(-x^2 + 1)^(1/3) +
2*sqrt(3)*(-1)^(2/3))) - 12*4^(1/3)*x^4*arctan(2/3*sqrt(3)*(-x^2 + 1)^(1/3) + 1/
3*sqrt(3)) + 3*4^(1/3)*sqrt(3)*(2*x^2 + 3)*(-x^2 + 1)^(2/3))/x^4

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{5} \sqrt [3]{- \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + 3\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**5/(-x**2+1)**(1/3)/(x**2+3),x)

[Out]

Integral(1/(x**5*(-(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^5),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError