3.1011 \(\int \frac{1}{x \sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx\)

Optimal. Leaf size=136 \[ \frac{\log \left (x^2+3\right )}{12\ 2^{2/3}}+\frac{1}{4} \log \left (1-\sqrt [3]{1-x^2}\right )-\frac{\log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{4\ 2^{2/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{2-2 x^2}+1}{\sqrt{3}}\right )}{2\ 2^{2/3} \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{1-x^2}+1}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{\log (x)}{6} \]

[Out]

-ArcTan[(1 + (2 - 2*x^2)^(1/3))/Sqrt[3]]/(2*2^(2/3)*Sqrt[3]) + ArcTan[(1 + 2*(1
- x^2)^(1/3))/Sqrt[3]]/(2*Sqrt[3]) - Log[x]/6 + Log[3 + x^2]/(12*2^(2/3)) + Log[
1 - (1 - x^2)^(1/3)]/4 - Log[2^(2/3) - (1 - x^2)^(1/3)]/(4*2^(2/3))

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Rubi [A]  time = 0.231748, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318 \[ \frac{\log \left (x^2+3\right )}{12\ 2^{2/3}}+\frac{1}{4} \log \left (1-\sqrt [3]{1-x^2}\right )-\frac{\log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{4\ 2^{2/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{2-2 x^2}+1}{\sqrt{3}}\right )}{2\ 2^{2/3} \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{1-x^2}+1}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{\log (x)}{6} \]

Antiderivative was successfully verified.

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

[Out]

-ArcTan[(1 + (2 - 2*x^2)^(1/3))/Sqrt[3]]/(2*2^(2/3)*Sqrt[3]) + ArcTan[(1 + 2*(1
- x^2)^(1/3))/Sqrt[3]]/(2*Sqrt[3]) - Log[x]/6 + Log[3 + x^2]/(12*2^(2/3)) + Log[
1 - (1 - x^2)^(1/3)]/4 - Log[2^(2/3) - (1 - x^2)^(1/3)]/(4*2^(2/3))

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Rubi in Sympy [A]  time = 13.6533, size = 121, normalized size = 0.89 \[ - \frac{\log{\left (x^{2} \right )}}{12} + \frac{\sqrt [3]{2} \log{\left (x^{2} + 3 \right )}}{24} + \frac{\log{\left (- \sqrt [3]{- x^{2} + 1} + 1 \right )}}{4} - \frac{\sqrt [3]{2} \log{\left (- \sqrt [3]{- x^{2} + 1} + 2^{\frac{2}{3}} \right )}}{8} - \frac{\sqrt [3]{2} \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{\sqrt [3]{2} \sqrt [3]{- x^{2} + 1}}{3} + \frac{1}{3}\right ) \right )}}{12} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{- x^{2} + 1}}{3} + \frac{1}{3}\right ) \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(x**2)/12 + 2**(1/3)*log(x**2 + 3)/24 + log(-(-x**2 + 1)**(1/3) + 1)/4 - 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))/12 + sqrt(3)*atan(sqrt(3)*(2*(-x**2 + 1)**(1/
3)/3 + 1/3))/6

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Mathematica [C]  time = 0.212917, size = 111, normalized size = 0.82 \[ -\frac{21 x^2 F_1\left (\frac{4}{3};\frac{1}{3},1;\frac{7}{3};\frac{1}{x^2},-\frac{3}{x^2}\right )}{8 \sqrt [3]{1-x^2} \left (x^2+3\right ) \left (7 x^2 F_1\left (\frac{4}{3};\frac{1}{3},1;\frac{7}{3};\frac{1}{x^2},-\frac{3}{x^2}\right )-9 F_1\left (\frac{7}{3};\frac{1}{3},2;\frac{10}{3};\frac{1}{x^2},-\frac{3}{x^2}\right )+F_1\left (\frac{7}{3};\frac{4}{3},1;\frac{10}{3};\frac{1}{x^2},-\frac{3}{x^2}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-21*x^2*AppellF1[4/3, 1/3, 1, 7/3, x^(-2), -3/x^2])/(8*(1 - x^2)^(1/3)*(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]))

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Maple [F]  time = 0.057, size = 0, normalized size = 0. \[ \int{\frac{1}{x \left ({x}^{2}+3 \right ) }{\frac{1}{\sqrt [3]{-{x}^{2}+1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.239655, size = 257, normalized size = 1.89 \[ -\frac{1}{144} \cdot 4^{\frac{2}{3}} \sqrt{3}{\left (\sqrt{3} \left (-1\right )^{\frac{1}{3}} \log \left (4^{\frac{2}{3}} \left (-1\right )^{\frac{2}{3}}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + 4^{\frac{1}{3}}{\left (-x^{2} + 1\right )}^{\frac{2}{3}} - 4 \, \left (-1\right )^{\frac{1}{3}}\right ) - 2 \, \sqrt{3} \left (-1\right )^{\frac{1}{3}} \log \left (4^{\frac{2}{3}}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} - 4 \, \left (-1\right )^{\frac{2}{3}}\right ) + 4^{\frac{1}{3}} \sqrt{3} \log \left ({\left (-x^{2} + 1\right )}^{\frac{2}{3}} +{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + 1\right ) - 2 \cdot 4^{\frac{1}{3}} \sqrt{3} \log \left ({\left (-x^{2} + 1\right )}^{\frac{1}{3}} - 1\right ) - 6 \, \left (-1\right )^{\frac{1}{3}} \arctan \left (-\frac{1}{6} \, \left (-1\right )^{\frac{1}{3}}{\left (4^{\frac{2}{3}} \sqrt{3}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + 2 \, \sqrt{3} \left (-1\right )^{\frac{2}{3}}\right )}\right ) - 6 \cdot 4^{\frac{1}{3}} \arctan \left (\frac{2}{3} \, \sqrt{3}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + \frac{1}{3} \, \sqrt{3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/144*4^(2/3)*sqrt(3)*(sqrt(3)*(-1)^(1/3)*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)*log(4^(2/3)
*(-x^2 + 1)^(1/3) - 4*(-1)^(2/3)) + 4^(1/3)*sqrt(3)*log((-x^2 + 1)^(2/3) + (-x^2
+ 1)^(1/3) + 1) - 2*4^(1/3)*sqrt(3)*log((-x^2 + 1)^(1/3) - 1) - 6*(-1)^(1/3)*ar
ctan(-1/6*(-1)^(1/3)*(4^(2/3)*sqrt(3)*(-x^2 + 1)^(1/3) + 2*sqrt(3)*(-1)^(2/3)))
- 6*4^(1/3)*arctan(2/3*sqrt(3)*(-x^2 + 1)^(1/3) + 1/3*sqrt(3)))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: NotImplementedError