3.1016 \(\int \frac{1}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx\)

Optimal. Leaf size=113 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{2\ 2^{2/3} \sqrt{3}}+\frac{\tanh ^{-1}\left (\frac{x}{\sqrt [3]{2} \sqrt [3]{1-x^2}+1}\right )}{2\ 2^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt{3}}{x}\right )}{2\ 2^{2/3} \sqrt{3}}-\frac{\tanh ^{-1}(x)}{6\ 2^{2/3}} \]

[Out]

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

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Rubi [A]  time = 0.0606573, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{2\ 2^{2/3} \sqrt{3}}+\frac{\tanh ^{-1}\left (\frac{x}{\sqrt [3]{2} \sqrt [3]{1-x^2}+1}\right )}{2\ 2^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt{3}}{x}\right )}{2\ 2^{2/3} \sqrt{3}}-\frac{\tanh ^{-1}(x)}{6\ 2^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.832, size = 144, normalized size = 1.27 \[ \frac{\sqrt [3]{2} \log{\left (\sqrt [3]{2} \sqrt [3]{- x + 1} + \left (x + 1\right )^{\frac{2}{3}} \right )}}{8} - \frac{\sqrt [3]{2} \log{\left (\left (- x + 1\right )^{\frac{2}{3}} + \sqrt [3]{2} \sqrt [3]{x + 1} \right )}}{8} - \frac{\sqrt [3]{2} \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3}}{3} - \frac{2^{\frac{2}{3}} \sqrt{3} \left (x + 1\right )^{\frac{2}{3}}}{3 \sqrt [3]{- x + 1}} \right )}}{12} - \frac{\sqrt [3]{2} \sqrt{3} \operatorname{atan}{\left (\frac{2^{\frac{2}{3}} \sqrt{3} \left (- x + 1\right )^{\frac{2}{3}}}{3 \sqrt [3]{x + 1}} - \frac{\sqrt{3}}{3} \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2**(1/3)*log(2**(1/3)*(-x + 1)**(1/3) + (x + 1)**(2/3))/8 - 2**(1/3)*log((-x + 1
)**(2/3) + 2**(1/3)*(x + 1)**(1/3))/8 - 2**(1/3)*sqrt(3)*atan(sqrt(3)/3 - 2**(2/
3)*sqrt(3)*(x + 1)**(2/3)/(3*(-x + 1)**(1/3)))/12 - 2**(1/3)*sqrt(3)*atan(2**(2/
3)*sqrt(3)*(-x + 1)**(2/3)/(3*(x + 1)**(1/3)) - sqrt(3)/3)/12

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Mathematica [C]  time = 0.0734627, size = 118, normalized size = 1.04 \[ -\frac{9 x F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};x^2,-\frac{x^2}{3}\right )}{\sqrt [3]{1-x^2} \left (x^2+3\right ) \left (2 x^2 \left (F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};x^2,-\frac{x^2}{3}\right )-F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right )\right )-9 F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};x^2,-\frac{x^2}{3}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-9*x*AppellF1[1/2, 1/3, 1, 3/2, x^2, -x^2/3])/((1 - x^2)^(1/3)*(3 + x^2)*(-9*Ap
pellF1[1/2, 1/3, 1, 3/2, x^2, -x^2/3] + 2*x^2*(AppellF1[3/2, 1/3, 2, 5/2, x^2, -
x^2/3] - AppellF1[3/2, 4/3, 1, 5/2, x^2, -x^2/3])))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-x^2+1)^(1/3)/(x^2+3),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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