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

Optimal. Leaf size=109 \[ \frac{\tan ^{-1}\left (\frac{x}{\sqrt [3]{2} \sqrt [3]{x^2+1}+1}\right )}{2\ 2^{2/3}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} \sqrt [3]{x^2+1}\right )}{x}\right )}{2\ 2^{2/3} \sqrt{3}}-\frac{\tan ^{-1}(x)}{6\ 2^{2/3}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{3}}{x}\right )}{2\ 2^{2/3} \sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.0456091, antiderivative size = 109, 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 \[ \frac{\tan ^{-1}\left (\frac{x}{\sqrt [3]{2} \sqrt [3]{x^2+1}+1}\right )}{2\ 2^{2/3}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} \sqrt [3]{x^2+1}\right )}{x}\right )}{2\ 2^{2/3} \sqrt{3}}-\frac{\tan ^{-1}(x)}{6\ 2^{2/3}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{3}}{x}\right )}{2\ 2^{2/3} \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 11.585, size = 192, normalized size = 1.76 \[ \frac{\sqrt [3]{2} \sqrt{3} \log{\left (- x + \sqrt{3} \right )}}{24} - \frac{\sqrt [3]{2} \sqrt{3} \log{\left (x + \sqrt{3} \right )}}{24} + \frac{\sqrt [3]{2} \sqrt{3} \log{\left (- x - \sqrt [3]{2} \sqrt{3} \sqrt [3]{x^{2} + 1} + \sqrt{3} \right )}}{24} - \frac{\sqrt [3]{2} \sqrt{3} \log{\left (x - \sqrt [3]{2} \sqrt{3} \sqrt [3]{x^{2} + 1} + \sqrt{3} \right )}}{24} - \frac{\sqrt [3]{2} \operatorname{atan}{\left (\frac{2^{\frac{2}{3}} \left (- x + \sqrt{3}\right )}{3 \sqrt [3]{x^{2} + 1}} + \frac{\sqrt{3}}{3} \right )}}{12} + \frac{\sqrt [3]{2} \operatorname{atan}{\left (\frac{2^{\frac{2}{3}} \left (x + \sqrt{3}\right )}{3 \sqrt [3]{x^{2} + 1}} + \frac{\sqrt{3}}{3} \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2**(1/3)*sqrt(3)*log(-x + sqrt(3))/24 - 2**(1/3)*sqrt(3)*log(x + sqrt(3))/24 + 2
**(1/3)*sqrt(3)*log(-x - 2**(1/3)*sqrt(3)*(x**2 + 1)**(1/3) + sqrt(3))/24 - 2**(
1/3)*sqrt(3)*log(x - 2**(1/3)*sqrt(3)*(x**2 + 1)**(1/3) + sqrt(3))/24 - 2**(1/3)
*atan(2**(2/3)*(-x + sqrt(3))/(3*(x**2 + 1)**(1/3)) + sqrt(3)/3)/12 + 2**(1/3)*a
tan(2**(2/3)*(x + sqrt(3))/(3*(x**2 + 1)**(1/3)) + sqrt(3)/3)/12

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Mathematica [C]  time = 0.0558719, size = 124, normalized size = 1.14 \[ -\frac{9 x F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-x^2,\frac{x^2}{3}\right )}{\left (x^2-3\right ) \sqrt [3]{x^2+1} \left (2 x^2 \left (F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};-x^2,\frac{x^2}{3}\right )-F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};-x^2,\frac{x^2}{3}\right )\right )+9 F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-x^2,\frac{x^2}{3}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-9*x*AppellF1[1/2, 1/3, 1, 3/2, -x^2, x^2/3])/((-3 + x^2)*(1 + x^2)^(1/3)*(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 = 0.069, 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+3)/(x^2+1)^(1/3),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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