3.703 \(\int \frac{1}{(3-x) \sqrt [3]{1-x^2}} \, dx\)

Optimal. Leaf size=78 \[ -\frac{1}{4} \log (3-x)+\frac{3}{8} \log \left (-\frac{1}{2} (x+1)^{2/3}-\sqrt [3]{1-x}\right )-\frac{1}{4} \sqrt{3} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{(x+1)^{2/3}}{\sqrt{3} \sqrt [3]{1-x}}\right ) \]

[Out]

-(Sqrt[3]*ArcTan[1/Sqrt[3] - (1 + x)^(2/3)/(Sqrt[3]*(1 - x)^(1/3))])/4 - Log[3 -
x]/4 + (3*Log[-(1 - x)^(1/3) - (1 + x)^(2/3)/2])/8

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Rubi [A]  time = 0.0749439, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{1}{4} \log (3-x)+\frac{3}{8} \log \left (-\frac{1}{2} (x+1)^{2/3}-\sqrt [3]{1-x}\right )-\frac{1}{4} \sqrt{3} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{(x+1)^{2/3}}{\sqrt{3} \sqrt [3]{1-x}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[3]*ArcTan[1/Sqrt[3] - (1 + x)^(2/3)/(Sqrt[3]*(1 - x)^(1/3))])/4 - Log[3 -
x]/4 + (3*Log[-(1 - x)^(1/3) - (1 + x)^(2/3)/2])/8

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Rubi in Sympy [A]  time = 9.59674, size = 63, normalized size = 0.81 \[ - \frac{\log{\left (- x + 3 \right )}}{4} + \frac{3 \log{\left (- \sqrt [3]{- x + 1} - \frac{\left (x + 1\right )^{\frac{2}{3}}}{2} \right )}}{8} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3}}{3} - \frac{\sqrt{3} \left (x + 1\right )^{\frac{2}{3}}}{3 \sqrt [3]{- x + 1}} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(-x + 3)/4 + 3*log(-(-x + 1)**(1/3) - (x + 1)**(2/3)/2)/8 - sqrt(3)*atan(sqr
t(3)/3 - sqrt(3)*(x + 1)**(2/3)/(3*(-x + 1)**(1/3)))/4

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Mathematica [C]  time = 0.281949, size = 139, normalized size = 1.78 \[ \frac{15 (x-3) F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};-\frac{4}{x-3},-\frac{2}{x-3}\right )}{2 \sqrt [3]{1-x^2} \left (5 (x-3) F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};-\frac{4}{x-3},-\frac{2}{x-3}\right )-2 \left (F_1\left (\frac{5}{3};\frac{1}{3},\frac{4}{3};\frac{8}{3};-\frac{4}{x-3},-\frac{2}{x-3}\right )+2 F_1\left (\frac{5}{3};\frac{4}{3},\frac{1}{3};\frac{8}{3};-\frac{4}{x-3},-\frac{2}{x-3}\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(15*(-3 + x)*AppellF1[2/3, 1/3, 1/3, 5/3, -4/(-3 + x), -2/(-3 + x)])/(2*(1 - x^2
)^(1/3)*(5*(-3 + x)*AppellF1[2/3, 1/3, 1/3, 5/3, -4/(-3 + x), -2/(-3 + x)] - 2*(
AppellF1[5/3, 1/3, 4/3, 8/3, -4/(-3 + x), -2/(-3 + x)] + 2*AppellF1[5/3, 4/3, 1/
3, 8/3, -4/(-3 + x), -2/(-3 + x)])))

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Maple [F]  time = 0.061, size = 0, normalized size = 0. \[ \int{\frac{1}{3-x}{\frac{1}{\sqrt [3]{-{x}^{2}+1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(3-x)/(-x^2+1)^(1/3),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(1/(x*(-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. \[ \int -\frac{1}{{\left (-x^{2} + 1\right )}^{\frac{1}{3}}{\left (x - 3\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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