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

Optimal. Leaf size=76 \[ \frac{1}{4} \log (x+3)-\frac{3}{8} \log \left (-\frac{1}{2} (1-x)^{2/3}-\sqrt [3]{x+1}\right )+\frac{1}{4} \sqrt{3} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{(1-x)^{2/3}}{\sqrt{3} \sqrt [3]{x+1}}\right ) \]

[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)^(2/3)/2 - (1 + x)^(1/3)])/8

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Rubi [A]  time = 0.0683714, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{1}{4} \log (x+3)-\frac{3}{8} \log \left (-\frac{1}{2} (1-x)^{2/3}-\sqrt [3]{x+1}\right )+\frac{1}{4} \sqrt{3} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{(1-x)^{2/3}}{\sqrt{3} \sqrt [3]{x+1}}\right ) \]

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)^(2/3)/2 - (1 + x)^(1/3)])/8

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Rubi in Sympy [A]  time = 7.64757, size = 63, normalized size = 0.83 \[ \frac{\log{\left (x + 3 \right )}}{4} - \frac{3 \log{\left (- \frac{\left (- x + 1\right )^{\frac{2}{3}}}{2} - \sqrt [3]{x + 1} \right )}}{8} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (- x + 1\right )^{\frac{2}{3}}}{3 \sqrt [3]{x + 1}} - \frac{\sqrt{3}}{3} \right )}}{4} \]

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

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Mathematica [C]  time = 0.0751944, size = 139, normalized size = 1.83 \[ -\frac{15 (x+3) F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{4}{x+3},\frac{2}{x+3}\right )}{2 \sqrt [3]{1-x^2} \left (5 (x+3) F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{4}{x+3},\frac{2}{x+3}\right )+2 \left (F_1\left (\frac{5}{3};\frac{1}{3},\frac{4}{3};\frac{8}{3};\frac{4}{x+3},\frac{2}{x+3}\right )+2 F_1\left (\frac{5}{3};\frac{4}{3},\frac{1}{3};\frac{8}{3};\frac{4}{x+3},\frac{2}{x+3}\right )\right )\right )} \]

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 = 0.047, 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. \[ \int \frac{1}{\sqrt [3]{- \left (x - 1\right ) \left (x + 1\right )} \left (x + 3\right )}\, dx \]

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 - 1)*(x + 1))**(1/3)*(x + 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)