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