Optimal. Leaf size=101 \[ -\frac{\left (1-x^2\right )^{2/3}}{8 \left (x^2+3\right )}-\frac{\log \left (x^2+3\right )}{48\ 2^{2/3}}+\frac{\log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{16\ 2^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{2-2 x^2}+1}{\sqrt{3}}\right )}{8\ 2^{2/3} \sqrt{3}} \]
[Out]
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Rubi [A] time = 0.158692, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ -\frac{\left (1-x^2\right )^{2/3}}{8 \left (x^2+3\right )}-\frac{\log \left (x^2+3\right )}{48\ 2^{2/3}}+\frac{\log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{16\ 2^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{2-2 x^2}+1}{\sqrt{3}}\right )}{8\ 2^{2/3} \sqrt{3}} \]
Antiderivative was successfully verified.
[In] Int[x/((1 - x^2)^(1/3)*(3 + x^2)^2),x]
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Rubi in Sympy [A] time = 10.1217, size = 87, normalized size = 0.86 \[ - \frac{\left (- x^{2} + 1\right )^{\frac{2}{3}}}{8 \left (x^{2} + 3\right )} - \frac{\sqrt [3]{2} \log{\left (x^{2} + 3 \right )}}{96} + \frac{\sqrt [3]{2} \log{\left (- \sqrt [3]{- x^{2} + 1} + 2^{\frac{2}{3}} \right )}}{32} + \frac{\sqrt [3]{2} \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{\sqrt [3]{2} \sqrt [3]{- x^{2} + 1}}{3} + \frac{1}{3}\right ) \right )}}{48} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(-x**2+1)**(1/3)/(x**2+3)**2,x)
[Out]
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Mathematica [C] time = 0.0433382, size = 70, normalized size = 0.69 \[ \frac{-\sqrt [3]{\frac{x^2-1}{x^2+3}} \left (x^2+3\right ) \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};\frac{4}{x^2+3}\right )+x^2-1}{8 \sqrt [3]{1-x^2} \left (x^2+3\right )} \]
Antiderivative was successfully verified.
[In] Integrate[x/((1 - x^2)^(1/3)*(3 + x^2)^2),x]
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Maple [F] time = 0.054, size = 0, normalized size = 0. \[ \int{\frac{x}{ \left ({x}^{2}+3 \right ) ^{2}}{\frac{1}{\sqrt [3]{-{x}^{2}+1}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(-x^2+1)^(1/3)/(x^2+3)^2,x)
[Out]
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Maxima [A] time = 1.51993, size = 140, normalized size = 1.39 \[ \frac{1}{96} \cdot 4^{\frac{2}{3}} \sqrt{3} \arctan \left (\frac{1}{12} \cdot 4^{\frac{2}{3}} \sqrt{3}{\left (4^{\frac{1}{3}} + 2 \,{\left (-x^{2} + 1\right )}^{\frac{1}{3}}\right )}\right ) - \frac{1}{192} \cdot 4^{\frac{2}{3}} \log \left (4^{\frac{2}{3}} + 4^{\frac{1}{3}}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} +{\left (-x^{2} + 1\right )}^{\frac{2}{3}}\right ) + \frac{1}{96} \cdot 4^{\frac{2}{3}} \log \left (-4^{\frac{1}{3}} +{\left (-x^{2} + 1\right )}^{\frac{1}{3}}\right ) - \frac{{\left (-x^{2} + 1\right )}^{\frac{2}{3}}}{8 \,{\left (x^{2} + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((x^2 + 3)^2*(-x^2 + 1)^(1/3)),x, algorithm="maxima")
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Fricas [A] time = 0.236092, size = 173, normalized size = 1.71 \[ -\frac{4^{\frac{2}{3}} \sqrt{3}{\left (\sqrt{3}{\left (x^{2} + 3\right )} \log \left (4^{\frac{2}{3}}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + 4^{\frac{1}{3}}{\left (-x^{2} + 1\right )}^{\frac{2}{3}} + 4\right ) - 2 \, \sqrt{3}{\left (x^{2} + 3\right )} \log \left (4^{\frac{2}{3}}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} - 4\right ) - 6 \,{\left (x^{2} + 3\right )} \arctan \left (\frac{1}{6} \cdot 4^{\frac{2}{3}} \sqrt{3}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + \frac{1}{3} \, \sqrt{3}\right ) + 6 \cdot 4^{\frac{1}{3}} \sqrt{3}{\left (-x^{2} + 1\right )}^{\frac{2}{3}}\right )}}{576 \,{\left (x^{2} + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((x^2 + 3)^2*(-x^2 + 1)^(1/3)),x, algorithm="fricas")
[Out]
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(-x**2+1)**(1/3)/(x**2+3)**2,x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((x^2 + 3)^2*(-x^2 + 1)^(1/3)),x, algorithm="giac")
[Out]