3.1021 \(\int \frac{x^3}{\sqrt [3]{1-x^2} \left (3+x^2\right )^2} \, dx\)

Optimal. Leaf size=101 \[ \frac{3 \left (1-x^2\right )^{2/3}}{8 \left (x^2+3\right )}-\frac{3 \log \left (x^2+3\right )}{16\ 2^{2/3}}+\frac{9 \log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{16\ 2^{2/3}}+\frac{3 \sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{2-2 x^2}+1}{\sqrt{3}}\right )}{8\ 2^{2/3}} \]

[Out]

(3*(1 - x^2)^(2/3))/(8*(3 + x^2)) + (3*Sqrt[3]*ArcTan[(1 + (2 - 2*x^2)^(1/3))/Sq
rt[3]])/(8*2^(2/3)) - (3*Log[3 + x^2])/(16*2^(2/3)) + (9*Log[2^(2/3) - (1 - x^2)
^(1/3)])/(16*2^(2/3))

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Rubi [A]  time = 0.183978, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{3 \left (1-x^2\right )^{2/3}}{8 \left (x^2+3\right )}-\frac{3 \log \left (x^2+3\right )}{16\ 2^{2/3}}+\frac{9 \log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{16\ 2^{2/3}}+\frac{3 \sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{2-2 x^2}+1}{\sqrt{3}}\right )}{8\ 2^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

(3*(1 - x^2)^(2/3))/(8*(3 + x^2)) + (3*Sqrt[3]*ArcTan[(1 + (2 - 2*x^2)^(1/3))/Sq
rt[3]])/(8*2^(2/3)) - (3*Log[3 + x^2])/(16*2^(2/3)) + (9*Log[2^(2/3) - (1 - x^2)
^(1/3)])/(16*2^(2/3))

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Rubi in Sympy [A]  time = 11.417, size = 94, normalized size = 0.93 \[ \frac{3 \left (- x^{2} + 1\right )^{\frac{2}{3}}}{8 \left (x^{2} + 3\right )} - \frac{3 \sqrt [3]{2} \log{\left (x^{2} + 3 \right )}}{32} + \frac{9 \sqrt [3]{2} \log{\left (- \sqrt [3]{- x^{2} + 1} + 2^{\frac{2}{3}} \right )}}{32} + \frac{3 \sqrt [3]{2} \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{\sqrt [3]{2} \sqrt [3]{- x^{2} + 1}}{3} + \frac{1}{3}\right ) \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*(-x**2 + 1)**(2/3)/(8*(x**2 + 3)) - 3*2**(1/3)*log(x**2 + 3)/32 + 9*2**(1/3)*l
og(-(-x**2 + 1)**(1/3) + 2**(2/3))/32 + 3*2**(1/3)*sqrt(3)*atan(sqrt(3)*(2**(1/3
)*(-x**2 + 1)**(1/3)/3 + 1/3))/16

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Mathematica [C]  time = 0.047249, size = 70, normalized size = 0.69 \[ -\frac{3 \left (3 \sqrt [3]{\frac{x^2-1}{x^2+3}} \left (x^2+3\right ) \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};\frac{4}{x^2+3}\right )+x^2-1\right )}{8 \sqrt [3]{1-x^2} \left (x^2+3\right )} \]

Antiderivative was successfully verified.

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

[Out]

(-3*(-1 + x^2 + 3*((-1 + x^2)/(3 + x^2))^(1/3)*(3 + x^2)*Hypergeometric2F1[1/3,
1/3, 4/3, 4/(3 + x^2)]))/(8*(1 - x^2)^(1/3)*(3 + x^2))

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Maple [F]  time = 0.064, size = 0, normalized size = 0. \[ \int{\frac{{x}^{3}}{ \left ({x}^{2}+3 \right ) ^{2}}{\frac{1}{\sqrt [3]{-{x}^{2}+1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.49939, size = 140, normalized size = 1.39 \[ \frac{3}{32} \cdot 4^{\frac{2}{3}} \sqrt{3} \arctan \left (\frac{1}{12} \cdot 4^{\frac{2}{3}} \sqrt{3}{\left (4^{\frac{1}{3}} + 2 \,{\left (-x^{2} + 1\right )}^{\frac{1}{3}}\right )}\right ) - \frac{3}{64} \cdot 4^{\frac{2}{3}} \log \left (4^{\frac{2}{3}} + 4^{\frac{1}{3}}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} +{\left (-x^{2} + 1\right )}^{\frac{2}{3}}\right ) + \frac{3}{32} \cdot 4^{\frac{2}{3}} \log \left (-4^{\frac{1}{3}} +{\left (-x^{2} + 1\right )}^{\frac{1}{3}}\right ) + \frac{3 \,{\left (-x^{2} + 1\right )}^{\frac{2}{3}}}{8 \,{\left (x^{2} + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/32*4^(2/3)*sqrt(3)*arctan(1/12*4^(2/3)*sqrt(3)*(4^(1/3) + 2*(-x^2 + 1)^(1/3)))
- 3/64*4^(2/3)*log(4^(2/3) + 4^(1/3)*(-x^2 + 1)^(1/3) + (-x^2 + 1)^(2/3)) + 3/3
2*4^(2/3)*log(-4^(1/3) + (-x^2 + 1)^(1/3)) + 3/8*(-x^2 + 1)^(2/3)/(x^2 + 3)

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Fricas [A]  time = 0.237465, size = 158, normalized size = 1.56 \[ \frac{3 \cdot 4^{\frac{2}{3}}{\left (2 \, \sqrt{3}{\left (x^{2} + 3\right )} \arctan \left (\frac{1}{6} \, \sqrt{3}{\left (4^{\frac{2}{3}}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + 2\right )}\right ) -{\left (x^{2} + 3\right )} \log \left (4^{\frac{2}{3}}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + 4^{\frac{1}{3}}{\left (-x^{2} + 1\right )}^{\frac{2}{3}} + 4\right ) + 2 \,{\left (x^{2} + 3\right )} \log \left (4^{\frac{2}{3}}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} - 4\right ) + 2 \cdot 4^{\frac{1}{3}}{\left (-x^{2} + 1\right )}^{\frac{2}{3}}\right )}}{64 \,{\left (x^{2} + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/64*4^(2/3)*(2*sqrt(3)*(x^2 + 3)*arctan(1/6*sqrt(3)*(4^(2/3)*(-x^2 + 1)^(1/3) +
2)) - (x^2 + 3)*log(4^(2/3)*(-x^2 + 1)^(1/3) + 4^(1/3)*(-x^2 + 1)^(2/3) + 4) +
2*(x^2 + 3)*log(4^(2/3)*(-x^2 + 1)^(1/3) - 4) + 2*4^(1/3)*(-x^2 + 1)^(2/3))/(x^2
+ 3)

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

[Out]

Exception raised: ValueError

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

[Out]

Exception raised: NotImplementedError