3.1009 \(\int \frac{x^3}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx\)

Optimal. Leaf size=94 \[ -\frac{3}{4} \left (1-x^2\right )^{2/3}+\frac{3 \log \left (x^2+3\right )}{4\ 2^{2/3}}-\frac{9 \log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{4\ 2^{2/3}}-\frac{3 \sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{2-2 x^2}+1}{\sqrt{3}}\right )}{2\ 2^{2/3}} \]

[Out]

(-3*(1 - x^2)^(2/3))/4 - (3*Sqrt[3]*ArcTan[(1 + (2 - 2*x^2)^(1/3))/Sqrt[3]])/(2*
2^(2/3)) + (3*Log[3 + x^2])/(4*2^(2/3)) - (9*Log[2^(2/3) - (1 - x^2)^(1/3)])/(4*
2^(2/3))

_______________________________________________________________________________________

Rubi [A]  time = 0.164891, antiderivative size = 94, 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 \[ -\frac{3}{4} \left (1-x^2\right )^{2/3}+\frac{3 \log \left (x^2+3\right )}{4\ 2^{2/3}}-\frac{9 \log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{4\ 2^{2/3}}-\frac{3 \sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{2-2 x^2}+1}{\sqrt{3}}\right )}{2\ 2^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

(-3*(1 - x^2)^(2/3))/4 - (3*Sqrt[3]*ArcTan[(1 + (2 - 2*x^2)^(1/3))/Sqrt[3]])/(2*
2^(2/3)) + (3*Log[3 + x^2])/(4*2^(2/3)) - (9*Log[2^(2/3) - (1 - x^2)^(1/3)])/(4*
2^(2/3))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 11.5047, size = 88, normalized size = 0.94 \[ - \frac{3 \left (- x^{2} + 1\right )^{\frac{2}{3}}}{4} + \frac{3 \sqrt [3]{2} \log{\left (x^{2} + 3 \right )}}{8} - \frac{9 \sqrt [3]{2} \log{\left (- \sqrt [3]{- x^{2} + 1} + 2^{\frac{2}{3}} \right )}}{8} - \frac{3 \sqrt [3]{2} \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{\sqrt [3]{2} \sqrt [3]{- x^{2} + 1}}{3} + \frac{1}{3}\right ) \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*(-x**2 + 1)**(2/3)/4 + 3*2**(1/3)*log(x**2 + 3)/8 - 9*2**(1/3)*log(-(-x**2 +
1)**(1/3) + 2**(2/3))/8 - 3*2**(1/3)*sqrt(3)*atan(sqrt(3)*(2**(1/3)*(-x**2 + 1)*
*(1/3)/3 + 1/3))/4

_______________________________________________________________________________________

Mathematica [C]  time = 0.0432703, size = 58, normalized size = 0.62 \[ \frac{3 \left (6 \sqrt [3]{\frac{x^2-1}{x^2+3}} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};\frac{4}{x^2+3}\right )+x^2-1\right )}{4 \sqrt [3]{1-x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(3*(-1 + x^2 + 6*((-1 + x^2)/(3 + x^2))^(1/3)*Hypergeometric2F1[1/3, 1/3, 4/3, 4
/(3 + x^2)]))/(4*(1 - x^2)^(1/3))

_______________________________________________________________________________________

Maple [F]  time = 0.057, size = 0, normalized size = 0. \[ \int{\frac{{x}^{3}}{{x}^{2}+3}{\frac{1}{\sqrt [3]{-{x}^{2}+1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.50275, size = 131, normalized size = 1.39 \[ -\frac{3}{8} \cdot 4^{\frac{2}{3}} \sqrt{3} \arctan \left (\frac{1}{12} \cdot 4^{\frac{2}{3}} \sqrt{3}{\left (4^{\frac{1}{3}} + 2 \,{\left (-x^{2} + 1\right )}^{\frac{1}{3}}\right )}\right ) + \frac{3}{16} \cdot 4^{\frac{2}{3}} \log \left (4^{\frac{2}{3}} + 4^{\frac{1}{3}}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} +{\left (-x^{2} + 1\right )}^{\frac{2}{3}}\right ) - \frac{3}{8} \cdot 4^{\frac{2}{3}} \log \left (-4^{\frac{1}{3}} +{\left (-x^{2} + 1\right )}^{\frac{1}{3}}\right ) - \frac{3}{4} \,{\left (-x^{2} + 1\right )}^{\frac{2}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/8*4^(2/3)*sqrt(3)*arctan(1/12*4^(2/3)*sqrt(3)*(4^(1/3) + 2*(-x^2 + 1)^(1/3)))
+ 3/16*4^(2/3)*log(4^(2/3) + 4^(1/3)*(-x^2 + 1)^(1/3) + (-x^2 + 1)^(2/3)) - 3/8
*4^(2/3)*log(-4^(1/3) + (-x^2 + 1)^(1/3)) - 3/4*(-x^2 + 1)^(2/3)

_______________________________________________________________________________________

Fricas [A]  time = 0.233519, size = 165, normalized size = 1.76 \[ \frac{3}{16} \cdot 4^{\frac{2}{3}}{\left (2 \, \sqrt{3} \left (-1\right )^{\frac{1}{3}} \arctan \left (-\frac{1}{6} \, \sqrt{3} \left (-1\right )^{\frac{1}{3}}{\left (4^{\frac{2}{3}}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + 2 \, \left (-1\right )^{\frac{2}{3}}\right )}\right ) - \left (-1\right )^{\frac{1}{3}} \log \left (4^{\frac{2}{3}} \left (-1\right )^{\frac{2}{3}}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + 4^{\frac{1}{3}}{\left (-x^{2} + 1\right )}^{\frac{2}{3}} - 4 \, \left (-1\right )^{\frac{1}{3}}\right ) + 2 \, \left (-1\right )^{\frac{1}{3}} \log \left (4^{\frac{2}{3}}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} - 4 \, \left (-1\right )^{\frac{2}{3}}\right ) - 4^{\frac{1}{3}}{\left (-x^{2} + 1\right )}^{\frac{2}{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/16*4^(2/3)*(2*sqrt(3)*(-1)^(1/3)*arctan(-1/6*sqrt(3)*(-1)^(1/3)*(4^(2/3)*(-x^2
+ 1)^(1/3) + 2*(-1)^(2/3))) - (-1)^(1/3)*log(4^(2/3)*(-1)^(2/3)*(-x^2 + 1)^(1/3
) + 4^(1/3)*(-x^2 + 1)^(2/3) - 4*(-1)^(1/3)) + 2*(-1)^(1/3)*log(4^(2/3)*(-x^2 +
1)^(1/3) - 4*(-1)^(2/3)) - 4^(1/3)*(-x^2 + 1)^(2/3))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

[Out]

Exception raised: NotImplementedError