3.1022 \(\int \frac{x}{\sqrt [3]{1-x^2} \left (3+x^2\right )^2} \, dx\)

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]

-(1 - x^2)^(2/3)/(8*(3 + x^2)) + ArcTan[(1 + (2 - 2*x^2)^(1/3))/Sqrt[3]]/(8*2^(2
/3)*Sqrt[3]) - Log[3 + x^2]/(48*2^(2/3)) + Log[2^(2/3) - (1 - x^2)^(1/3)]/(16*2^
(2/3))

_______________________________________________________________________________________

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]

[Out]

-(1 - x^2)^(2/3)/(8*(3 + x^2)) + ArcTan[(1 + (2 - 2*x^2)^(1/3))/Sqrt[3]]/(8*2^(2
/3)*Sqrt[3]) - Log[3 + x^2]/(48*2^(2/3)) + Log[2^(2/3) - (1 - x^2)^(1/3)]/(16*2^
(2/3))

_______________________________________________________________________________________

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]

-(-x**2 + 1)**(2/3)/(8*(x**2 + 3)) - 2**(1/3)*log(x**2 + 3)/96 + 2**(1/3)*log(-(
-x**2 + 1)**(1/3) + 2**(2/3))/32 + 2**(1/3)*sqrt(3)*atan(sqrt(3)*(2**(1/3)*(-x**
2 + 1)**(1/3)/3 + 1/3))/48

_______________________________________________________________________________________

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]

[Out]

(-1 + x^2 - ((-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))

_______________________________________________________________________________________

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]

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

_______________________________________________________________________________________

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")

[Out]

1/96*4^(2/3)*sqrt(3)*arctan(1/12*4^(2/3)*sqrt(3)*(4^(1/3) + 2*(-x^2 + 1)^(1/3)))
- 1/192*4^(2/3)*log(4^(2/3) + 4^(1/3)*(-x^2 + 1)^(1/3) + (-x^2 + 1)^(2/3)) + 1/
96*4^(2/3)*log(-4^(1/3) + (-x^2 + 1)^(1/3)) - 1/8*(-x^2 + 1)^(2/3)/(x^2 + 3)

_______________________________________________________________________________________

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]

-1/576*4^(2/3)*sqrt(3)*(sqrt(3)*(x^2 + 3)*log(4^(2/3)*(-x^2 + 1)^(1/3) + 4^(1/3)
*(-x^2 + 1)^(2/3) + 4) - 2*sqrt(3)*(x^2 + 3)*log(4^(2/3)*(-x^2 + 1)^(1/3) - 4) -
6*(x^2 + 3)*arctan(1/6*4^(2/3)*sqrt(3)*(-x^2 + 1)^(1/3) + 1/3*sqrt(3)) + 6*4^(1
/3)*sqrt(3)*(-x^2 + 1)^(2/3))/(x^2 + 3)

_______________________________________________________________________________________

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]

Exception raised: ValueError

_______________________________________________________________________________________

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]

Exception raised: NotImplementedError