3.1018 \(\int \frac{1}{x^4 \sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx\)

Optimal. Leaf size=556 \[ \frac{2 x}{27 \left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )}-\frac{2 \left (1-x^2\right )^{2/3}}{27 x}+\frac{\tan ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{18\ 2^{2/3} \sqrt{3}}+\frac{\tanh ^{-1}\left (\frac{x}{\sqrt [3]{2} \sqrt [3]{1-x^2}+1}\right )}{18\ 2^{2/3}}-\frac{2 \sqrt{2} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt{\frac{\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{1-x^2}+\sqrt{3}+1}{-\sqrt [3]{1-x^2}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{27 \sqrt [4]{3} \sqrt{-\frac{1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} x}+\frac{\sqrt{2+\sqrt{3}} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt{\frac{\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-\sqrt [3]{1-x^2}+\sqrt{3}+1}{-\sqrt [3]{1-x^2}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{9\ 3^{3/4} \sqrt{-\frac{1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} x}-\frac{\left (1-x^2\right )^{2/3}}{9 x^3}+\frac{\tan ^{-1}\left (\frac{\sqrt{3}}{x}\right )}{18\ 2^{2/3} \sqrt{3}}-\frac{\tanh ^{-1}(x)}{54\ 2^{2/3}} \]

[Out]

-(1 - x^2)^(2/3)/(9*x^3) - (2*(1 - x^2)^(2/3))/(27*x) + (2*x)/(27*(1 - Sqrt[3] -
(1 - x^2)^(1/3))) + ArcTan[Sqrt[3]/x]/(18*2^(2/3)*Sqrt[3]) + ArcTan[(Sqrt[3]*(1
- 2^(1/3)*(1 - x^2)^(1/3)))/x]/(18*2^(2/3)*Sqrt[3]) - ArcTanh[x]/(54*2^(2/3)) +
ArcTanh[x/(1 + 2^(1/3)*(1 - x^2)^(1/3))]/(18*2^(2/3)) + (Sqrt[2 + Sqrt[3]]*(1 -
(1 - x^2)^(1/3))*Sqrt[(1 + (1 - x^2)^(1/3) + (1 - x^2)^(2/3))/(1 - Sqrt[3] - (1
- x^2)^(1/3))^2]*EllipticE[ArcSin[(1 + Sqrt[3] - (1 - x^2)^(1/3))/(1 - Sqrt[3]
- (1 - x^2)^(1/3))], -7 + 4*Sqrt[3]])/(9*3^(3/4)*x*Sqrt[-((1 - (1 - x^2)^(1/3))/
(1 - Sqrt[3] - (1 - x^2)^(1/3))^2)]) - (2*Sqrt[2]*(1 - (1 - x^2)^(1/3))*Sqrt[(1
+ (1 - x^2)^(1/3) + (1 - x^2)^(2/3))/(1 - Sqrt[3] - (1 - x^2)^(1/3))^2]*Elliptic
F[ArcSin[(1 + Sqrt[3] - (1 - x^2)^(1/3))/(1 - Sqrt[3] - (1 - x^2)^(1/3))], -7 +
4*Sqrt[3]])/(27*3^(1/4)*x*Sqrt[-((1 - (1 - x^2)^(1/3))/(1 - Sqrt[3] - (1 - x^2)^
(1/3))^2)])

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Rubi [A]  time = 0.810555, antiderivative size = 556, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ \frac{2 x}{27 \left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )}-\frac{2 \left (1-x^2\right )^{2/3}}{27 x}+\frac{\tan ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{18\ 2^{2/3} \sqrt{3}}+\frac{\tanh ^{-1}\left (\frac{x}{\sqrt [3]{2} \sqrt [3]{1-x^2}+1}\right )}{18\ 2^{2/3}}-\frac{2 \sqrt{2} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt{\frac{\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{1-x^2}+\sqrt{3}+1}{-\sqrt [3]{1-x^2}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{27 \sqrt [4]{3} \sqrt{-\frac{1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} x}+\frac{\sqrt{2+\sqrt{3}} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt{\frac{\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-\sqrt [3]{1-x^2}+\sqrt{3}+1}{-\sqrt [3]{1-x^2}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{9\ 3^{3/4} \sqrt{-\frac{1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} x}-\frac{\left (1-x^2\right )^{2/3}}{9 x^3}+\frac{\tan ^{-1}\left (\frac{\sqrt{3}}{x}\right )}{18\ 2^{2/3} \sqrt{3}}-\frac{\tanh ^{-1}(x)}{54\ 2^{2/3}} \]

Warning: Unable to verify antiderivative.

[In]  Int[1/(x^4*(1 - x^2)^(1/3)*(3 + x^2)),x]

[Out]

-(1 - x^2)^(2/3)/(9*x^3) - (2*(1 - x^2)^(2/3))/(27*x) + (2*x)/(27*(1 - Sqrt[3] -
(1 - x^2)^(1/3))) + ArcTan[Sqrt[3]/x]/(18*2^(2/3)*Sqrt[3]) + ArcTan[(Sqrt[3]*(1
- 2^(1/3)*(1 - x^2)^(1/3)))/x]/(18*2^(2/3)*Sqrt[3]) - ArcTanh[x]/(54*2^(2/3)) +
ArcTanh[x/(1 + 2^(1/3)*(1 - x^2)^(1/3))]/(18*2^(2/3)) + (Sqrt[2 + Sqrt[3]]*(1 -
(1 - x^2)^(1/3))*Sqrt[(1 + (1 - x^2)^(1/3) + (1 - x^2)^(2/3))/(1 - Sqrt[3] - (1
- x^2)^(1/3))^2]*EllipticE[ArcSin[(1 + Sqrt[3] - (1 - x^2)^(1/3))/(1 - Sqrt[3]
- (1 - x^2)^(1/3))], -7 + 4*Sqrt[3]])/(9*3^(3/4)*x*Sqrt[-((1 - (1 - x^2)^(1/3))/
(1 - Sqrt[3] - (1 - x^2)^(1/3))^2)]) - (2*Sqrt[2]*(1 - (1 - x^2)^(1/3))*Sqrt[(1
+ (1 - x^2)^(1/3) + (1 - x^2)^(2/3))/(1 - Sqrt[3] - (1 - x^2)^(1/3))^2]*Elliptic
F[ArcSin[(1 + Sqrt[3] - (1 - x^2)^(1/3))/(1 - Sqrt[3] - (1 - x^2)^(1/3))], -7 +
4*Sqrt[3]])/(27*3^(1/4)*x*Sqrt[-((1 - (1 - x^2)^(1/3))/(1 - Sqrt[3] - (1 - x^2)^
(1/3))^2)])

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Rubi in Sympy [A]  time = 8.05951, size = 24, normalized size = 0.04 \[ - \frac{\operatorname{appellf_{1}}{\left (- \frac{3}{2},\frac{1}{3},1,- \frac{1}{2},x^{2},- \frac{x^{2}}{3} \right )}}{9 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**4/(-x**2+1)**(1/3)/(x**2+3),x)

[Out]

-appellf1(-3/2, 1/3, 1, -1/2, x**2, -x**2/3)/(9*x**3)

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Mathematica [C]  time = 0.268365, size = 245, normalized size = 0.44 \[ \frac{-\frac{27 x F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};x^2,-\frac{x^2}{3}\right )}{\left (x^2+3\right ) \left (2 x^2 \left (F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};x^2,-\frac{x^2}{3}\right )-F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right )\right )-9 F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};x^2,-\frac{x^2}{3}\right )\right )}+\frac{10 x^3 F_1\left (\frac{3}{2};\frac{1}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right )}{\left (x^2+3\right ) \left (2 x^2 \left (F_1\left (\frac{5}{2};\frac{1}{3},2;\frac{7}{2};x^2,-\frac{x^2}{3}\right )-F_1\left (\frac{5}{2};\frac{4}{3},1;\frac{7}{2};x^2,-\frac{x^2}{3}\right )\right )-15 F_1\left (\frac{3}{2};\frac{1}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right )\right )}-\frac{9}{x^3}+6 x+\frac{3}{x}}{81 \sqrt [3]{1-x^2}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/(x^4*(1 - x^2)^(1/3)*(3 + x^2)),x]

[Out]

(-9/x^3 + 3/x + 6*x - (27*x*AppellF1[1/2, 1/3, 1, 3/2, x^2, -x^2/3])/((3 + x^2)*
(-9*AppellF1[1/2, 1/3, 1, 3/2, x^2, -x^2/3] + 2*x^2*(AppellF1[3/2, 1/3, 2, 5/2,
x^2, -x^2/3] - AppellF1[3/2, 4/3, 1, 5/2, x^2, -x^2/3]))) + (10*x^3*AppellF1[3/2
, 1/3, 1, 5/2, x^2, -x^2/3])/((3 + x^2)*(-15*AppellF1[3/2, 1/3, 1, 5/2, x^2, -x^
2/3] + 2*x^2*(AppellF1[5/2, 1/3, 2, 7/2, x^2, -x^2/3] - AppellF1[5/2, 4/3, 1, 7/
2, x^2, -x^2/3]))))/(81*(1 - x^2)^(1/3))

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Maple [F]  time = 0.075, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{4} \left ({x}^{2}+3 \right ) }{\frac{1}{\sqrt [3]{-{x}^{2}+1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(-x^2+1)^(1/3)/(x^2+3),x)

[Out]

int(1/x^4/(-x^2+1)^(1/3)/(x^2+3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{2} + 3\right )}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^2 + 3)*(-x^2 + 1)^(1/3)*x^4),x, algorithm="maxima")

[Out]

integrate(1/((x^2 + 3)*(-x^2 + 1)^(1/3)*x^4), x)

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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 + 3)*(-x^2 + 1)^(1/3)*x^4),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{4} \sqrt [3]{- \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + 3\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**4/(-x**2+1)**(1/3)/(x**2+3),x)

[Out]

Integral(1/(x**4*(-(x - 1)*(x + 1))**(1/3)*(x**2 + 3)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{2} + 3\right )}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^2 + 3)*(-x^2 + 1)^(1/3)*x^4),x, algorithm="giac")

[Out]

integrate(1/((x^2 + 3)*(-x^2 + 1)^(1/3)*x^4), x)